Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using High Permeability Shells

s t r at e g i c w h i t e pa p e r

Approaching the Lower Bounds
on Q for Electrically Small
Electric-Dipole Antennas Using
High Permeability Shells
Howard R. Stuart Arthur D. Yaghjian
Member, IEEE Life Fellow, IEEE

Abstract—We study the effect of surrounding electrically
small, top-loaded, electric-dipole antennas with a thin
shell of high-permeability magnetic material. The magnetic
polarization currents induced in the thin shell of magnetic
material reduce the internal stored energy, resulting in a
lower Q as compared to conventional designs. The simulated
Q of thin-magnetic-shell cylindrical and spherical antennas
are compared to recently derived lower bounds on Q. The
high permeability shells reduce the Q of these antennas to
values below the lower bounds for purely global electric
current sources. In the case of the spherical electric-dipole
antenna, a sufficiently large value of permeability enables
the Q to be reduced to a value that is only 1.11 times the Chu
lower bound.

Index Terms—Dipole antennas, electrically small antennas,
lower bounds, quality factor.

Manuscript received December 17, 2009; revised March 15, 2010; accepted May 24, 2010. Date of publication September 23, 2010; date of current version
November 30, 2010.
This work was supported in part by the Defense Advanced Research Projects Agency (DARPA) and in part by the U.S. Air Force Office of Scientific Research (AFOSR).
H. R. Stuart is with LGS Innovations, Florham Park, NJ 07932 USA.
A. D. Yaghjian is a Research Consultant at Concord, MA 01742 USA.
Digital Object Identifier 10.1109/TAP.2010.2078466

S t r at e g i c w h i t e p a p e r
Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells

table of contentS
[ 1 ] INTRODUCTION 01

[ 2 ] CY L INDRICA L CAPPED MONOPO L E 02

A. Resonant Modes 02

[ 3 ] S PHERICA L CAPPED MONOPO L E 09

A. Resonant Modes 09

B. Matching to 50 ohms 10

C. Electric Permittivity 11

[4 ] S UMMARY 12

[ 5 ] REFERENCE S 13

[ 6 ] B i o g r a p h i e s 14

ii

1. introduction
The lower bounds on the radiation quality factor Q of electrically small antennas were first derived by Chu [1] (see also [2]–
[6]) for a spherical surface circumscribing an antenna under the assumption that there is no stored energy inside the sphere
except possibly for the stored energy needed to tune to zero the reactance produced by the stored energy outside the sphere.
It is generally assumed there will be additional stored energy inside an actual antenna that will raise the Q above the
Chu lower bound. Wheeler [7] was the first to observe that, for the case of a small spherical magnetic-dipole antenna with
electric currents confined to its surface, filling the antenna volume with a material of infinite magnetic permeability would
reduce the stored magnetic energy inside the sphere to zero, enabling the Chu lower bound to be achieved theoretically.
Notwithstanding the practical issues of internal resonances and producing the required low-loss magnetic material,
this result provides an approach for reducing the internal energy and lowering the Q of small magnetic-dipole antennas
using permeable materials [8]. Wheeler offered no analogous method for reducing the internal stored electric energy of an
electrically small spherical electric-dipole antenna. (It should be noted that a high-permeability material filling the antenna
volume is not equivalent to filling the volume with a perfect magnetic conductor (PMC) because the electric field, unlike the
magnetic field, does not approach zero inside a high-permeability core.)

Chu’s lower bound calculation was modified by Thal [9] in a manner that accounts for the stored energy inside the bounding
sphere when the antenna consists of global1 electric currents confined to the spherical surface. The Thal lower bound for
a small electric-dipole antenna is 1.5/(ka)3, a factor of 1.5 above the Chu lower bound for antennas with ka % 1 (k is the
wavenumber and a is the radius of the antenna’s circumscribing sphere). This result concurs with the known performance
of several optimized electrically small spherical electric-dipole antennas [10]–[13], and is generally thought to represent the
best achievable performance for these electric-dipole antennas.

