Simple Formulas for Near-Field Transmission, Gain, and Fields

Abstract

We review previous work on modifying the far-field transmission formula to describe near-field links between electrically-small antennas. Then, we derive a simple and useful formula for the antenna gain (G) of a small loop antenna as a function of size (R), wave number (k) and loaded quality factor (QL), specifically: G = C QL (kR)^3 where C is a constant dependent upon shape factors, loading, and the electric or magnetic nature of the antenna. Finally, conventional formulas for the near fields describe dipole sources in terms of dipole moments defined by currents and charges -quantities that are difficult to determine in practical applications. We demonstrate how to define the magnetic dipole moment in terms of the Effective Isotropic Radiated Power (EIRP).

Full text

Simple Formulas for
Near-Field Transmission, Gain, and Fields
Hans G. Schantz and Alfred Hans Unden
Q-Track Corporation
2223 Drake Avenue SW 1st Floor
Huntsville, AL 35805
M. A. Nikravan and Do-Hoon Kwon
Department of Electrical and Computer Engineering
University of Massachusetts Amherst
Amherst, MA 01003
Abstract: We review previous work on modifying the far-field transmission formula
to describe near-field links between electrically-small antennas. Then, we derive a
simple and useful formula for the antenna gain (G) of a small loop antenna as a
function of size (R), wave number (k) and loaded quality factor (QL), specifically:
G = C QL (kR)3 where C is a constant dependent upon shape factors, loading, and the
electric or magnetic nature of the antenna. Finally, conventional formulas for the
near fields describe dipole sources in terms of dipole moments defined by currents
and charges - quantities that are difficult to determine in practical applications. We
demonstrate how to define the magnetic dipole moment in terms of the Effective
Isotropic Radiated Power (EIRP).
1 Introduction
Most conventional wireless systems today operate on far-field principles, with many
wavelengths separating each end of the link. However, the earliest commercial wireless
systems from the 1880s and 1890s operated on near-field principles, employing magnetic
[1] or electric [2] near-field coupling to enable communication between a moving train
and infrastructure along the track [3]. The limited commercial success of these systems
was soon overshadowed by the much greater opportunities inherent in long-distance, far-
field wireless [4].
These early far-field wireless systems operated with low frequencies and long
wavelengths, so many of the earliest antennas were “electrically small” [5]. “Electrically
small” antennas are those of a size less than the radiansphere radius of R =

/2

or
equivalently kR = 1 where k = 2

/

is the wave number, and R is the radius of the sphere
bounding the antenna. The physics underlying these electrically small antennas was
imperfectly understood at first. As late as the 1930s, the leading textbook, Radio
Engineering, could justly observe “An understanding of the mechanism by which energy

is radiated from a circuit and the derivation of equations for expressing this radiation
involve conceptions which are unfamiliar to the ordinary engineer” [6]. In the 1930s,
however, Maxwell’s equations began to enter engineering practice with profound
implications for RF engineering in general [7]. But, Maxwell’s equations began to be
more broadly understood and applied at precisely the time when RF electronics had
advanced to the practical implementation of VHF, UHF, and microwave systems. Thus,
theoretical attention concentrated on these more useful and important aspects of RF
engineering, and many implications of electrically-small antennas and near-field links
were ignored. Even today in the realm of electrically-small antennas, researchers expend
considerable effort optimizing antennas that just barely qualify as electrically small, i.e.
kR  1.
High frequencies and short wavelengths support considerable bandwidth. This makes
them ideal for high-data-rate communications. For many applications, however, other
considerations take precedence. If one desires a short-range wireless link, robust with
respect to blockage and multipath, and if one has modest requirements with respect to
data rates, low-frequency links are highly attractive. Many technologies are emerging to
exploit the fundamental physics of near-field wireless, including [8]:
Near-Field Communications (NFC) offering 106 kbit/s to 424 kbit/s while
operating at 13.56MHz [ISO/IEC 18000-3] [9],
Near-Field Electromagnetic Ranging (NFER) offering 30cm-1m indoor
location accuracy at ~1MHz [10],
Radio-Frequency Identification (RFID) encompassing a wide variety of
techniques including 10cm range detection at 120-150kHz and 10cm-1m
detection at 13.56MHz [11], and
RuBee offering a 1200 baud data link at 131kHz [IEEE 1902.1] [12].
These applications require electrically-small antennas with kR as small as 10-4 to 10-6 or
smaller. In addition, these applications require operation at distances less than a
wavelength, often well within near-field range.
We review previous work on how the far-field transmission formula must be modified to
describe near-field links between electrically-small antennas. Then, this paper surveys
experimental measurements of electrically-small antenna gain and derives a useful
formula for antenna gain as a function of size and quality factor. Conventional formulas
for the near fields describe dipole sources in terms of dipole moments defined by currents
and charges –– quantities that are difficult to determine in practical applications. We
demonstrate how to define dipole moments in terms of the Effective Isotropic Radiated
Power (EIRP) of the magnetic dipole moment. Finally, this paper presents a variety of
numerical simulations to validate link, gain, and field relations.


