Internal Energy, Q-Energy, Poynting's Theorem, and the ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 6, JUNE 2007

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Internal Energy, Q-Energy, Poynting’s Theorem, and the Stress Dyadic in Dispersive Material Arthur D. Yaghjian, Fellow, IEEE

The work presented in this paper for the Special Issue devoted to Prof. Leo Felsen benefitted greatly from the lucid treatment of electromagnetic energy density and power flow given in Felsen and Marcuvitz, Radiation and Scattering of Waves, New York: IEEE Press, Sec. 1.5(a), 1994.

Abstract—General expressions are derived for time-domain energy density and the time integral of the Poynting vector that are related to the kinetic, potential, and heat energy densities of the bound charge-polarization carriers and the stored electromagnetic field energy density in passive, nonlinear or linear, lossy or lossless, temporally and/or spatially dispersive polarized media. In the most general linear, lossless, spatially nondispersive media, the energy expressions reveal non-negative quadratic forms for the frequency-domain internal energy densities that are used in expressions for the quality factor ( ) of antennas containing lossless dispersive material. The analysis also reveals useful inequalities that imply that the magnitude of the group velocity in lossless material is always less than or equal to the speed of light. The energy expressions do not, however, predict an internal energy in highly lossy temporally dispersive media that can be used to improve upon the expressions for the of antennas in such of antennas with media. To improve upon the accuracy of the highly lossy dispersive media, a non-negative “Q-energy” density is found that maintains the accuracy of the inverse relationship between and matched VSWR half-power fractional bandwidth for antennas containing highly lossy dispersive material. Lower are found in terms of previously bounds for this improved determined lower bounds. The paper also confirms the result that, for general linear lossy or lossless dispersive material, the steady state time averages of the electromagnetic power density, force density, and stress dyadic with sinusoidal time dependence that turns on at some finite time in the past, unlike the internal energy, does not contain derivatives with respect to frequency. Index Terms—Dispersive media, internal energy, Poynting’s theorem, Q-energy, quality factor, stress dyadic.

I. INTRODUCTION ONSIDER a medium in a volumetric region that contains a given reservoir (no carriers added or subtracted) of “microscopic” charge and polarization carriers that can produce “macroscopic” charge, current, and electric and magnetic polar] at any point ization densities [ belonging to at time . These charge, current, and polarization source densities in along with the source densities outside produce the macroscopic electric and magnetic fields, and

C

Manuscript received September 1, 2006; revised November 14, 2006. This work was supported by the U.S. Air Force Office of Scientific Research (AFOSR). The author is with the AFRL/SNHA, Hanscom AFB, MA 01731 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2007.897350

, within . Assume that the effective operational bandwidth of the frequency spectra of these sources and fields is finite and that the frequencies of all vibrational and radiative heat energy lies in a spectrum outside this “operational bandwidth.” Also, assume that the medium is “passive” in that any forces, with frequency spectra within the operational bandwidth, exerted by the noncarrier particles of the medium on the charge and polarization carriers subtract rather than add energy to the carriers. These dissipative forces are called “frictional forces” and any energy lost to the frictional forces is assumed converted into “heat” energy (that is, energy outside the operational bandwidth). In this medium, Maxwell’s differential equations for the macroscopic sources and fields can be written as (1a) (1b) (2a) (2b) for in the passive medium contained in , where and are the permittivity and permeability of free space. A combination and manipulation of Maxwell’s equations in (1) and (2) yields a generalization of Poynting’s theorem in the differential form ([1, (2.174)])

(3) where is the Poynting vector. The is an electromagnetic power density whose quantity time integral, as shown in the following sections, leads to useful theorems and definitions of time-domain and frequency-domain electromagnetic energies in lossy or lossless material. The constitutive relations (4a) (4b) recast (3) into the form

(5) so that

0018-926X/$25.00 © 2007 IEEE

(6)

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In a simple linear lossless medium characterized by a dispersionless scalar permittivity and permeability such that and over their operational frequency bandwidth of interest, (6) becomes (7) is often regarded The non-negative quantity in as the difference in internal energy per unit volume at the medium with and without the electromagnetic field. In a dispersive and/or lossy medium, a similar interpretation of internal energy per unit volume at the position and time is not generally possible ([2], p. 272) and thus we return to (3) or (5) as the fundamental electromagnetic power equation in an arbitrary macroscopic distribution of current and polarization. II. ENERGY THEOREM FOR A PASSIVE MEDIUM WITH BOUND COLLISIONLESS CARRIERS , It was proven in [1, Sec. 2.1.10] that is defined in (3) or (5), is equal to the where and polarizapower supplied to the current tion in the macroscopically small (containing the point ) by the electrovolume element at the time .1 Expressing magnetic fields the electromagnetic power defined in (3) and (5) as a time at each fixed position , that is, derivative of a function , and integrating from the remote to the present time at each gives past