In a recent paper [14], we have derived general expressions for the lower bounds on the Q of electric- and magnetic-dipole
antennas in arbitrarily shaped electrically small volumes. For the case of a spherical electric dipole excited by global
electric current sources, the lower bound matches Thal’s result of 1.5/(ka)3. For the more general case of both electric
and magnetic current sources, the spherical electric-dipole lower bound is reduced to 1/(ka)3, matching the Chu bound.
A similar result follows for arbitrary geometries. That is, allowing for the possibility of magnetic current sources yields
reduced lower bounds as compared to the case of global electric current sources alone. Because magnetic charge does not
exist, achieving this general source condition in practice requires the use of magnetic polarization currents, such as those
arising in permeable magnetic materials. These new lower-bound equations therefore imply it is possible to use magnetic
materials to improve the Q of electrically small electric-dipole antennas. The derivation in [14] also suggests the form and
physical mechanism by which these improvements occur. In particular, a suitably driven thin shell of permeable material
surrounding an electric-dipole antenna can produce magnetic polarization currents that reduce the stored electric energy
inside the antenna, thereby improving the Q.

In this paper we illustrate antenna designs incorporating thin shells of high relative magnetic permeability to improve the
Q of electrically small electric-dipole antennas. We treat top-loaded dipole antennas (or equivalently, monopoles on infinite
ground planes) confined to cylindrical (Section II) and spherical (Section III) volumes. The simulated quality factors are
compared to the general lower bounds found in [14] in order to assess the optimality of the designs. For the cylindrical
antennas, some basic design equations are provided to estimate the minimum value of relative permeability required to see
a notable reduction in Q.

1 By “global” electric currents we mean electric currents other than those on small Amperian current loops (or possibly those on slotted electric conductors) that produce
effective magnetic current or polarization.

1

2. CYLINDRICAL CAPPED MONOPOLE
2 .1 Resonant Modes

We consider the top-loaded monopole structure, shown in cross-section in Figure 1(a), consisting of a short straight
conducting wire with a circular conducting cap. When placed over a large ground plane, the structure forms a scatterer that
resonates like a small electric dipole with the lowest resonant frequency occurring at a wavelength that is typically much
larger than the height of the straight wire because of the capacitive loading of the circular cap. As an example, consider
a monopole height of 100 mm with a circular cap radius of 150 mm, a center conductor radius of 7.5 mm, and a top plate
thickness of 2 mm. With no dielectric or permeable material present, the fundamental resonant frequency of the structure
shorted to ground is 154 MHz and it has a Q of 14.4, both of which are determined numerically using an eigenmode
simulation with a perfectly matched layer at the outer boundaries [13], [15]. The height of the monopole at this frequency is
slightly larger than 1/20 of a wavelength. The electric field of the resonant mode is shown in Figure 2(a). Underneath the
cap, the electric field is predominantly z-polarized and uniform along the z-direction, deviating from this behavior most
notably near the edge of the cap.

We next consider how the mode structure is affected by introducing a thin shell with high magnetic permeability
surrounding the cylindrical volume formed by the projection of the circular cap. The thin shell sits just under the outer
edge of the circular cap [see Figure 1(b)] such that the overall circular cylindrical volume occupied by the open resonator
(scatterer) does not protrude beyond the cap. The equations relating the electric and magnetic time-harmonic
(exp(–i~t), ~ >0) fields E and H and can be written as

(1)

(2)

where the magnetic induction B is related to H by the magnetic permeability through B = n0nr H, J is the current density
on the conductors, which are assumed to be perfectly electrically conducting, and e0 and n0 are the permittivity and
permeability of free space. Because of the axial symmetry, the fundamental mode oscillates with z-polarized magnetic
fields (B, H) and with an r- and z-polarized electric field E for cylindrical coordinates (r, z, z), where z points upwards in
Figs. 1, 2, and 3.

Figure 1

Two-dimensional cross-section of the cylindrical capped monopole on an infinite ground plane (a) in free space (b) with
a thin cylindrical shell of high magnetic permeability surrounding the cylindrical volume formed by the projection of the
circular cap.

2

Figure 2

Electric field of the fundamental resonant mode of the capped monopole (a) in free space (b) with a 2 mm thick shell of
magnetic permeability nr = 472 and (c) with a 2 mm thick shell of magnetic permeability nr = 8000. The thin high-nr
layer in (b) makes the electric field equal to zero near the bottom outside edge and substantially reduces the electric
field everywhere beneath the cap. By further increasing nr, the Ez = 0 point moves to a higher position on the inside of
the shell layer, and the electric field beneath the cap is further reduced.

