IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 6, JUNE 2007
1495
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: arthur.yaghjian@hanscom.af.mil).
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
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(6)
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 6, JUNE 2007
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
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( )
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.
1497
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)
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YAGHJIAN: INTERNAL ENERGY, Q-ENERGY, POYNTING’S THEOREM, AND THE STRESS DYADIC
and the reality relations [1, p. 95]
1499
) 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,
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(43b)
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 6, JUNE 2007
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
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
(47)
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(52)
YAGHJIAN: INTERNAL ENERGY, Q-ENERGY, POYNTING’S THEOREM, AND THE STRESS DYADIC
1501
(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
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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)
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YAGHJIAN: INTERNAL ENERGY, Q-ENERGY, POYNTING’S THEOREM, AND THE STRESS DYADIC
(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)
1503
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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1504
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 6, JUNE 2007
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
^
and
For dispersive materials in which
time-harmonic stress dyadic in (83) becomes
(85a)
and
^
1
, the
(85b)
which, when inserted into (84), gives the same results as those
obtained in [6].
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YAGHJIAN: INTERNAL ENERGY, Q-ENERGY, POYNTING’S THEOREM, AND THE STRESS DYADIC
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.
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[17] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New
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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.
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