1. IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 1
Radiation Q Bounds for Small Electric Dipoles
Over a Conducting Ground Plane
Hsieh-Chi Chang, Yong Heui Cho, Member, IEEE, and Do-Hoon Kwon, Senior Member, IEEE
Abstract—The radiation quality factors of vertically and hor-
izontally polarized single-mode dipole antennas over a ground
plane are investigated and compared with their free-space coun-
terparts. The theoretical Q results are validated using simulation
and measurement results of small spherical helix dipole antennas.
For vertically polarized antennas, the bandwidth can be enhanced
approximately by a factor of two.
Index Terms—Antenna bandwidth, dipole antennas, electri-
cally small antennas, quality factor.
I. INTRODUCTION
ELECTRICALLY small antennas has been an important
research topic for many years, with the bandwidth per-
formance characterized by the radiation quality factor Q.
For small dipoles, use of the fundamental spherical mode
current can describe antenna characteristics accurately. Using
the fundamental TM01 spherical mode, the Chu bound [1]
QChu
fs =
1
ka
+
1
(ka)3
(1)
is the wave physics-imposed fundamental limit, where k =
2π/λ is the free-space wavenumber in terms of the wavelength
λ and a is the radius of the smallest circumscribing sphere. In
(1), the subscript ‘fs’ signifies that the Chu bound applies to
antennas in free space. Since the energy inside the sphere is ig-
nored in the Chu limit, practical antennas cannot achieve QChu
fs .
For air-core spherical antennas with an electric source over the
sphere surface, Thal derived a new bound [2] that includes
the internal energy and thus may be closely approached in
practice. An approximate expression was developed by Hansen
and Collin [3] and it is equal to
QThal
fs =
0.71327
ka
+
1.49589
(ka)3
. (2)
Radiation Q of spherical antennas with material cores excited
by electric or magnetic surface currents was reported in [4].
For small antennas of arbitrary shape, theoretical bandwidth
lower bounds have been developed based on quasi-electrostatic
and quasi-magnetostatic scattering properties of antenna vol-
umes [5]–[7].
Antennas having bandwidths that closely approach the the-
oretical bounds are being actively investigated. Best reported
This work was supported by the SI Organization, Inc.
H.-C. Chang and D.-H. Kwon are with Department of Electri-
cal and Computer Engineering, University of Massachusetts Amherst,
Amherst, Massachusetts 01003, USA (e-mail: hchang@ecs.umass.edu; dhk-
won@ecs.umass.edu).
Y. H. Cho is with the School of Information and Communication Engi-
neering, Mokwon University, Daejeon, 302-729, Korea (e-mail: yongheui-
cho@gmail.com).
spherical helix electric monopole/dipole [8], [9] and mag-
netic dipole [10] antennas with air core, where bandwidths
consistent with the Thal bounds for electric and magnetic
antennas were obtained. A low-Q spherical helix antenna
design with tunability was reported in [11]. In [12], [13], small
magnetic dipole antennas with air or material cores have been
studied and they were verified to have bandwidths close to the
theoretical limits. A capped monopole antenna design using
high permeability shells for reducing the internal stored energy
and thereby closely approaching QChu
fs was reported in [14].
Comprehensive reviews of fundamental limits and electrically
small antenna designs are available in [15]–[17].
These theoretical Q bounds and the associated antenna
designs are for antennas in free space. A conductor-backed
scenario is another practical antenna operating environment.
Antenna bandwidths can be significantly affected by a ground
plane and thus the associated Q bounds are expected to be
different from those for free space antennas. However, how a
small antenna’s Q bound changes with the introduction of a
ground plane has not been quantified thus far. In [18], Sten et
al. presented approximate closed-form expressions for the Q of
conductor-backed dipoles by considering a sphere in free space
that contains both the original and image antennas. Since the
energy internal to this large sphere was not included in the Q
computation, their bounds tend to be overly conservative and
thus may not be closely approached using practical antennas.
In this paper, fundamental Q bounds for small spherical
dipole antennas in vertical and horizontal polarizations over a
ground plane are investigated. Two quality factors — QChu
gnd and
QThal
gnd — are computed as a function of the antenna size and the
ground separation, following the definitions by Chu and Thal.
The theoretical Q results are validated using simulation and
measurement of spherical helix dipole antennas [9] placed over
a ground plane, where an approximate value of Q is obtained
using the frequency-swept driving point impedance [19]. For
horizontally polarized dipoles, Q increases significantly from
the free-space values as the ground separation is reduced, as
expected. For vertically polarized dipoles, it is found that Q
can decrease by approximately a factor of two, i.e. doubling
of the bandwidth, compared with the free-space antenna of
the same size. This anticipated bandwidth enhancement is
supported by antenna simulation and measurement.
In the following development, an ejωt
time convention is
assumed and omitted throughout.