Figure 3

The thin, high-nr layer introduces a discontinuity in the tangential component of the electric field across the boundary.
For 9l> 0 and finite nr, Ez is continuous, though rapidly changing, inside the thin layer—the discontinuity referring to
the values of Ez on either side of the layer. In the figure, a positive z-directed magnetic field points into the page.

The thin shell is assumed to have a relative magnetic permeability nr& 1. Because the surface layer is everywhere tangential
to the z and z directions and normal to the r direction, a jump in value is introduced in the z-directed electric field across
this layer. The size of this jump is determined using (1) and Stokes theorem to relate the line integral of the electric field
along the contour shown in Figure 3 to the surface integral of the magnetic induction Bz on the surface enclosed by the
contour. Letting ∆S go the zero causes the Er contributions on the ends of the contour to cancel, and letting ∆l approach the
thickness d of the shell yields the following equation for the difference of E z across the shell in terms of Bz or Hz in the shell

(3)

(4)

where h = (n 0 /e 0 )1/2 is the characteristic impedance of free space (120r ohms), k = ~(n 0 /e 0 )1/2 is the free-space wavenumber,
and (Bz, Hz) are approximately uniform across the thin shell. In the limit as d becomes vanishingly small but n r is made
large enough that the quantity n r d is nonzero, (4) illustrates that the high-n r layer behaves like a magnetic surface current,
inducing a discontinuity in the tangential component of the electric field across the thin shell.

3

We shall now show that it is possible to reduce the E z field under the cap to practically zero and thus reduce the Q of the
electric-dipole radiator to a value close to the smallest possible lower bound for a radiator confined to the given cylindrical
volume. If E z is successfully reduced to a value that is practically zero everywhere inside the cylinder, the magnetic field
H] inside the cylinder is given from the integral form of (2) as I0/(2rr), where I0 is the electric current along the center
conductor. (The top loading produces a fundamental resonant mode that has an approximately uniform current along z in
the center conductor.) At the outer surface of the cylinder, Hz = I0/(2rrc), where rc is the cylinder radius. The value of the
electric field Ez at the outer surface can be estimated from the static capacitance C DC of the capped monopole structure,
O

where the capacitance can be determined numerically or estimated from known formulas [16]. The resulting electric field
O
is Ez = –iI0/(~C DCh), where h is the height of the cylinder above the ground plane. (In practice EO will not be perfectly
z
uniform but this relationship provides a reasonable estimate.) Using (4), we then determine the following approximate
condition needed to make E z = 0 inside the cylinder

(5)

Figure 4

The magnitude of the electric field versus position along the ground plane for the fundamental resonant mode using a
2 mm thick n r = 472 shell.

For the dimensions of the structure considered here, the capacitance is determined by numerical simulation to be 18.6 pF.
The required n r d depends upon the resonant frequency, which itself will move to lower frequencies due to the presence
of the high-n r shell. For a resonant frequency of about 104 MHz and a shell thickness of d = 2 mm, (5) yields n r = 472. A
numerical eigenmode simulation of the shorted capped monopole with this shell configuration finds a resonant frequency of
103.9 MHz and a Q = 34.7. For this case, the electric field indeed goes to zero near the ground plane just inside the cylinder
surface [see Figure 2(b)]. This effect is illustrated more clearly in Figure 4. The shell material (positioned at 150 mm) induces
a large change in the electric field such that the field is zero on the ground plane just inside the surface of the material. It
is not zero everywhere underneath the cap, however. This is due, in part, to the fact that the field is only purely z -polarized
at the ground plane; a small r -polarized component becomes more pronounced above the ground. (This component is
normal to the shell surface and unaffected by the high-n r material.) Also, E z increases with height above the ground plane,
such that the chosen value n r of does not work ideally at all positions (a larger value is required to zero the field at higher
positions). Nevertheless, there is a pronounced reduction in the size of the electric field throughout the region beneath the
cap. The effect becomes more dramatic if we further increase n r [see Figure 2(c)], with larger reductions in stored energy
accompanying larger values of n r . In practice, there will, of course, be a limitation on the maximum obtainable value of n r .
Equation (5) is useful for providing a rough estimate for the minimum value of n r d necessary to observe a strong effect.