(8) It is assumed that , and are zero in the remote past so that (8) implies . that is not equal In general, even in a passive medium, (or whatever later time the sources to the change from and fields become nonzero) in the per unit volume of reversible , and kinetic and potential energy of the carriers of that reside at the position at time plus the irreversible energy lost by the carriers to frictional forces (and converted to vibrational and radiated heat energy) for three reasons: 1) The carriers may drift so that the ones that reside in a macroscopically small volume element at at some time are not the same

1

1The only proviso for this result to hold is that the surface of V is assumed to lie in free space just outside of the material so that the surface polarization charge n and surface magnetization current r; t n (with n denoting the surface unit normal) is included in the integration over V by means of delta functions in the spatial derivatives of the field and polarization densities across the surface of V ; see [1, Sec. 2.1.10]. Note that since = = = and these last two terms are zero = when integrating in free space, we can just as well choose (5)–(7) over a volume with its surface in free space.

^ 1P

B(

carriers that reside in that volume at the same but at another time ; 2) the kinetic energy within the operational frequency bandwidth can be transferred from carriers at position to carriers at another position through direct collisions with each other; and 3) the electromagnetic power supplied to the charge and polarization carriers in each macroscopically small volume is converted to radiation reaction energy in addielement tion to kinetic and potential energy of the carriers [3]. The first reason does not apply if we assume not only that the is zero for all and for all ,2 but also current that the medium can be modeled by the carriers “bound” by infinitesimal restoring “springs” (which can be nonlinear and lossy) to a fixed lattice (as in stationary undeformable solids that nonetheless can be inhomogeneous and anisotropic) such that the carrier drift is negligible. If the heat energy generated in the restoring springs is transferred to the carriers and other noncarrier particles of the medium, this transferred energy is assumed to end up as heat energy (that is, energy outside the operational bandwidth). The second reason also does not apply if it is assumed that the spring-bound carriers are collisionless so that carriers in neighboring macroscopically small volumes do not interfere with each other at frequencies within the operational bandwidth (other than through the macroscopic fields they produce). In general, the medium can be both temporally dispersive (or, equivalently, frequency dispersive) and spatially dispersive. It turns out that the third reason is not a real issue because the radiation reaction energy of the carriers in a macroscopically is proportional to the amount of ensmall volume element ergy this macroscopically small volume of and would radiate if it were alone in free space, that is, an amount proportional to , which is higher order than and thus does not contribute to the local per unit volume energy [1, p. 46], [3]. , which begins at a value of zero, Consequently, equals (for this media model) the total reversible energy change up to the time plus the frictional energy loss per unit volume of and , and thus can never the carriers between the time of in this passive, lossy or lossbe negative; that is, less, linear or nonlinear, inhomogeneous anisotropic medium of bound collisionless charge-polarization carriers, and we can conclude from (8) and (5) that

1

) 0DD 2M M 0PP 2 (B

^ M( ) 2 ^ 1 E 2H = D 2 0M M) S = D 2BB ( )

(9)

J( ) 1

2For example, if r; t were not zero, a beam of charged particles could enter a small volume V (located at r) in which the electric field r; t brought the charged particles to rest. Then W r; t would have a negative value in V even though there is no change in kinetic and potential energy or frictional losses in V .

1

1

( )

E( )

YAGHJIAN: INTERNAL ENERGY, Q-ENERGY, POYNTING’S THEOREM, AND THE STRESS DYADIC

for all and for all . The inequality in (9) was first presented in ([4], Appendix B) but we have not seen it stated or derived elsewhere in the published literature, although Tonning [5] concluded that the integral in the second line of (9) is the electromagnetic energy “absorbed” per unit volume in a “reversible,” lossless medium. In [6] Figotin and Schenker introduce auxiliary variables into Maxwell’s equations in order to that define a manifestly non-negative energy density equals . The non-negative energy density in (9) multiplied by at would equal the actual energy change a point (kinetic and potential plus heat energy) in the material contained (see Footnote 1) between the remote past and time if the in medium were adiabatic, that is, if no heat were transferred into . If the medium is not adiabatic, then or out of equals the total change in kinetic and potential energy of the charge and polarization carriers at time plus the frictional energy loss (heat) that was generated up to time but not necesat each point . This nonadiabatic heat sarily retained in energy can be conducted and radiated throughout (or out of) the volume of the passive medium. Of course, heat energy may also enter from outside the volume of a nonadiabatic passive medium.