2. 2 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
x
y
z
O
h
(a)
x
y
z
O
h
h−
(b)
J
aO ax
az
(c)
aO
J
ax
az
(d)
2a
Infinite PEC ground
ss
Fig. 1. Problem configuration of a spherical antenna over an infinite PEC
ground plane. (a) The problem geometry. (b) The free-space configuration
after applying image theory. (c) Electric surface current over the antenna
surface for vertical polarization given by Js = −ˆθaJ0 sin θa. (d) Surface
current for horizontal polarization given by Js = − ˆψaJ0 sin ψa.
II. METHOD OF ANALYSIS
A. Conductor-Backed Small Antennas
In Fig. 1(a), a spherical antenna over a perfect electric
conductor (PEC) ground plane is illustrated. The image theory
is applied and an image spherical antenna is placed below
the plane of the original ground, as illustrating in Fig. 1(b).
The radius of the spherical antenna and the sphere center-
to-ground separation are equal to a and h, characterized in
terms of wavelength via ka and kh, respectively. Referenced
to the antenna center Oa, a fundamental-mode electric surface
current flows over the original antenna surface. For a vertically
polarized antenna, Fig. 1(c) shows the electric surface current
given by Js = −ˆθaJ0 sin θa, where θa is the angle measured
from the +za axis and J0 is a constant. For a horizontally
polarized antenna, the current is equal to Js = − ˆψaJ0 sin ψa
in term of the angle ψ measured from the +xa axis, as
illustrated in Fig. 1(d).
In the presence of a ground plane of finite or infinite extent,
part or all of the ground plane is sometimes considered to be
part of the overall antenna. However, in this study, an antenna
specifically refers to the spherical radiating structure only,
without any part of the ground plane included. The energy
stored inside the antenna refers to the non-propagating energy
inside the original spherical antenna volume. It will be the
differentiating quantity in computing QChu
gnd and QThal
gnd .
In order to obtain a tight bound on Q, all non-propagating
energy stored in all space should be accurately evaluated and
accounted for. To this end, inclusion of the stored energy
inside the sphere that tightly encloses both the original and
image antennas is the key. For spherical antennas that excite
only the dominant spherical mode, M and N spherical vector
wave functions [20], [21] are the best suited for this purpose.
Whenever possible, volume energy densities and radiation
power densities as well as their volume and surface integrals
will be represented in terms of vector wave functions.
B. Field Expressions for a Single Spherical Antenna
Outside the source volume, the electric and magnetic fields
radiated by a current distribution are expanded in terms of the
M and N vector functions by
E =
∞
n=1
n
m=−n
AmnM(4)
mn + BmnN(4)
mn (3)
H =
j
η
∞
n=1
n
m=−n
AmnN(4)
mn + BmnM(4)
mn (4)
where η is the free-space intrinsic impedance. The expansion
coefficients Amn and Bmn can be obtained using an inner
product between the source and mode functions. For the
electric surface current under consideration, they are given by
Amn
Bmn
= −
k2
η
2λmn
M
(1)∗
mn
N
(1)∗
mn
· Jsds′
(5)
where
λmn =
2πn(n + 1)
2n + 1
(n + m)!
(n − m)!
. (6)
In (3)–(5), superscripts ‘(1)’ and ‘(4)’ refer to the radial
dependence of the first and fourth kinds [20], represented
by the spherical Bessel function jn(kr) and the spherical
Hankel function of the second kind h
(2)
n (kr), respectively.
The current over the original antenna surface is designed to
radiate the fundamental spherical mode, which is a reasonable
approximation for small dipole antennas. The fields outside
the sphere can be matched with those of a Hertzian dipole
located at the sphere center having a proper dipole moment.
The fields internal to the source distribution are obtained by
the same set of equations (3)–(5) after the superscripts ‘(1)’
and ‘(4)’ are interchanged.
The spherical mode of a ˆz-directed Hertzian dipole that
generates the same fields as the spherical antenna outside the
sphere in Fig. 1(c) is the TM01 mode. The modes of an ˆx-
directed Hertzian dipole that generate the same fields as the
current in Fig. 1(d) are a combination of TM11 and TM−11
modes.
C. Vector Addition Theorem
When a problem configuration involves multiple sources at
different locations, it is not easy to analyze them in mixed
coordinates. To compute the total fields from multiple sources,
we can represent the fields from each source with respect to
a common origin. To move the origin away from the sphere
center, a vector addition theorem [21] can be employed. It is
applied to the fields of both antennas in Fig. 1(b) to express the
total fields referenced at O, which is the mid-point between
the two sphere centers.

3. CHANG et al.: RADIATION Q BOUNDS FOR SMALL ELECTRIC DIPOLES OVER A CONDUCTING GROUND PLANE 3
x
Original
antenna
Image
antenna
aO
iO
O
z
1 2 3 4
5
5
a
Fig. 2. Division of the entire space into five different regions having volumes
Vi (i = 1, 2, . . . , 5).
D. Division of Space and Computation of Q
Since the equivalent two-sphere configuration in Fig. 1(b)
does not possess a spherical symmetry, it is not easy to
preform integrations involving the surface and volume of a
sphere having the center away from O. To circumvent this
difficulty, the entire space is split into five regions, as indicated
in Fig. 2.1
Since only non-propagating energy should be taken
into account when computing the radiation Q, different regions
of space need to be identified depending on the presence or
absence of energy associated with radiation.