4

The measure of whether a high-n r material is successful in significantly reducing the overall stored energy in the mode
is the value of Q. The Q increases as a result of adding a high-n r d shell because of the associated lowering of the resonant
frequency. Consequently, for a proper evaluation, the Q must be compared to known lower bounds on Q for the particular
electrical size and shape of the volume to which the antenna is confined. We use the convenient expressions for the lower
bounds on the Q of electrically small dipole antennas derived for the case of solely global electric current excitations [14, Eq.
(29b)]

(6)

and for the case of both electric and magnetic current excitations [14, Eq. (21b)]

(7)

where the electrostatic polarizability ae in [14] has been expressed as l (the polarizability normalized to the volume) times
the volume V of the antenna. The parameter l depends solely upon the shape of the antenna volume (l $ 1) and serves as a
convenient parameter for comparing the performance of various structures. (We refer to l as the shape factor, following the
terminology of Wheeler, who used a similar shape factor in characterizing the capacitance/inductance of electrically small
cylindrical antennas [17]).

For the cylindrical volume considered here, the shape factor l = 2.88 (determined numerically from COMSOL Multi-
physics), which corresponds to lower bounds on Q in (6) of 2.71/(ka)3 for global electric current excitations alone, and
1.77/(ka)3 in (7) for both electric and magnetic current excitations. Thus, the Q can be improved, in principal, by a factor of
about 1.5 by allowing both electric and magnetic currents. From numerical simulation, we determine the resonant frequency
and Q of the fundamental mode of our cylindrical-volume resonator for a range of values of magnetic permeability and shell
thickness; the results are shown in Figure 5. For the free-space case (n r = 1), the Q is slightly above the lower bound for
global electric current excitations [15]. As the shell permeability increases (introducing magnetic currents), the Q crosses the
electric-current lower bound and begins to decrease. After transitioning to lower values, the Q reaches a plateau such that
further increases in n r offer little additional reductions in Q. The lowest Q is obtained for the thinnest shell (2 mm) with
very high permeability; in this case we reach a Q = 1.84(ka)3, a value only 4% above the smallest possible lower bound of
1.77(ka)3. Increasing the shell thickness d lowers the value of the permeability required to see an effect.

However, the minimum achievable Q is higher for thicker shells (which allow more field penetration beneath the cap,
thereby increasing the stored energy). Filling the entire cylinder with magnetic material lowers the Q of electrically small
electric-dipole antennas very little (unlike electrically small magnetic-dipole antennas). The value of n r d required to bring
the n r into the “transition” region of the curve can be estimated using (5) with the frequency set to the resonant frequency
(~f s = 154 MHz) of the free-space (n r = 1) antenna. (The values predicted in this way are shown in Figure 5 along the
0
x-axis.) This is convenient, as the shift in resonant frequency induced by the introduction of the high-n r shell is not
typically known a priori for a given structure.

5

Figure 5

The resonant frequency and Q of the fundamental resonant mode of the cylindrical capped monopole [normalized by
(ka)3] versus relative permeability of the shell layer for a range of shell thicknesses. The lower bounds for solely global
electric current excitations, and for both electric and magnetic current excitations, are also shown. By applying (5)
f
with the free-space resonants frequency (~ 0 = 154 MHz), we can estimate the permeability value required to reach the
transition portion of the curve for each case.

2 .2 The Corresponding Cylindrical Antenna

The capped monopole open resonator (scatterer) discussed in the previous section can be converted into an electric-dipole
antenna by driving the base of the center conductor through the ground plane. For a d = 2 mm shell of n r = 100, the
eigenmode analysis predicts a resonant frequency of 80.8 MHz and a Q = 69.4(Q(ka)3 = 1.97). The time-harmonic simulation
of the driven antenna produces the impedance shown in Figure 6. The Q of the antenna is estimated from the frequency
derivative of the impedance using the formula derived by Yaghjian and Best [18, Eq. (96)]

(8)

where the reactance is assumed to be tuned to zero at each frequency ~ and R(~) and X(~) are the resistance and reactance
of the untuned antenna. The resulting Q z near the resonance is shown in Figure 7. These results confirm that the high-n r
layer produces an antenna exceeding the performance bounds for electric-dipole antennas with solely global electric current
excitations and achieves a Q that is only about 11% higher than the smallest possible lower bound for antennas with both
electric and magnetic current excitations. Figs. 6 and 7 illustrate the accuracy of the eigenmode simulation in predicting
the Q of the driven antenna at resonance. Similar results are observed for the other resonator configurations studied in the
eigenmode analysis.