2 Links
This section discusses the theory of ideal near-field link behavior as well as the
performance of near-field links in practical settings.
2.1 Theory of Near-Field Links
Harald Friis presented his “simple” transmission formula in 1947 [13]. The ratio of
received power (PRX) to transmitted power (PTX) in a far-field RF link is:

22
2
2222
2
22
2
4
4
1616
r
AA
r
AG
kr
GG
fr
cGG
r
GG
P
P
RXTX
RXTX
RXTXRXTXRXTX
TX
RX







(1)
where GTX is the transmit antenna gain, GRX is the receive antenna gain,

is the
wavelength, f is the frequency, c =

f is the speed of light, r is the range between
antennas, and the wave number is: k = 2/

. Although modern practice usually writes this
formula in terms of gain, Friis preferred the formulation of his link law in terms of
“aperture.” Aperture (A), the effective capture area of a typical (i.e. not “supergain”)
antenna, is related to gain as follows:


4
2G
A. (2)
In fact, a more intuitive approach to Friis’ link law would express it in terms of transmit
antenna gain and receive antenna aperture. For instance, in the case of ultra-wideband
(UWB) links, an antenna with constant gain with respect to frequency transmits a faithful
copy of an incident voltage waveform while a constant aperture antenna receives a
faithful copy of an incident electromagnetic field waveform [14]. This follows from an
asymmetry in the principle of reciprocity – the transmitting transfer function of an
antenna is proportional to the time derivative of the receiving transfer function [15]. In
the frequency domain, this amounts to a ninety degree phase delay in the transmission of
a signal. This in turn is a result of a basic principle of antenna physics – the storage of
electromagnetic energy in the near-field of a small transmit antenna for about a quarter
period (in the time harmonic case) before the energy decouples and radiates away [16].
Further implications and problems inherent to the transmit-gain-receive-gain formulation
of the link law will be more evident shortly.
In the near-field, the link formula diverges depending on whether the antennas at either
end of the link are electric (like whips or dipoles) or magnetic (like loops or loopsticks)
[17]. One relation describes links with “like” antennas, i.e. both electric or both magnetic
as in Fig. 1a and 1b. The other describes links with “unlike” antennas as in Fig. 1c. In the
transmit-gain-receive-gain formulation, these near-field link laws become:

R1
R2
r
R1
z
y
x
z
y
x
x
y
z
r
x
y
z
r
x
y
z
z
y
x
R1
r
z
y
x
z
y
x
R1R2
R2
R2
Fig. 1(a) Like magnetic link, (b) unlike link, (c) like electric link, (d) longitudinal link.
(a)
(b)
(c)
(d)

  
 










unlike
krkr
like
krkrkr
GG
P
PRXTX
TX
RX
42
642
11
111
4
(3)
Another feature of near-field links is an additional polarization, longitudinal polarization
or “L-POL.” An example of an L-POL link is that between two loops aligned on a
common axis, as in Fig. 1d. Conventional practice defines polarization in terms of E-field
orientation. Since a small loop is a TE1 source, we use “L-POL” to denote the link arising
from the longitudinally-oriented magnetic near-field component. L-POL links tend to be
about 6dB better than comparable near-field links, but do not extend into the far-field:
 







 64
11
krkr
GG
P
P
RXTX
TX
RX
(4)
Figure 2 illustrates the behavior of these near-field links for unity gain antennas. Far-field
links vary as -20dB/decade with respect to range due to their inverse r2 dependence. In
the vicinity of the “radiansphere,” the r =

/ 2

or k r = 1 range from a small antenna at
which near and far fields have the same amplitudes, the far-field link behavior transitions
to near-field behavior [18]. L-POL links transition from -60dB/decade to -40dB per
decade variation with respect to range due to their inverse r6 dependence within the
radiansphere and inverse r4 dependence outside the radiansphere [19]. Like-antenna near-
field links vary as -60dB/decade with respect to range due to their inverse r6 dependence
within the radiansphere. The behavior of like-antenna links and L-POL links within the
radiansphere is very similar except that L-POL links are 6dB stronger than like-antenna
links. Unlike-antenna near-field links vary as -40dB/decade with respect to range due to
their inverse r4 dependence within the radiansphere. Note, kr = 1 implies r =

/2.
These link laws assume “weak coupling” between antennas on either side of a link, or in
other words PRX/PTX << 1. In the strong coupling limit, PRX/PTX  1, each antenna
introduces a significant load on the other, decreasing Q as well as gain, and introducing a
mismatch. In any case, conservation of energy requires that PRX/PTX  1.
Fig.2 Link response for unity gain antennas in various near-field links versus radian distance k r.