At any point outside a spherical antenna surface, the surface
current distribution can be replaced by an equivalent Hertzian
dipole located at the sphere center. Hence, at any field point
outside the two antenna spheres in Fig. 2, the total fields can be
obtained by a vector sum of fields generated by two equivalent
Hertzian dipoles located at (x, y, z) = (0, 0, h) and (0, 0, −h).
This makes the spherical surface indicated by a black dashed
contour the boundary between two volumes having standing-
wave (inside) and propagating-wave (outside) characteristics.
The red dashed circles indicate region boundaries for using
different methods used for evaluating volume integrals. Vol-
ume integrations for energy computation can be obtained in a
closed form as an infinite series inside the small red dashed
circle (region 1), outside the large red dashed circle (region
4), and inside the two spheres (region 5). In the two remaining
regions (2 and 3), closed-form expressions cannot be obtained
for the stored energies and thus volume integrals need to be
evaluated numerically. The non-propagating stored electric and
1Stored energies in each region Wi
m and Wi
e (i = 1, 2, . . . , 5) as well
as the radiated power P in the following development will account only
for the z > 0 range. However, due to the relative simplicity of power and
energy evaluations over the entire angular range of 0 ≤ θ ≤ π, volume Vi
will include both the z > 0 portion and its image in the z < 0 range. A
multiplication of a resulting integral by 1/2 will give appropriate values for
the energies and power.
magnetic energies are
We =
1
2 V
wedv′
(7)
Wm =
1
2 V
wmdv′
(8)
where we and wm represent the electric and magnetic volume
energy densities. They are equal to
we =
ǫ
4
|E|
2
− wr
e (9)
wm =
µ
4
|H|
2
− wr
m (10)
where wr
e and wr
m are the electric and magnetic volume energy
densities associated with radiation, respectively. In (7)–(8), the
integration volume V is all space and the factor of 1/2 is due
to the validity of the free-space configuration being limited to
z > 0.
For the stored energy in region 5, we choose the center of
the original antenna sphere as the common origin and use the
vector addition theorem to express the total fields due to both
antennas. A volume integration over the original volume can
be performed to find the energy stored inside the spherical
antennas, which results in a covering infinite series.
Finally, the radiated power of the original antenna above the
ground plane is
P =
1
2
Re
r=h+a
1
2
E × H∗
· ds′
(11)
where again the extra factor 1/2 is due to the limited validity
of the two-sphere system. Any spherical surface of radius r >
a + h can be used for closed-form evaluation of (11).
At this point, all needed quantities for Q evaluation are
computed. We can substitute all quantities into the definition
of the quality factor
Q =
2ω max {We, Wm}
P
. (12)
In (12), omission or inclusion of the stored energies in region
5 leads to QChu
gnd or QThal
gnd as appropriate.
E. Non-Propagating Stored Energies and Radiated Power
1) Region 1: The radial function is the spherical Bessel
function. The electric and magnetic fields in this region
represent standing waves, so there is no energy associated with
radiation to the far zone. Therefore, both wr
e and wr
m are equal
to zero. Expressions for non-propagating, stored electric and
magnetic energies are given in Appendix A for both vertically
and horizontally polarized spherical antennas.
2) Region 2: Here, the integration volume does not have
boundaries that conform to constant-coordinate surfaces, re-
quiring numerical integration. The fields are expressed by a
superposition of contributions made by the two equivalent
Hertzian dipoles, referenced at Oa and Oi as illustrated in
Fig. 3. The coordinate transformations between (ra, θa, φa)
referenced at Oa and (r, θ, φ) referenced at O are given by
ra = r2 − 2rh cos θ + h2 (13)
θa = cos−1 r cos θ − h
ra
. (14)

4. 4 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
, ,a iz z z
aO
O
iO
ax
ix
x
aθ
iθ
θ
ar
ir
r
P
Fig. 3. Coordinate transformation between three systems having origins at
Oa, Oi, and O.
Similarly, the transformations between the two coordinate
systems centered at Oi and O are
ri = r2 + 2rh cos θ + h2 (15)
θi = cos−1 r cos θ + h
ri
. (16)
The electric and magnetic fields can be represented by
E(r, θ, φ) = Ea(ra, θa, φa) + Ei(ri, θi, φi) (17)
H(r, θ, φ) = Ha(ra, θa, φa) + Hi(ri, θi, φi). (18)
The fields in this region are standing waves. Hence, both
wr
e and wr
m are equal to zero. The non-propagating, stored
energies can be computed numerically using integrations (7)–
(8) over the region volume.
3) Region 3: The fields in this region represent propagating
waves, necessitating subtraction of the radiation energies from
the total energies to obtain the non-propagating, stored electric
and magnetic energies. Following the approach by Collin and
Rothschild [22], the difference between stored magnetic and
electric energies is found from the complex Poynting theorem
[23] as
W3
m − W3
e =
1
2
Im
1
2ω ∂V3
1
2
E × H∗
· ds′
(19)
where the surface integral is over the bounding surface ∂V3
of the volume V3. The total energy density associated with
radiation is equal to the real part of the radial component of
the complex Poynting vector divided by the speed of energy
flow, which is the speed of the light. Hence, the sum of electric
and magnetic volume energy densities is
wr
e + wr
m = ˆr ·
√
µǫRe
1
2
E × H∗
. (20)
The sum of the stored energies can be obtained by first
subtracting (20) from the total energy density and performing
a volume integration over V3, i.e.