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of the material by quickly isolating energy from a volume in free space, collecting the radiated energy emitted by as the sources and fields reduce to zero, then absorbing the adiabatically generated heat energy in a bath at the original remote-past equilibrium temperature of the material. The non-negative time-integral form of Poynting’s theorem in (10) cannot be applied directly to obtain information about single frequency sinusoidal (time harmonic) fields because to these fields, if truly single frequency, exist from and thus are not zero in the remote past as required in the derivation of the inequality in (10). If, on the other hand, we assume that the electric field, for example, has a sinusoidal in the past, time dependence that turns on at some time that is3 (11) where

is the unit step function that turns on just before and a phase constant, then the frequency spectrum has a finite bandwidth and the time dependence of of will not, in general, because of frequency dispersion, be simply a sinusoid that turns on at . An exception occurs, however, for the case of a linear medium with a complex frequency-domain permittivity given by (12)

III. TIME INTEGRAL OF POYNTING’S THEOREM FOR A PASSIVE MEDIUM WITH BOUND COLLISIONLESS CARRIERS The time-integral of Poynting’s differential theorem can be expressed from (9) as

for time dependence, where and are constants. (Note that in a linear medium, the electrical conductivity can be taken into account by the imaginary part of the permittivity.) The permittivity in (12) satisfies the Kramers-Kronig dispersion relations [1, p. 98] and its time-domain counterpart is given by (13)

(10) with We showed in Section II, that for a passive medium with bound collisionless charge-polarization carriers, the time integral on the right-hand side of the equation in (10) is the change within the operational frequency bandwidth (at ) of reversible kinetic and potential energy density of the carriers between the nullfield state in the remote past and the time , plus the irreversible frictional energy density within the operational bandwidth lost by the carriers to heat (energy outside the operational bandwidth) up to the time . This change in kinetic and potential plus heat energy densities is always greater than or equal to , which zero. The energy density is also greater than or equal to zero, is the change within the operational frequency bandwidth in per unit volume (at ) of reversible “stored electromagnetic field energy” between the null-field state in the remote past and the time . Therefore, within this type of media, the negative of the time integral of the divergence of the Poynting vector [integral on the left-hand side of (10)] is equal to the sum of these kinetic, potential, heat, and stored electromagnetic field energy densities. This sum is, of course, greater than or equal to zero since the sum of the kinetic, potential, and heat energy densities as well as the stored electromagnetic field energy density are each greater than or equal to zero. In an adiabatic medium, one can imagine extracting this

being the delta function, so that [1, p. 96]

(14) in (14), substituting the Taking the time derivative of result into the second line of (9), and performing the integration over time while making use of the distribution identity leads to

(15) for all and any . We have assumed that at each chosen point at which the above derivation has been performed, the sources outside a macroscopically small volume . containing are chosen to maintain 3The superscript on 0 means just before t = 0 and is inserted to clarify the derivation.

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The term in the brackets multiplying in (15) can be shown to be greater than or equal to zero, and since the inequality in (15) holds for all time , it implies that (16a) and

, and are the permeability where dyadic, the permittivity dyadic, and the magneto-electric dyadics. Like the fields, they are, in general, functions of frequency and position within the media.4 To illustrate the derivation, which is based on evaluating the second line of (9), we begin by evaluating

(16b) The inequality in (16b) confirms that the electrical conductivity cannot be negative in a passive material. The inequality in (16a) can be proven for a lossless medium from the KramersKronig dispersion relations [2, footnote, p. 287]. Here we have shown that it also holds in a lossy medium with the imaginary . part of the frequency-domain permittivity given by A similar analysis with the field in a medium with the frequency-domain permeability given by (17) with

and

constants, yields

(22) for an applied sinusoidal electric field whose magnitude begins and gradually increases to a constant magnitude, at zero at specifically (23) where is a constant that eventually will be allowed to approach zero while in the upper limit of integration in (22) approaches infinity such that also approaches infinity. To further simplify the initial derivation, assume a scalar permittivity such that (24)

(18a) where

and (18b) For a medium characterized by the permittivity and permeability given in (12) and (17), respectively, the average “internal energy density” , defined as the sum of kinetic, potential, and stored electromagnetic field energy densities, is given by the time average of the term with the coefficient in (15) [and the time average in the corresponding magnetic-field equation], namely (19) where we have written the sinusoidal fields in the usual phasor notation . The result in time-harmonic (19) agrees with the time average of sinusoidal fields obtained from (7). The corresponding time-average energy density and converted into heat is given dissipated per unit time by the coefficient of the term in (15) (and in the corresponding magnetic-field equation) that grows linearly with time, namely

(25) The simple integrations in (25) evaluate as

(26) from (26) into (24) and taking the inverse Fourier Inserting transform gives

(27) Applying the method of residues to evaluate the integral in (27), one obtains5

(20) a result which agrees with the time-average power density loss obtained for sinusoidal fields in a medium with electric and magnetic conductive heat losses.