As discussed in [15], the impedance properties of a single resonance top-loaded monopole antenna are well described over
a wide frequency range using the equivalent circuit model shown in Figure 8. The lumped element values are determined
from four parameters: the resonant frequency ~ 0 , the Q of the fundamental resonant mode, the radiation resistance R 0 of
the fundamental mode at its resonant frequency, and the static capacitance C D C of the antenna, through the equations
C=1/~ 0 QR 0 , L=QR 0 /~ 0 , R=1/[R 0 (~ 0 C)2] and C res=C D C − C . The three parameters ~ 0 , Q, and C D C were determined
numerically in the previous section and we shall next derive an equation for determining R 0 .

6

Figure 6 Figure 7

Impedance versus frequency for the cylindrical capped Qz versus frequency for the cylindrical capped monopole
monopole antenna with a 2 mm shell of n = 1000. The with a 2 mm shell of n = 1000.
parameters of the lumped element model are derived
from the resonant frequency, the Q, the resonant
radiation resistance of the fundamental mode, and the
static capacitance.

For excitations with global electric currents alone, the radiation resistance at resonance can be determined from the well-
known equation for a small monopole of uniform current and height h over an infinite ground plane [19]

(9)

where m 0 is the resonant wavelength. Equation (9) was shown in [15] to provide a highly accurate value for the radiation
resistance of the fundamental mode of the capped monopole antenna (with no dielectric or permeable materials). Equation
(9) is no longer accurate once the high-n r shell is added to the structure. This is because the magnetic currents induced in
the high-n r shell, in addition to greatly reducing the electric fields interior to the shell, also provide a contribution to the
total dipole moment of the resonance beyond that of the electric currents alone.

To account for this effect, we first write the electric-current contribution to the electric-dipole moment ( ) as [20]

(10)

where the factor of 2 accounts for the presence of the ground plane. In order to determine the magnetic-current contribution
to the electric dipole, we first rewrite (1) in the form

(11)

where the magnetization M is related to the magnetic induction B through the constitutive relation B = n 0 (H+M). The
magnetization M = M ] ] is nonzero only inside the high-n r material. For a very thin shell, M ] is approximately uniform
across the shell and equals (n r -1 ) H ] , where H ] is the value of the approximately uniform magnetic field in the shell. The
right side of (11) can be interpreted as a magnetic-current source Jm analogous to the electric-current source J in (2), where
Jm= - i~n 0 M . Taking the magnetic-current analog of the more familiar relationship in which a circulating electric current
produces a magnetic-dipole moment, we write the magnetic-current contribution to the electric-dipole moment as ( ) as

(12)

7

where A is the cross-sectional area of the circular cylinder (rrc2) and Im is the magnetic current in the high-n r shell. The
total magnetic current circulating inside the shell material can be expressed as

(13)

where we have assumed a uniform H ] along the entire height of the cylinder with the factor of 2 again accounting for the
ground plane. Because Im is a magnetic current, it has units of volts (not amperes), such that the units in (12) are that of an
electric-dipole moment (ampere-second-meters), that is, identical to the units in (10). The two electric-dipole moments add in
phase to give the total dipole moment, which produces a total radiated power of [21, p. 437]

(14)

where the factor of 1/2 accounts for the infinite ground plane. Combining (10) with (12)–(14) and using the relationships and
H ] = I0/(2rrc) and R0= 2P/|I0 | 2 yields the following expression for the resonant radiation resistance

(15)

where we have let (n r -1 ) . n r because n r & 1 . The second term in the last factor of (15) accounts for the increase in the
radiation resistance due to the presence of the high-n r shell, and illustrates how the magnetic currents directly assist in
polarizing the radiating dipole (this term, without the approximation, goes to zero when n r =1 ). It is important to recall
:
that m0 is itself a function of n r (m0 increases as n r in the limit of high n r) such that the resonant radiation resistance and
the ratio of the dipole moments (pmc/ pec ) approach limiting values as n r becomes very large. This is implied by Q reaching
e e
a plateau for large values of n r .