W3
m + W3
e =
1
2 V3
ǫ
4
|E|
2
+
µ
4
|H|
2
− (wr
m + wr
e) dv′
.
(21)
From (19) and (21), non-propagating electric and magnetic
energies in region 3 are found.
aO
iO
Original antenna
Image antenna
Fig. 4. Transformation of the coordinate system from Oi to Oa.
4) Region 4: This region extends out to infinity and the
fields represent propagating waves. Hence, the energies asso-
ciated with radiation should be subtracted from total energies.
The same approach as for region 3 can be applied, as was also
done by Fante [24]. Closed-form expressions for W4
e + W4
m
and W4
m − W4
e in an infinite series format lead to individual
stored energies.
5) Region 5: As illustrated in Fig. 4, the fields generated
by the image antenna can be expanded around the center Oa
of the original antenna. The total fields are then represented
by a superposition of spherical vector waves referenced at Oa.
The fields in this region are standing waves, so wr
e and wr
m
are both equal to zero. Expressions for the fields and energies
for vertically and horizontally polarized antennas summarized
in Appendix B.
6) Radiated Power: The total fields are available in region
4, expanded in spherical wave functions referenced at O.
Owing to orthogonality between different spherical modes, the
total radiated power is the sum of radiated powers of individual
modes. The radiated power of a vertically polarized conductor-
backed spherical antenna for the radial mode index v is written
as
Pv =
1
2
4π
2v + 1
|αout
E |
2
2k2η
v(v + 1) A01
0v
2
(22)
where αout
E is defined in (29). The radiated power of a
horizontally polarized conductor-backed spherical antenna for
mode v is
Pv =
1
2
4π
2v + 1
|αout
E |
2
2k2η
1
2
A11
1v
2
+
1
2
B11
1v
2
[v(v + 1)]
2
+ A−11
−1v
2
+ B−11
−1v
2
. (23)
Then, the total radiated power is
P =
∞
v=1
Pv. (24)
III. THEORETICAL RESULTS
Stored energies, radiated power, and radiation Q are ana-
lyzed for small antenna dimensions (ka ≤ 0.5) over a ground

5. CHANG et al.: RADIATION Q BOUNDS FOR SMALL ELECTRIC DIPOLES OVER A CONDUCTING GROUND PLANE 5
0.1 0.2 0.3 0.4 0.5
10
0
10
1
10
2
10
3
10
4
ka (kh=0.5252)
Q
Qfs
Thal
Qfs
Chu
Qgnd
Thal
Qgnd
Chu
[18]
Ant. sim.
0.1 0.2 0.3 0.4 0.5
10
0
10
1
10
2
10
3
10
4
ka (kh=2ka)
Q
Qfs
Thal
Qfs
Chu
Qgnd
Thal
Qgnd
Chu
[18]
Ant.sim.
(a)
(b)
Fig. 5. Radiation Q for vertically polarized antennas over a ground plane
with respect to ka, together with those of the same-size antennas in free
space. (a) For a fixed ground separation kh = 0.5252. (b) For a fixed ratio
of ground separation to antenna size kh = 2ka. In both cases, the quality
factor from [18] is also shown for comparison.
separation up to one wavelength (kh ≤ 2π). For both antenna
polarizations, it was found that the electric energy dominates
for all combinations of ka and kh considered.
A. Vertically Polarized Spherical Antenna
Fig. 5 compares five different Q values over 0.1 ≤ ka ≤ 0.5
for a vertically polarized spherical antenna. In Fig. 5(a), the
ground separation is fixed (kh = 0.5252), which makes
the horizontal axis correspond to antenna size at a fixed
height over the ground at a given frequency. In Fig. 5(b),
the ratio between the antenna size and ground separation
is fixed (kh = 2ka). In this case, the horizontal axis is
proportional to frequency for fixed physical antenna size and
ground separation. In each case, a Q curve based on the result
from [18, Eq. (25)]2
is displayed as a reference, which is
overly conservative (too low). Too low a minimum Q from
[18] results from the omission of stored energies inside the
imaginary sphere containing both the antenna and its image
(the outer boundary of region 3 in Fig. 2 in our configuration),
where high energy densities are expected. Hence, the minimum
Q in [18] cannot be approached using practical antennas. In
2Note that the length a in [18] is the radius of the sphere that encloses
both the original and image antennas. It corresponds to h + a in this work.
0 1 2 3 4 5 6
0.4
0.6
0.8
1
1.2
1.4
kh
Q
gnd
Chu
/Q
fs
Chu
ka=0.5
0.4
0.3
0.2626
0.2
0.1
0 1 2 3 4 5 6
0.4
0.6
0.8
1
1.2
1.4
kh
Q
gnd
Thal
/Q
fs
Thal
Ant. sim.