(28) To further evaluate (28), use the power series expansion

IV. APPLICATION OF ENERGY THEOREM TO LINEAR, PASSIVE, SPATIALLY NONDISPERSIVE MEDIA In this section, the energy theorem derived in Section II is applied to a linear, passive, spatially nondispersive medium to obtain frequency-domain expressions for internal energy densities in lossless media and inequalities that the linear constitutive relations must obey in lossless media. The most general linear, spatially nondispersive constitutive relations are given in the frequency domain as

(29) 4Finite bandwidth fields in linear, passive media satisfying the constitutive relations in (21) can be modeled by a multitude of spring-bound collisionless charge-polarization carriers; for example, by an unlimited number of Lorentz models with electric and magnetic charges having different masses and coupling. Although the actual medium may not conform to this idealized model, it is sufficient that the frequency dependent constitutive parameters can be obtained over the operational bandwidth with the idealized model for (9) to apply. 5The only contribution to the integral in (27) is from the residues of the poles within the parentheses of the integrand of (27) because this quantity within the parentheses approaches zero as approaches zero for all complex values of ! except for ! near ! . Alternatively, the integral can be evaluated directly along the real axis by expanding in power series about ! to arrive at the same result given in (31).

6

(21)

6


and the reality relations [1, p. 95]

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) in a lossless medium with sinusoidal fields is given (30)

by (38)

to get

(31) where the primes denote derivatives with respect to angular frequency and

Taking the time derivative of in (23) yields

Note that if is not equal to zero, the time average of in (34) gives very little information because it includes, in addition to the conductive loss term (the term linear in ), the term, which diverges as . and anChoosing a time that makes , (37) produces the other time that makes inequalities

(32)

(39)

in (31) and dotting it into

These two inequalities in (39) that hold in lossless media are equivalent to the two inequalities in [2, Eqs. (84.1) and (84.2)] where they are proven from the Kramers-Kronig dispersion reis zero. lations for frequencies in windows where the loss Here we have shown, as we did previously in [4], that these inequalities can be proven from energy considerations as well. With no dispersion, the right-hand side of (38) reduces to , the per unit volume “average reactive energy,” so can be viewed as the per unit volume inthat crease in the kinetic and potential energy of the carriers as the sinusoidal fields are built up in a lossless medium from an amplitude of zero to their final amplitude. This result in (38) was first derived by Brillouin [7], although he did not prove the inequalities in (39). If the lossless medium (in the frequency window of interest) is also characterized by a scalar permeability, that is

(33) which when substituted into (22) and integrated over time produces the result (with for all at each chosen )

(40) then the internal energy density becomes (in the usual time-harmonic phasor notation) (41)

(34) provided is allowed to get arbitrarily small while maintaining . For a lossy medium , the inequality in (34) is dominated by the loss term that increases linearly with time and thus this inequality reduces to merely (35) which tells us that (36) in a lossy passive medium. in a frequency window In a lossless medium such that about , the inequality in (34) becomes

and, in addition to the inequalities in (39), we have (42) The inequalities in (39) and (42) imply that the magnitude of the group velocity, [8, Sec. 5.17], in a linear lossless medium characterized by a scalar permittivity and permeability (in the frequency window of interest) is equal to or less than the speed of light. To prove this, begin by differentiating the square of the propagation constant with respect to frequency and then take the absolute value to get (43a) Since

and

, we have from (43a) that

(37) which reveals that the average internal electric energy density (kinetic potential stored electric field energy density,

(43b)

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where the last inequality is obtained by noting that the minimum for is 2. Taking the inverse of value of in (43b) produces the desired result (43c)

, and and are real where , begin at a value of zero at functions that are zero for , and equal unity after some finite time . The time-average of this function over any one cycle beginning after a time has passed is

Of course, in a lossy dispersive medium, the magnitude of the group velocity may be greater than the speed of light [8, p. 334], [9]. For a lossless medium obeying the general linear constitutive relations in (21), a derivation similar to the above derivation for scalar permittivity and permeability was performed in [4, Appendix B] to obtain the internal energy density

(44) and the inequalities (in tensor notation)

(48) the usual time-harmonic result [1, Eq. (2.347)], [2, footnote, Sec. 59], [8, Sec. 2.20]. Applying this result to (3) and (5), we have for the time-average electromagnetic power density in any linear medium satisfying Maxwell’s equations in (1) and fields [1, 2.349], [2, footnote, (2) with time-harmonic Sec. 59], [8, Sec. 2.20]

(45a) (45b) (45c) (45d) (From hereon out, we omit the subscript on the frequencydomain constitutive parameters.) and , the inequality in (44) reduces For to the inequality given in Felsen and Marcuvitz [10, p. 81]. By or , (44) yields letting (46a) (46b) However, (44) does not imply that .

(49) which does not involve derivatives with respect to frequency. If and , then (49) becomes (50) the familiar electric and magnetic conductive heat losses found and . The time averages of in (20) with time derivatives of periodic functions vanish and thus the electromagnetic field energy term in (3) or (5) does not contribute to (49). V. RELATIONSHIP BETWEEN INTERNAL ENERGY AND QUALITY-FACTOR ENERGY

A. Time-Average Electromagnetic Power Density and Poynting’s Vector in a Linear Dispersive Medium With Sinusoidal Fields The time-harmonic internal energy density in (44) contains derivatives of the constitutive parameters with respect to frequency. However, it is a trivial matter to show that the time-average of the electromagnetic power density in (3) or (5) does not contain frequency derivatives for any linear material with sinu. soidal time-dependent fields that turn on at some time This, of course, would be expected because the electromagnetic power density, unlike the internal energy density, at each point in time does not depend on its time history (that is, does not depend on an integral over the previous times). To see this, note that each term in (3) or (5) is quadratic in the fields. Thus, for sinusoidal source densities and fields, each term in (3) or (5) at a point will have a time dependence of the form