Equation (15) is used in combination with the modal analysis and the static capacitance to determine values for the
lumped element parameters in Figure 8. The resulting model produces the solid lines shown in Figure 6. Although the
lumped element model overestimates the radiation resistance at resonance by approximately 8% (not surprising given
the approximation used to derive (15) that H ] is uniform along z), the model nonetheless closely matches the simulated
impedance over two octaves around the resonance. A similarly close correspondence between the lumped element model
and the simulated impedance is seen for other values of n r , with the correspondence becoming slightly worse for smaller
n r . This degradation is caused by the greater inaccuracy of (15) when larger electric fields exist beneath the cap (as happens
mc
for both smaller n r and thicker shells). Likewise, (12) overestimates p e for thick shells, implying that (15) is most accurate
for very thin shells. Nonetheless, for a 50 mm shell with n r =100 (15) only overestimates R0 by 16%, close enough that the
impedance curves of the model match the time harmonic simulation reasonably well at and below resonance. However,
when the entire cylindrical volume is filled with the high- n r material, the discrepancy between (15) and the observed R0
becomes too large (greater than a factor of two) to provide a useful estimate.

FIGURE 8

Lumped element model for short top-loaded monopoles.

8
S T R AT E G I C W H I T E P A P E R
APPROACHING THE LOWER BOUNDS ON Q FOR ELECTRICALLY SMALL ELECTRIC-DIPOLE ANTENNAS USING HIGH PERMEABILITY SHELLS

3. SPHERICAL CAPPED MONOPOLE
3.1 Resonant Modes

Although spherical antennas may be less practical than cylindrical antennas, they are useful in determining how closely
the original Chu lower bound can be approached. In this section we study the spherical capped monopole shown in cross-
section in Figure 9(a) (the spherical analog of the cylindrical antenna discussed in the previous section). The antenna
considered here has a radius of 100 mm with a center conductor radius of 7.5 mm. The shell thickness will be varied, but
in each case the thickness of the electrically conducting portion of the shell (upper region) is made equal to the thickness
of the high-n r shell (lower region), forming a solid hemispherical shell of uniform thickness. The conducting portion of the
shell extends from the top of the sphere down to an angle of 50 degrees from the top (this is roughly the optimal angle for
minimizing the Q at resonance relative to lower bounds for the free space antenna). The entire shell conforms to a spherical
bounding surface in order to minimize the Q with respect to the original Chu lower bound.

With no dielectric or permeable material present in the lower portion of the shell, the antenna shorted to ground has a
resonant frequency of 249.3 MHz (ka = 0.522) and a Q of 11.9 for a conducting shell thickness of 1 mm. This is a factor of
1.34 above the Chu lower bound2 [1/(ka)3 + 1(ka)], and a factor of 1.008 above the lower bound 1.33[1/(ka)3 + 1(ka)] of Thal [9,
Table I] for spherical antennas with global electric currents confined to the surface of the sphere. Increasing the thickness of
the conducting shell raises the resonant frequency (253.6 MHz for a 10 mm thickness), and also raises the Q relative to the
lower bounds very slightly (1.35 times Chu lower bound and 1.024 times Thal lower bound at a 10 mm thickness).

We next introduce magnetic material into the lower portion of the shell. Figure 10 illustrates the effect of increasing relative
permeability values on the resonant frequency and Q of the fundamental resonant mode. In the limit of a thin shell with
very high permeability, the fundamental mode has a Q that is only a factor of 1.11 above the Chu lower bound.

The physical mechanism enabling the Q to closely approach the lower bound is identical to that seen previously in the
cylindrical antenna. A magnetic polarization current is induced in the high-n r shell by the magnetic field of the resonant
mode. This polarization acts as a magnetic surface current that reduces the magnitude of the electric field inside the sphere
and thus the stored electric energy inside the sphere. For very high values of relative permeability, a small amount of
electric field energy is still present inside the sphere (concentrated predominantly near the junction between the high- and
conducting regions of the shell), such that the Q does not reach the Chu lower bound but achieves a value that is 11% above
the Chu lower bound.

Figure 9

(a) Cross-section of the spherical capped monopole antenna on a large ground plane. A thin, high-n r shell is used to com-
plete the sphere, in a geometry analogous to the cylindrical antenna. The center conductor of the antenna is fed through
the ground plane. (b) Splitting the feed into two asymmetric posts and grounding one allows for matching to 50 ohms.
2 Because ka = 0.522 is not < 1, we include the 1/(ka) term in addition to the 1/(ka)3 term in the Chu lower bound.