(a)
(b)
Fig. 6. The Q lower bounds for vertical dipole antennas over a ground plane
normalized to the bounds in free space. (a) QChu
gnd /QChu
fs for the Chu bound.
(b) QThal
gnd /QThal
fs for the Thal bound.
contrast, by including all (for QThal
gnd ) or part (for QChu
gnd ) of the
energies inside the same sphere, the Q values in this study are
tighter bounds that can be closely approached in practice.
Curves for QChu
fs and QThal
fs are identical in Figs. 5(a)–5(b).
For a vertically polarized antenna, it is observed that QChu
gnd
and QThal
gnd are both significantly lower than their free-space
counterparts. In other words, placing a spherical dipole of
vertical polarization over a PEC ground plane can decrease the
quality factor. This implies that the bandwidth will become
broader than that of a same-sized antenna in free space. It
is interesting to note that QThal
gnd < QChu
fs ; the effect of a
ground plane is significant enough to make the realizable
Q of a conductor-backed dipole lower than the fundamental
limit for a free-space antenna of the same size. Fig. 5 and
several following figures also have circled data points marked
“antenna simulation.” They are Q values for real antenna
designs associated with a given set of ka and kh values, which
will be discussed later in Section IV.
For several values of ka as a parameter, Fig. 6 presents
the Q bounds for vertically polarized dipoles over a ground
plane as a function of kh. The Chu bound QChu
gnd and the Thal
bound QThal
gnd are normalized to their free-space bounds (1)
and (2), respectively. Only the fundamental spherical spherical
mode current is assumed over the antenna surface in deriving
these two bounds. Since excitation of any high-order mode
contributes only to increasing Q, QChu
gnd in Fig. 6(a) represents

6. 6 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
0 1 2 3 4 5 6
0.8
1
1.2
1.4
1.6
1.8
2
kh
ka=0.5
0.4
0.3
0.2626
0.2
0.1
gnd
Thal/Wfs
Thal
P
W
Thal/Pfs
Thal
gnd
Fig. 7. The radiated power and stored energy for vertical polarization with
respect to kh normalized to their free-space values.
the fundamental bound that no Q of any passive physical
antenna can fall below. The Thal bound QThal
gnd in Fig. 6(b)
is the fundamental bound for spherical electrical dipoles with
air core excited by electric sources. It is expected that the
two ratios in Fig. 6 approach unity for kh ≫ ka as the
antenna bandwidth will not be significantly affected by a
ground plane at a large electrical distance. This is observed
for the two small antenna sizes ka = 0.1 and 0.2626 in the
range of kh considered. There is a wide range of kh on the
lower end where QChu
gnd /QChu
fs < 1, QThal
gnd /QThal
fs < 1, and they
monotonically decrease with respect to decreasing kh. They
reach values 0.5 or lower for the two small antenna sizes,
which correspond to bandwidth enhancement by a factor of
two or larger.
In order to understand the reduction of Q better, Fig. 7
shows ratios for the stored energy WThal
gnd /WThal
fs and the
radiated power PThal
gnd /PThal
fs that combine to describe the Q
behavior. Here, we focus on the quantities for the Thal bound
because it is QThal
gnd that will be tested and validated in Section
IV. It can be observed that WThal
gnd is not significantly different
from WThal
fs . The radiated power is a stronger function of kh
and PThal
gnd /PThal
fs → 2 as kh → 0 because the fields from the
original and image antennas add constructively in all directions
in the upper hemisphere in the far zone. This increase in the
radiate power is the primary reason for the decrease in Q for
small ground separations relative to the free-space case.
B. Horizontally Polarized Spherical Antenna
Fig. 8 compares different quality factors for a conductor-
backed horizontally polarized spherical dipole antenna. The
ground separation is fixed at kh = 0.5252 in Fig. 8(a) and the
ground separation-to-antenna size ratio is fixed at kh = 2ka
in Fig. 8(b). Both QChu
gnd and QThal
gnd are significantly higher than
QChu
fs and QThal
fs . The Q results from [18] show significantly low
bounds in this horizontal polarization case as in the vertical
polarization case. For a fixed ground separation in Fig. 8(a),
Q increases by approximately a constant factor with the intro-
duction of a ground plane for all ka considered. For a fixed
ratio kh/ka in Fig. 8(b), it is noted that QChu
gnd , QThal
gnd approach
QChu
fs , QThal
fs as the ground separation is increased. Hence, it
0.1 0.2 0.3 0.4 0.5
10
0
10
1
10
2
10
3
10
4
ka (kh=0.5252)
Q
Q
gnd
Thal
Q
gnd
Chu
Q
fs
Thal
Q
fs
Chu
[18]
Ant. sim.
0.1 0.2 0.3 0.4 0.5
10
0
10
1
10
2
10
3
10
4
ka (kh=2ka)
Q
Q
gnd
Thal
Q
gnd
Chu
Q
fs
Thal
Q
fs
Chu
[18]
Ant. sim.