(47)

In [4] it was shown that the internal energy density in (44) arises naturally from Maxwell’s equations and their frequency of anderivatives in the determination of a quality factor tennas that is approximately equal to twice the inverse of the matched VSWR (voltage standing wave ratio) half-power fractional impedance bandwidth of antennas. In particular, the of a one-port, linear, passive, lossy or lossless antenna tuned at a (so that the input reactance ) to resofrequency or antiresonance was given nance in [4] as (51) where is the power accepted (power radiated plus power loss) by the antenna and the internal energy is found from (52)


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(56b) (56c) For material that is lossless in a frequency window about , (44) shows that the volume integrals in (53) or (56) add positively to the of the antenna (whether or not the values of the constitutive parameters are positive or negative). We showed in [4] that the in (51), which depends on the definition of the internal energy in (53) or (56), was approximately equal to the matched VSWR half-power fractional band, that is width (57)

Fig. 1. One-port, linear, passive antenna with feed and shielded power supply.

with

(53a)

(53b) (53c) and, as usual, primes denoting differentiation with respect to the angular frequency . As shown in Fig. 1, is the volume of the antenna material that lies outside the shielded power supply and reference plane . The tuning capacitor or inductor is included in . , which includes the volume , is the entire The volume volume outside the shielded power supply and reference plane out to a large sphere in free space of radius that surrounds , the volume becomes inthe antenna system. As finite. The solid angle integration element is with being the usual spherical coordinates of the position vector , and the complex far electric field is defined by pattern (54) For the simple scalar constitutive relations (55) and the magnetic, electric, and magnetoelectric internal energies in 53(a)–(c) reduce to (56a)

for a sufficiently isolated6 resonance or antiresonance with ( often suffices), except when the antenna was dominated by highly lossy dispersive materials; see, for example, [4, Fig. 19]. We emphasize, however, that constant (independent of freand/or ) quency) conductivity materials ( should not be included in the exceptions because these constant conductivities do not add to the internal energy. They merely in (57) to maintain the high accuracy of change the the inverse relationship in (57) between bandwidth and . (Indeed, Maxwell’s frequency-domain equations are not dispersive and . This is further corrobowith respect to constant rated in the improved formulas (60) where a frequency indeor does not contribute to the Q-energy because pendent or .) Unfortunately, the non-negative expression (9) for the incannot reveal the modification to ternal energy density needed to improve the accuracy of the internal energy (57) for antennas containing highly lossy dispersive materials diverges because, as shown in (34), the internal energy with time in lossy media. Nonetheless, in [11] it was found , that a quality-factor energy,7 or simply Q-energy, given in the following formulas proved to be the replacement that produced a which maintained the accuracy for of the relationship between and matched VSWR half-power fractional bandwidth in (57) for antennas containing highly lossy dispersive material (58) 6The approximation in (57) can become very inaccurate near very closely spaced resonances and antiresonances—a limitation that remains even after j j is replaced by to obtain (58). 7The term “quality-factor energy” or simply “Q-energy” is introduced as an alternative to the term “internal energy” to describe the generalized formulas applied to lossy dispersive media because these formulas involve dissipative energy as well as stored energy per unit volume. The purpose here is to define energy densities, which when integrated, will produce a total Q-energy that determines with reasonable accuracy the inverse-bandwidth of antennas including those that contain highly lossy dispersive materials. Unlike previous energy densities defined for lossy dispersive media [12]–[15], the quality-factor energy densities defined here are not model dependent but depend only on the macroscopic constitutive parameters and fields of the antenna media (and thus are useful for antenna design).

W

W

Q

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with

(59) instead of (52)-(53),8 and with

(60a)

efficiency of the antenna. The and are defined in terms of the energies stored in the reactive electric and magnetic fields (total minus radiation fields) in the free-space region outside the minimum radius of the sphere that circumscribes the antenna. They are given explicitly by [19], [4, Appendix C, Eq. (C.5)], and (62), shown at the bottom of the page. In [4, Appendix C], and . The contribution to it is proven that the quality factor from inside the circumscribing sphere of radius is shown in (63) at the bottom of the page. The antenna is assumed to be resonant at the frequency , that is, it’s input reactance is zero either naturally or because of a series tuning of aninductor or capacitor (considered part of the volume tenna material). The input reactance of an antenna can be written explicitly in terms of the fields of the antenna as [4, Eq. (53)]