9

Figure 10

The resonant frequency and Q-factor of the fundamental resonant mode of the shorted spherical capped monopole
antenna versus the relative permeability of the shell layer, normalized to the Chu lower bound [1/(ka)3 + 1/(ka)] . For
very thin, very high permeability shells, the Q is 1.11 times the Chu lower bound

3 . 2 M at c h i n g t o 5 0 o h m s

The shorted structure of Figure 9(a) can be made into an antenna by driving the conducting post through the ground plane.
Numerical simulations confirm that the resulting antennas have resonant frequencies and Q’s corresponding to those
predicted by the eigenmode simulations. As in the case of the cylindrical antennas, the radiation resistance is typically
small at resonance (< 10 ohms). However, there are well known feeding techniques that can be applied to matching these
antennas (both the spherical and cylindrical versions) to 50 ohms. One example is shown in Figure 9(b); the feed post
is divided asymmetrically into two arms, with the wider arm shorted to ground and the narrow arm fed through the
underlying ground plane. Simulations of this geometry confirm that the spherical antenna can be matched to a 50 ohm
impedance (see Figure 11). In this case, the sphere radius is 100 mm, and the two posts are composed of flat plates with
widths of 2.5 mm (feed post) and 7.5 mm (shorted post), separated by 2.5 mm. The resulting antenna is matched to 50
ohms at 163 MHz. The magnetic shell is 5 mm thick with n r = 100; in this configuration Figure 10 predicts a Q at 1.15
times the Chu lower bound, or equivalently, a -3 dB matched fractional bandwidth of 6.2% (10 MHz) for ka = 0.341. This
performance is observed in the simulated antenna (see inset of Figure 11).

10

Figure 11

The simulated impedance (solid line: resistance, dashed line: reactance) for the spherical antenna structure shown in
Figure 9(b), with a 5 mm magnetic shell of n r = 100. The antenna is matched to 50 ohms with bandwidth performance
corresponding to a Q at 1.15 times the Chu lower bound.

3 . 3 El e c t r i c P e r m i t t i v i t y

In all of the cases discussed so far, the relative electric permittivity in the shell material has been assumed to be 1. Because
this is unlikely to occur in real materials, Figure 12 illustrates the effect of setting the electric permittivity of the magnetic
material to values greater than 1 for one particular structure, namely the spherical antenna with a 2.5 mm shell thickness.
The baseline structure in air achieves a Q just slightly above the Thal lower bound, and increasing the electric permittivity
of the shell increases the Q relative to this bound. A correspondingly higher value of n r is then required to reduce the Q to
values appreciably below this bound. Figure 12 makes clear the importance of minimizing the electric permittivity of the
magnetic material in order to obtain the optimal performance.

It is important to note that the simultaneous requirement of high relative permeability and low relative permittivity do
not apply along the same axes. For the antennas considered here, the high-n r is required only in the azimuthal direction,
whereas the low-e r is required only in the r- and z-directions. The use of anisotropic magnetic materials may enable some
mitigation of the effects seen in Figure 12.

Figure 12

The Q-factor versus shell permeability of the fundamental resonant mode normalized to the Thal lower bound for the
spherical capped monopole with a 2.5 mm shell. Increasing the relative electric permittivity er above 1 degrades the
performance.
11

4. SUMMARY
We have illustrated how thin shells of high magnetic permeability can be utilized to reduce the radiation Q of electrically
small, top-loaded, electric-dipole antennas. The Q’s of these antennas approach the recently derived general lower bounds
on Q for antennas with both electric and magnetic current excitations [14], and for the case of spherical antennas, the Q’s
approach the Chu lower bound. A relative permeability of n r = 100 would enable the realization of antennas with notably
improved performance, provided the electric permittivity could be kept relatively small. The practical challenge of realizing
these designs is in the development of low-loss materials with high magnetic permeability and relatively low electric
permittivity. Finally, we assumed in all simulations that no frequency dispersion is present in the magnetic material,
as there is no fundamental limitation preventing negligible dispersion for high permeability materials over the antenna
bandwidths. However, any frequency dispersion that may be present could reduce the achievable bandwidth.