(a)
(b)
Fig. 8. Radiation Q factors for horizontally polarized antennas over a ground
plane with respect to ka, together with those of the same-sized antennas in
free space. (a) For fixed ground separation kh = 0.5252. (b) For a fixed ratio
of ground separation to antenna size kh = 2ka. In both cases, the quality
factor from [18] is also shown for comparison.
is anticipated that Q of a conductor-backed horizontal dipole
may become lower than that of a free-space dipole for some
kh.
Fig. 9 presents the new bounds QChu
gnd and QThal
gnd for horizon-
tally polarized dipoles over the ground plane as a function of
kh for several electrically small values of ka. Similarly to the
vertical polarization case, they represent fundamental bounds
for an arbitrary dipole antenna in horizontal polarization that is
contained in a sphere of radius a and for a spherical antenna
of radius a with air core excited by an electric source. Q
increases dramatically as kh → 0, but there exists a broad
range of ground separation around kh = 2 in which Q is lower
than the free-space value for both Chu and Thal bounds. A
horizontal dipole that is quarter-wavelength above the ground
plane belongs in this range. For small ka and large kh, both
Q’s are not noticeably affected by the presence of the ground
plane as expected.
From Fig. 10, it is seen that WThal
gnd /WThal
fs is a slowly
varying function of kh, especially for small ka. In contrast,
PThal
gnd /PThal
fs is more sensitive to the ground separation and
it approaches zero as kh → 0. This is the reason for rapidly
increasing Q as kh is reduced. Since the image antenna points
in the opposite direction from the original antenna, the dipole
mode is suppressed and the quadrupole mode becomes the

7. CHANG et al.: RADIATION Q BOUNDS FOR SMALL ELECTRIC DIPOLES OVER A CONDUCTING GROUND PLANE 7
0 1 2 3 4 5 6
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
kh
Q
gnd
Chu
/Q
fs
Chu
ka=0.5
0.4
0.3
0.2626
0.2
0.1
0 1 2 3 4 5 6
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
kh
Q
gnd
Thal
/Q
fs
Thal
Ant. sim.
(a)
(b)
Fig. 9. The Q lower bounds for horizontal dipole antennas over a ground
plane normalized to the bounds in free space. (a) QChu
gnd /QChu
fs for the Chu
bound. (b) QThal
gnd /QThal
fs for the Thal bound.
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
1.2
kh
ka=0.5
0.4
0.3
0.2626
0.2
0.1
gnd
Thal/Wfs
Thal
P
W
Thal/Pfs
Thal
gnd
Fig. 10. The radiated power and stored energy for horizontal polarization
with respect to kh normalized to their free-space values.
dominant radiation mechanism with an associated higher mode
Q. A surge of PThal
gnd /PThal
fs above unity around kh = 2 and the
associated range of kh for PThal
gnd /PThal
fs > WThal
gnd /WThal
fs define
the range of ground separation that allows a wider bandwidth
for a conductor-backed antenna.
IV. ANTENNA SIMULATION AND MEASUREMENT
To validate the Q results for conductor-backed dipoles in
Section III, several folded spherical helix wire dipoles based
TABLE I
RESONANT VERTICALLY POLARIZED ANTENNA PROPERTIES OVER A PEC
GROUND PLANE FOR DIFFERENT GROUND SEPARATIONS
No. of Armsa No. of Turnsa kh Q
imp
gnd QThal
gnd
3 1.59 0.5252 47.76 46.21
4 1.64 π/2 68.07 65.77
4 1.635 π 95.99 92.49
4 1.633 4.5 86.66 82.58
4 1.63 2π 88.77 88.56
a The exact geometrical parameters of the spherical helix antennas
in this study are given by [8, Eqs. (1)–(5)] using these parameters
together with the radius of the sphere.
TABLE II
RESONANT HORIZONTALLY POLARIZED ANTENNA PROPERTIES OVER A
PEC GROUND PLANE FOR DIFFERENT GROUND SEPARATIONS
No. of Arms No. of turns kh Q
imp
gnd QThal
gnd
8 1.634 0.5252 399.77 371.64
4 1.631 π/2 77.11 72.05
4 1.675 π 92.94 88.37
4 1.637 2π 90.55 89.57
on [8], [9] were designed for matching to a 50-Ω transmis-
sion line. Full-wave simulation results were obtained for the
frequency-swept input impedance using a method-of-moments
based tool FEKO version 6.1 from EMSS. An antenna Q
was computed from the input impedance [19, Eq. (96)] at the
design frequency and it is denoted by Qimp
gnd.
An electrically small antenna dimension ka = 0.2626 was
chosen at the design frequency of 300 MHz (antenna diameter
2a = 84 mm) and several ground separations ranging from
kh = 0.5252 to 2π were selected. For a given polarization
and ground separation, the number of helical arms and the
number of turns were optimized to obtain an input resistance at
resonance close to 50 Ω. The reference antenna for comparison
is the same-sized spherical dipole having four helical arms of
1.588 turns each from [9], which is designed for free space.
The associated reference Q values are found to be Qimp
fs =
90.17 from the simulated input impedance and QThal
fs = 85.32
from (2).