(60b) (60c) instead of (56). Unlike , the internal energy densities within the integrals of defined in (60) cannot be related directly to the sum of kinetic, potential, and store electromagnetic and are zero. field energy densities unless A. Lower Bounds on Quality Factor The question often arises as to whether the lower bounds on antenna quality factor determined by Chu and others [16]–[20] remain valid for electrically small antennas containing highly dispersive material. To answer this question, we begin with an exact relationship that in (58) satisfies

(61) which can be derived similarly to the corresponding relationship in [4, Appendix C]. The symbol denotes the radiation the medium is lossless in a frequency window about ! , not only does ; = , and = . Then = , but also = (59) and (60) reduce to (52)-(53), and (56), respectively. This can be proven by showing that the imaginary parts of 1 (! ) 1 ; 1 (! ) 1 , and 1 (! ) 1 + 1 (! ) 1 are zero, and the real parts of 1 (! ) 1 1 (! ) 1 cannot be negative because of (46). The [ 1 (! ) 1 + and 1 (! ) 1 ] term in (59) was mistakenly written as j 1 [(! ( + )] 1 j in [11, Eqs. (13) and (17a)]. (The imaginary part of 1 [(! ( + )] 1 does not generally equal zero in lossless media.) 8If

= ; = , and

E E H

H H E E

E

H

HE

E

E

E

H

E

H H H H

(64) Generally, approximated by

and (61) can be well

(65) Since , and are all greater than or equal to zero, the minimum in (65) would occur for a and if . In the previous given determinations of the lower bounds on quality factor [16]–[20], could be zero only if the fields it was assumed that of the antenna inside the circumscribing sphere of radius , fields in (63), shown at the bottom of the that is, the and page, were zero. But then (64) and (62), shown at the bottom could not be zero unof the page, would imply that less . Thus, in general, it was assumed that could not be zero but had to be at least as large as the quality factor of a tuning inductor or capacitor that would make , and thus the lower bound on quality factor (often called the Chu lower bound) was assumed to be (66) For lossy dispersive materials, however, it is presumably possible for , and to all be zero in

(62a)

(62b)

(63)


(63), and thus , without and being zero incan be zero and yet side the sphere of radius . That is, there can be contributions from inside the sphere of radius to the input reactance (64) of the antenna that allow the antenna to may be tuned (have zero reactance at ) even though . Thus, for antennas that contain lossy dispernot equal sive materials, the minimum possible value (lower bound) for the in (58) is simply

(67)

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zero. It will now be shown, however, that this is impossible, at least for antennas that contain only lossless isotropic materials with and zero). (scalar and For lossless isotropic materials and , it is not possible to make the input reactance in (64) zero while keeping in (63) negligible unless and . But it can be proven9 that these two inequalities are incompatible with the inequalities for lossless isotropic materials in (39) and (42). Thus, for antennas that contain only lossless isotropic material, the internal energy in must be large enough to tune the antenna, and consequently the lower bound on approximately equals the Chu lower bound; that is

if , or if , we see from (66) and (67) that the lower bound on the improved in (58) is related to the Chu lower bound by Writing

(72) For electric and magnetic dipole fields outside the circumscribing sphere of radius , (72) combines with (69) [with ] to give

(68) (73) in (67) and (68) because the antenna material inwhere side the sphere of radius has to be lossy for (67) and (68) to hold. For electric and magnetic dipole fields outside the circum, we have scribing sphere of radius and

since the stored energy in electric and magnetic dipoles are predominantly in the electric and magnetic fields, respectively, whereas the power radiated by the dipoles depends equally on and . (For example, the values of if or , and if [16]–[20].) Then (68) gives

The expressions in (67), (68), and (72) give the lower bounds on the improved of (58) for lossy dispersive antennas and lossless dispersive antennas (containing only isotropic materials), respectively. To achieve the lossy lower bound in (67) and (68) , efficiency would have to be sacrificed—and to what for extent, especially for , is not known. , the lower bound on For antennas with is reached if the fields inside the circumscribing sphere of radius are zero. This minimum quality factor can, in principle, be attained for any antenna by exciting these fields with electric and , on the surface of and magnetic surface currents, this sphere given by

(70)

(74)

(69)

For tuned antennas that are lossless in a frequency window about , the inequality in (44) combines with (63) to show that

(71) From (61) and (71) we see that the lower bound on is again , which can now occur only if the given by (67) if antenna fields are zero inside the sphere of radius . Since the antenna is tuned , however, and the antenna fields are zero inside the radius , all the input reactance of the antenna comes from the antenna fields that lie outside the sphere of radius . Thus, from (62) and (64), the lower bound in (67) can only be perfectly realized for lossless antennas if . On the other hand, one could argue that it may be possible for to be negligible if the integral in the numerator of (71) is negligible while the material constitutive parameters are so large in the antenna material that the input reactance in (64) can be made

where is the outward normal to the sphere [21]. The Stratton-Chu formulas [8, Sec. 8.14] ensure that the fields produced by these surface currents will be zero inside the sphere (a result sometimes referred to as the “extinction theorem”). in (67)-(68), and (72) hold for all The lower bounds on one-port, linear, passive, antennas containing the most general spatially nondispersive lossy material or lossless isotropic material, respectively, whose constitutive parameters can be either negative or positive (or zero). Of course, if nonlinear or active material is allowed in the antenna proper or in its tuning elements, then twice the inverse of the matched VSWR half-power fractional bandwidth is not limited by these lower bounds on . VI. ELECTROMAGNETIC FORCE DENSITY AND STRESS DYADIC IN A GENERAL POLARIZABLE MEDIUM The electromagnetic force density exerted by the electromagnetic fields on the charge, current, and polarization

9If 0 then ! < 0, which violates the second inequality in (39). 0 then 2 > ! , which violates the first inequality in (39); and If similarly for .