12

5. REFERENCES
[1]
L. J. Chu, “Physical limitations on omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948.
[2]
R. F. Harrington, “Effect of antenna size on gain, bandwidth, and efficiency,” J. Res. Nat. Bureau Stand., vol. 64D, p. 112, Jan. 1960.
[3]
R. E. Collin and S. Rothschild, “Evaluation of antenna Q,” IEEE Trans. Antennas Propag., vol. 17, pp. 23–27, Jan. 1964.
[4]
R. L. Fante, “Quality factor of general ideal antennas,” IEEE Trans. Antennas Propag., vol. 17, pp. 151–155, Mar. 1969.
[5]
J. S. Mclean, “A re-examination of the fundamental limits on the radiation-Q of electrically small antennas,” IEEE Trans. Antennas Propag., vol.
44, pp. 672–676, May 1996.
[6]
A. D. Yaghjian, “Internal energy, Q-energy, Poyntings theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antennas Propag., vol.
55, pp. 1495–1505, Jun. 2007.
[7]
H. A. Wheeler, “The spherical coil as an inductor, shield, or antenna,” Proc. IRE, vol. 46, pp. 1595–1602, Sep. 1958.
[8]
O. S. Kim, O. Breinbjerg, and A. D. Yaghjian, “Electrically small magnetic dipole antennas with quality factors approaching the Chu lower
bound,” IEEE Trans. Antennas Propag., to be published.
[9]
H. L. Thal, “New radiation Q limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2757–2763, Oct. 2006.
[10]
S. R. Best, “The radiation properties of electrically small folded spherical helix antennas,” IEEE Trans. Antennas Propag., vol. 52, pp. 953–960,
Apr. 2004.
[11]
H. R. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antennas Propag.,
vol. 54, pp. 1644–1653, Jun. 2006.
[12]
H. R. Stuart and S. R. Best, “A small wideband multimode antenna,” presented at the IEEE Antennas Propag. Soc. Int. Symp., San Diego, CA, 2008.
[13]
H. R. Stuart, “Eigenmode analysis of small multielement spherical antennas,” IEEE Trans. Antennas Propag., vol. 56, pp. 2841–2851, Sep. 2008.
[14]
A. D. Yaghjian and H. R. Stuart, “Lower bounds on the Q of electrically small dipole antennas,” IEEE Trans. Antennas Propag., vol. 58, pp. 3114–
3121, Oct. 2010.
[15]
H. R. Stuart, “Eigenmode analysis of a two element segmented capped monopole antenna,” IEEE Trans. Antennas Propag., vol. 57, pp. 2980–2988,
Oct. 2009.
[16]
H. A. Wheeler, “A simple formula for the capacitance of a disc on dielectric on a plane,” IEEE Trans. Antennas Propag., vol. 30, pp. 2050–2054,
Nov. 1982.
[17]
H. A. Wheeler, “Fundamental limitations of small antennas,” Proc. IRE, vol. 35, pp. 1479–1484, Dec. 1947.
[18]
A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag.,vol. 53, pp. 1298–1324, Apr. 2005.
[19]
C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997.
[20]
H. A. Wheeler, “A helical antenna for circular polarization,” Proc. IRE, vol. 35, pp. 1484–1488, Dec. 1947.

[21]
J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941.

13

6. Biographies
Howard R. Stuart (M’98) received the S.B. and S.M. degrees in electrical engineering from the Massachusetts Institute of
Technology, Cambridge, in 1988 and 1990, respectively, and the Ph.D. degree in optics from the University of Rochester,
Rochester, NY, in 1998.

From 1990 to 1993, he worked as a Research Scientist for the Polaroid Corporation, Cambridge, MA. In 1998, he joined Bell
Laboratories, Lucent Technologies as a Member of Technical Staff in the Advanced Photonics Research Department,

Holmdel, NJ. Since 2003, he has worked in the Bell Labs Government Communications Laboratory, which became part of
LGS Innovations in 2007. He has published papers on a variety of research topics, including small resonant antennas, metal
nanoparticle enhanced photodetection, multimode optical fiber transmission, optical waveguide interactions and devices,
optical MEMS, and optical performance monitoring.

Dr. Stuart served as the Integrated Optics Topical Editor for the OSA journal Applied Optics from 2002–2008.

Arthur D. Yaghjian (S’68–M’69–SM’84–F’93–LF’09) received the B.S., M.S., and Ph.D. degrees in electrical engineering from
Brown University, Providence, RI, in 1964, 1966, and 1969.

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 materials and “metamaterials,” the development of exact, numerical, and high-frequency methods
for predicting and measuring the near and far fields of antennas and scatterers, the design of electrically small supergain
arrays, and the reformulation of the classical equations of motion of charged particles.

Dr. Yaghjian is a Life Fellow of the IEEE, has served as an Associate Editor for the IEEE and URSI, and is a member of Sigma
Xi. He has received best paper awards from the IEEE, NIST, and AFRL.

14
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