Tables I and II list the design parameters and Q values
for each case. For all antenna designs, a good agreement was
obtained between QThal
gnd and Qimp
gnd and this validates the new
Q values for conductor-backed small electric dipole antennas.
For kh = 2ka = 0.5252, the values of Qimp
gnd are indicated by
circles in Figs. 5 and 8, which shows an excellent agreement
with QThal
gnd for both polarizations. Data points from Tables
I and II indicated by circles in Figs. 6(b) and 9(b) closely
follow the curves associated with ka = 0.2626 for all values
of kh considered. In both figures, the data points from antenna
simulations represent normalized values to QThal
fs .
Among the designs reported above, one vertically polarized
and one horizontally polarized spherical antennas were fabri-
cated and their input reflection coefficients were measured.
Copper wire of 2.6-mm diameter was bent to form each
helical arm. For vertical polarization, the kh = 0.5252 variant
(1st entry in Table I) was chosen. An aluminum plate of
0.9 m × 0.9 m was used as the ground plane after confirming
by simulation that the ground size does not significantly affect
the bandwidth and resonance frequency compared with the

8. 8 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
250 260 270 280 290 300 310 320 330 340 350
−20
−15
−10
−5
0
Frequency (MHz)
|S
11
|(dB)
Measurement
Simulation
Sim: ref. ant.
250 260 270 280 290 300 310 320 330 340 350
−20
−15
−10
−5
0
Frequency (MHz)
|S
11
|(dB)
Measurement
Simulation
Sim: ref. ant.
(b)
(a)
Mirror
Ground
spacer
Foam
Ground
Fig. 11. Reflection coefficients of spherical antennas over a ground plane.
(a) Vertically polarized dipole corresponding to the 1st entry in Table I. (b)
Monopole version of the horizontally polarized dipole corresponding to the
2nd entry in Table II. For S11 associated with the monopole measurement, a
reference impedance of 25 Ω was used.
infinite ground case. In order to avoid the adverse effect
of cable leakage current, an indirect impedance measure-
ment technique without direct antenna excitation [25] was
employed. Fig. 11(a) compares the simulated and measured
input reflection coefficient with that of the free-space reference
dipole. The indirect measurement approach requires high-SNR
signals for accurate results and reliable result is obtained
around the resonance frequency. Simulated and measured
responses show an excellent agreement around the design
frequency and the bandwidth is distinctly broader than that
of the reference antenna, validating the prediction by the Q
theory [Fig. 6(b)]. For horizontal polarization, the kh = π/2
version (2nd entry in Table II) was selected. In this case,
symmetry in geometry, excitation, and polarization allows a
monopole configuration using a conducting electric mirror in
the symmetry plane of the antenna. Hence, a monopole version
was fabricated. It was mounted on a 0.9 m × 0.9 m mirror
(positioned horizontally) and then placed on a 1.2 m × 1.2 m
ground plane (positioned vertically), both of aluminum plates.
The measured reflection coefficient is compared in Fig. 11(b)
with the simulation result together with the reference free-
space case. The measured and simulated responses show an
excellent agreement and the bandwidth is similar to that of the
free-space antenna of the same size, as predicted by QThal
gnd in
Fig. 9(b).
V. CONCLUSION
Fundamental Chu radiation Q lower bounds for conductor-
backed small electric dipole antennas have been established
for vertical and horizontal polarizations. Assuming a funda-
mental spherical mode radiation from a dipole over a ground
plane, the fundamental Q bound was computed from the
radiated power and the stored energies outside the antenna-
circumscribing spheres in a free-space environment after the
application of image theory. Due to portions of volume and
surface integration bounds that do not conform to constant-
coordinate surfaces and contours in spherical coordinates, the
final results for Q are numerical in nature with the antenna size
and ground separation as parameters. In applicable volumes,
spherical vector wave functions and vector addition theorems
were used to find the stored energies in a closed-form series
format. Appropriate for air-core spherical antennas excited by
an electric source over the sphere surface, Thal bounds on
radiation Q were found for the same conductor-backed dipoles
by including the stored energies within the spherical antenna
volume. As were confirmed using antenna simulation and
measurement, they may be closely approached using simple,
practical antennas unlike the Chu bounds.
For vertically polarized small dipoles over a ground plane,
it was found that the Q may decrease by a factor around
0.5 for small ground separations, signifying the possibility
of doubling the bandwidth compared with an antenna of the
same size in free space. For horizontally polarized dipoles, the
radiation Q dramatically increases as the ground separation is
reduced, which is a qualitatively well-known phenomenon. As
the ground separation is increased, there is a broad range of
antenna height over the ground plane where a wider bandwidth
is expected than in free space.
Using simulation and measurement of of folded spherical
helix dipole antennas of 84-mm diameter for 300 MHz over
a ground plane, the newly established Thal bounds for air-
core spherical dipole antennas were validated. The Q values
computed from frequency-swept input impedance showed an
excellent agreement with the predicted Thal bounds.