0

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densities [1, Sec. 2.1.10, Eq. (2.140)]

can be expressed as

power density and unlike the internal energy density, the electromagnetic force density at each point in time does not depend on its time history (that is, does not depend on an integral over previous times). Next consider the symmetric time-domain stress dyadic for a general polarizable media given by [22, Eq. (10.66)]

(75)

(79) The general constitutive relations in (4) show that in (76) by related to the stress dyadic

with the electromagnetic stress dyadic (tensor) given by

is

(76) often referred to as the electromagand netic momentum density stored in the fields or simply the electromagnetic field momentum density. In other words, it was is equal to the force proven in [1, Sec. 2.1.10] that , current , and polarexerted on the charge in the macroscopically small ization (containing the point ) by the electromagvolume element netic fields at the time .10 Regardless of how complicated the fields and sources or what the constitutive parameters, (75) reveals that the electromagnetic force density can be determined solely, with the help of the electromagnetic stress dyadic defined by (76), in terms of and . A. Time-Average Electromagnetic Force Density and Stress Dyadic in a Linear Dispersive Medium With Sinusoidal Fields Sinusoidal fields can exist in a linear medium, but they must turn on at some time in the past. Thus, let us apply (75) to sinusoidal fields that turn on at some time and reach steady state after a time . We can then use the formula (48) to immediately obtain the time average of (75) for general time-harmonic sources and fields, namely the time-average force density

(80) If one takes the divergence of both sides of (80) and integrates over a macroscopic volume element such that the surface is taken as free space just outside (see Footnote 10), of then the term on the right-hand side of (80) that is multiplied by 1/2 vanishes because the divergence of this term can be conwhere verted to a surface integral in the free space outside and are zero. In other words, the electromagnetic force for in the density obtained by substituting electromagnetic force (75) gives the identical force on a volume of material when integrated over that volume. Thus, for the sake of determining the electromagnetic force on any volume, we can write (81) where indicates that these two dyadics are interchangeable for the purpose of determining the force on any volume of material (see Footnote 10). For single-frequency sinusoidal fields time averaged over one cycle, (81) becomes (82) with

given in (78) and

(77) where is the time-harmonic complex electromagnetic stress dyadic defined as [1, (2.350)]

(83) Thus, the time-harmonic electromagnetic force density in (77) can be rewritten as (84)

(78) The time averages of time derivatives of periodic functions vanish and thus the Poynting vector term in (75) does not contribute to (77). The formulas in (77) and (78) do not contain derivatives with respect to frequency and are identical to those obtained by averaging the electromagnetic force over any one cycle of single-frequency sinusoidal fields [1, Sec. 2.3.9]. This result is not unexpected because, like the electromagnetic 10Again, as in Footnote 1, the only proviso for this result to hold is that the surface of V is assumed to lie in free space just outside of the material so that the surface polarization charge n 1 P and surface magnetization current M r; t 2 n (with n denoting the surface unit normal) is included in the integration over V by means of delta functions in the spatial derivatives of the field and polarization densities across the surface of V ; see [1, Sec. 2.1.10]

1 ( ) ^

1

^

^

1

and For dispersive materials in which time-harmonic stress dyadic in (83) becomes

, the

(85a) and

(85b) which, when inserted into (84), gives the same results as those obtained in [6].


VII. CONCLUDING SUMMARY A general expression is derived for a non-negative time-domain energy density in passive, nonlinear or linear, lossy or lossless, temporally and/or spatially dispersive polarized media. This energy is related to the kinetic, potential, and heat energy densities of the bound collisionless charge-polarization carriers and the stored electromagnetic field energy density. It is shown that the negative of the divergence of the time integral of the Poynting vector equals the sum of the kinetic, potential, heat, and stored electromagnetic energy densities. In a linear material characterized by scalar constant permittivity, permeability, and electric and magnetic conductivities, the energy theorem predicts the expected “internal energy” density and time-average dissipated energy density for frequency-domain fields. In the most general linear, lossless, spatially nondispersive media, the energy theorem reveals non-negative quadratic forms for the frequency-domain internal energy densities that are used of antennas conin the expressions for the quality factor taining lossless (except for conductivity) dispersive material. The analysis leading to the expressions for internal energy and quality factor also reveals useful inequalities that the constitutive parameters for the general linear, lossless, spatially nondispersive materials must satisfy. These inequalities are shown to predict that the magnitude of the group velocity in lossless material is always less than or equal to the speed of light. It is further shown, however, that the general theorem does not predict an internal energy in highly lossy dispersive media that can be used to improve upon the expressions for the of antennas in such media. We nonetheless determine a non-negative “Q-energy” density that improves upon the previous exin that the improved maintains the accupressions for racy of its inverse relationship to matched VSWR half-power fractional bandwidth for antennas containing highly lossy dispersive material. Lower bounds for this improved are found in terms of previously determined lower bounds [16]–[20]. For antennas containing only lossless isotropic materials, the new lower bounds are identical to the previous ones. The paper also confirms the result that, for general linear lossy or lossless dispersive material, the steady state time-averages of the electromagnetic power density, force density, and stress dyadic with sinusoidal time dependence that turns on at some , unlike the internal energy, does not contain derivatime tives with respect to frequency but simply equals the one cycle average of the frequency-domain electromagnetic power density, force density, and stress dyadic. ACKNOWLEDGMENT The author thanks Prof. A. Figotin of the University of California at Irvine for his thoughtful review of this work. REFERENCES [1] T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications. Piscataway, NJ: IEEE/Wiley Press, 1999. [2] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. Oxford, U.K.: Butterworth-Heinemann, 1984.

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[3] A. D. Yaghjian, Relativistic Dynamics of a Charged Sphere: Updating the Lorentz-Abraham Model, 2nd ed. New York: Springer-Verlag, 2005. [4] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 1298–1324, Apr. 2005. [5] A. Tonning, “Energy density in continuous electromagnetic media,” IRE Trans. Antennas Propag., vol. AP-8, pp. 428–434, Jul. 1960. [6] A. Figotin and J. Schenker, “Hamiltonian treatment of time dispersive and dissipative media within the linear response theory,” J. Comp. Appl. Math. Phys. Aug. 2006, also see arXiv:math-ph/0608003 v1, 1. [7] L. Brillouin, Wave Propagation and Group Velocity. New York: Academic Press, 1960. [8] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [9] M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A, vol. 4, pp. 2104–2108, Nov. 1971. [10] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. New York: IEEE Press, 1994. [11] A. D. Yaghjian, “Improved formulas for the Q of antennas with highly lossy dispersive materials,” IEEE Antennas Wireless Propag. Lett., vol. 5, Aug. 2006, Online. [12] R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A, vol. 3, pp. 233–245, May 1970, “Corrigendum” vol. 4, p. 450, Jul. 1970. [13] R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A, vol. 299, pp. 309–312, Jul. 2002. [14] S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A, vol. 343, pp. 231–237, Jun. 2005. [15] A. D. Boardman, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B, vol. 73, pp. 165110-1–165110-7, Apr. 2006. [16] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [17] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961, sec. 6–13. [18] R. E. Collin and S. Rothschild, “Evaluation of antenna Q,” IEEE Trans. Antennas Propag., vol. AP-12, pp. 23–27, Jan. 1964. [19] R. L. Fante, “Quality factor of general ideal antennas,” IEEE Trans. Antennas Propag., vol. AP-17, pp. 151–155, Mar. 1969. [20] J. S. McLean, “A re-examination of the fundamental limits on the radiation of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, pp. 672–676, May 1996. [21] H. A. Wheeler, “Small Antennas,” in Antenna Engineering Handbook, R. C. Johnson and H. Jasik, Eds., 2nd ed. New York: McGraw-Hill, 1984, sec. 6–5. [22] F. N. H. Robinson, Macroscopic Electromagnetism. Oxford, U.K.: Pergamon, 1973.

Q

Arthur D. Yaghjian (S’68–M’69–SM’84–F’93) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Brown University, Providence, RI, in 1964, 1966, and 1969, respectively. During the spring semester of 1967, he taught mathematics at Tougaloo College, MS. After receiving the Ph.D. degree he taught mathematics and physics for a year at Hampton University, VA, and in 1971 he joined the research staff of the Electromagnetics Division of the National Institute of Standards and Technology (NIST), Boulder, CO. He transferred in 1983 to the Electromagnetics Directorate of the Air Force Research Laboratory (AFRL), Hanscom AFB, MA, where he was employed as a Research Scientist until 1996. In 1989, he took an eight-month leave of absence to accept a visiting professorship in the Electromagnetics Institute of the Technical University of Denmark. He presently works as an Independent Consultant in electromagnetics. His research in electromagnetics has led to the determination of electromagnetic fields in continuous media, the development of exact, numerical, and high-frequency methods for predicting and measuring the near and far fields of antennas and scatterers, and the reformulation of the classical equations of motion of charged particles. Dr. Yaghjian is a Member of Sigma Xi. He has served as an Associate Editor for the IEEE and the International Radio Scientific Union (URSI). He has received Best Paper Awards from the IEEE, NIST, and AFRL.