APPENDIX A
NON-PROPAGATING, STORED ENERGIES IN REGION 1
The expansion coefficients Amn
uv and Bmn
uv are
Amn
uv = Amn
uv,a + Amn
uv,i (25)
Bmn
uv = Bmn
uv,a + Bmn
uv,i (26)
where Amn
uv,a and Bmn
uv,a are the expansion coefficients due
to the original antenna; Amn
uv,i and Bmn
uv,i are the expansion
coefficients due to the image antenna. All coefficients are
referenced to O. When referenced individually to Oa and
Oi for the original and image antennas, only one (vertical
polarization) or two (horizontal polarization) spherical modes
are excited. Translation of the origin for the associated M and
N functions from Oa and Oi to O results in an infinite number
of modes.

9. CHANG et al.: RADIATION Q BOUNDS FOR SMALL ELECTRIC DIPOLES OVER A CONDUCTING GROUND PLANE 9
Specifically, Amn
uv,a and Bmn
uv,a are given by Cruzan [21,
Fig. 1 and Eq. (26)] using a ← h and θ0 ← π. Similarly,
Amn
uv,i and Bmn
uv,i are obtained using a ← h and θ0 ← 0 in the
same equation.
A. Vertically Polarized Spherical Antenna
The field expressions in region 1 are
E = αout
E
∞
v=1
A01
0vN
(1)
0v (r, θ, φ) (27)
H = αout
H
∞
v=1
A01
0vM
(1)
0v (r, θ, φ) (28)
where
αout
E = jηJ0
ka
3
[2j0(ka) − j2(ka)] (29)
αout
H =
j
η
αout
E . (30)
The non-propagating, electric energy is
W1
e =
1
2
∞
v=1
ǫ
4
αout
E A01
0v
2
V1
N
(1)
0v
2
dv′
. (31)
The non-propagating, magnetic energy is
W1
m =
1
2
∞
v=1
µ
4
αout
H A01
0v
2
V1
M
(1)
0v
2
dv′
. (32)
B. Horizontally Polarized Spherical Antenna
The field expressions in region 1 are
E = αout
E
∞
v=1
1
2
A11
1vN
(1)
1v (r, θ, φ) + B11
1vM
(1)
1v (r, θ, φ)
− A−11
−1vN
(1)
−1v(r, θ, φ) − B−11
−1vM
(1)
−1v(r, θ, φ) (33)
H = αout
H
∞
v=1
1
2
A11
1vM
(1)
1v (r, θ, φ) + B11
1vN
(1)
1v (r, θ, φ)
− A−11
−1vM
(1)
−1v(r, θ, φ) − B−11
−1vN
(1)
−1v (r, θ, φ) . (34)
The non-propagating, electric and magnetic energies are given
by (35) and (36), respectively, shown on the following page.
APPENDIX B
NON-PROPAGATING, STORED ENERGIES IN REGION 5
Here, the expansion coefficients are referenced to Oa. The
coefficients Amn
uv,i and Bmn
uv,i are given by Cruzan [21, Fig. 1
and Eq. (26)] using a ← 2h and θ0 ← 0.
A. Vertically Polarized Spherical Antenna
The field expressions in region 5 are
E =
∞
v=1
αout
E A01
0v,i + δ1vαin
E N
(1)
0v (ra, θa, φa) (37)
H =
∞
v=1
αout
H A01
0v,i + δ1vαin
H M
(1)
0v (ra, θa, φa) (38)
where αin
E and αin
H are the same coefficients with αout
E and αout
H
in (29)–(30) with the radial function changed to Hankel func-
tion of the second kind. The symbol δ1v refers to Kronecker
delta.
The non-propagating, stored electric energy is
W5
e =
1
2
∞
v=1
ǫ
4
αout
E A01
0v,i + δ1vαin
E
2
V5
N
(1)
0v
2
dv′
. (39)
The non-propagating, stored magnetic energy is
W5
m =
1
2
∞
v=1
ǫ
4
αout
H A01
0v,i + δ1vαin
H
2
V5
M
(1)
0v
2
dv′
. (40)
B. Horizontally Polarized Spherical Antenna
The field expressions in region 5 are
E =
∞
v=1
1
2
αout
E A11
1v,i − αin
Eδ1v N
(1)
1v + B11
1v,iM
(1)
1v
− αout
E A−11
−1v,i − αin
Eδ1v N
(1)
−1v + B−11
−1v,iM
(1)
−1v (41)
H =
∞
v=1
1
2
αout
H A11
1v,i − αin
Hδ1v M
(1)
1v + B11
1v,iN
(1)
1v
− αout
H A−11
−1v,i − αin
Hδ1v M
(1)
−1v + B−11
−1v,iN
(1)
−1v . (42)
The non-propagating, stored, electric energy is
W5
e =
1
2 V5
ǫ
4
|E|
2
dv′
(43)
and the non-propagating, stored, magnetic energy is
W5
m =
1
2 V5
µ
4
|H|
2
dv′
. (44)
ACKNOWLEDGMENT
The authors would like to thank Yutong Yang for fabrication
and measurement of the antennas.
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W1
e =
1
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ǫ
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E
2
V1
1
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1v
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N
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