Fig.3 Link response versus radian distance k r for unity gain antennas in various near-field links
normalized to a far-field link.
In the transmit-gain-receive-aperture formulation, these near-field link laws may be
compared and normalized with respect to far-field propagation. In this approach, the link
laws become:
 











unlike
kr
like
krkr
r
AG
P
P
RXTX
TX
RX
2
42
2
1
1
11
1
4

(5)
And the L-POL near-field link law becomes:
 








422
44
4krkr
r
AG
P
P
RXTX
TX
RX

(6)
In this formulation, far-field path loss is 0dB by definition, as seen in Figure 3. This
figure highlights one remarkable feature of near-field links: how much better they work
at short ranges than comparable far-field links.
One antenna fallacy ascribes a mysterious frequency-dependent loss to the inverse f
2
behavior in the frequency-dependent form of Friis’ Law in (1). Of course, this is just an
artifact of using antenna gain to characterize antenna performance in a link. One little
appreciated aspect of link behavior is that the “path loss” actually becomes “path gain”
for links between electrically-small antennas within each other’s near-fields. In the strong
coupling limit, as the path gain approaches one, these link laws no longer apply.
Just as Friis’ law is merely a starting point to understanding far-field propagation in real-
world applications, so also are these near-field link laws an approximation to the more
complicated behavior of near-field links in practical settings – the subject of the
following section.

Fig. 4 Layout of Q-Track’s facility showing Locator-Receiver deployments (seven gray squares) and
transmitter locations (28 blue squares). The map spans 174ft x 87ft.
2.2 Near-Field Links in Practice
To evaluate near-field propagation at 1034kHz, we deployed seven QT-550 Locator-
Receivers (as shown in Fig 5a) at the gray squares denoted in Fig. 4. Each Locator
Receiver has three channels: two orthogonal magnetic antennas and one vertical whip.
We calibrated the Received Signal Strength Indicator (RSSI). We took link data
including received power and phase difference measurements between channels to
characterize electric and magnetic links. The QT-640 tag of Figs. 5b and 5c provided a
magnetic transmitter source. This transmitter comprises two orthogonal magnetic
antennas fed in quadrature. Our electric transmitter was a GME SG-10A DDS Signal
Generator with a Mini-Circuits ZHL-6A Amplifier feeding an Empire LA-105 Electric
(whip) Antenna. Fig. 5d shows this system. Table 1 summarizes the transmit parameters
for the magnetic and electric systems. We took data at 28 points shown in blue in Fig. 4.
With two transmitters (electric and magnetic), three channels per Locator Receiver, seven
Locator Receivers deployed, and 28 transmitter locations, we took a total of 1176 data
points.
(c)
Fig. 5(a) QT-550 Locator-Receiver, (b) QT-640 tag rendering, (c) QT-640 magnetic transmit tag photo, (d)
electric transmitter system.
(b)
(d)
(a)

The indoor propagation channel is much more
restrictive than an ideal free space link. Typical
metal pan ceilings and rebar-embedded
concrete floors behave as a parallel-plate
waveguide, confining near-field signals to the
plane of a particular structure level. Thus near-
field propagation tends to be enhanced in indoor
propagation channels [20]. Far-field links tend to be attenuated by indoor propagation
(except in exceptional cases like ducting down a hallway).
Our test included four combinations of like and unlike antenna links of vertical E and
horizontal and longitudinal H polarizations. The magnetic transmitter used quadrature-fed
orthogonal magnetic antennas, so the magnetic signal included a mix of transverse and L-
POL signals. We found enhanced propagation in virtually all our links. On average, the
free-space, inverse r6 “like” links became r-4.2 for magnetic-magnetic links and r-3.3 for
electric-electric links. The free-space, inverse r4 “unlike” links became r-3.1 for magnetic-
electric links and r-3.7 for electric-magnetic links. We defined the range as the intersection
of the best fit power law roll-off with 3dB above the noise floor. Typical range was about
30m or better. Worst case range was about 20m and the best links exhibited up to 70-80m
ranges. Table 2 summarizes the results. Interestingly, both far-field and near-field
propagation tend to converge around a power law of about 1/r3 for short-range links [21,
22].
Magnetic Electric
PTX(dBm) 20.0 26.4
Gain (dBi) -89.1 -63.5
EIRP (dBm) -69.1 -37.1
Power Law Exponent
Magnetic TX Ideal Actual Range (m)
Mag-Mag Avg -6.0 -4.2 31.4
Max -2.9 60.1
Min -5.1 20.8
Mag-Elec Avg -4.0 -3.1 31.5
Max -1.7 71.8
Min -4.0 19.7
Power Law Exponent
Electric TX Ideal Actual Range (m)
Elec-Mag Avg -4.0 -3.7 36.4
Max -2.7 67.9
Min -5.5 22.7
Elec-Elec Avg -6.0 -3.3 36.2
Max -2.1 82.0
Min -4.3 19.3
Table 1: Transmit properties of electric and
magnetic sources.
Table 2: Power law results for an indoor propagation test comparing “like” and “unlike” links.

Fig. 6: Deviation of near-field propagation from free-space ideal.
Actual link behavior begins to deviate from free-space behavior at ranges comparable to
half the ceiling height. Beyond that range, the floor and ceiling begin to behave like a
waveguide, confining the signals and resulting in more gradual roll-off than the inverse r6
one would expect for free space. To demonstrate this, we evaluated a link between two
identical 30cm diameter magnetic reference antennas (Empire/Singer LP-105 loop
antennas) for ranges from 30cm to 5m. The 60cm : 90cm slope is 60.8dB/decade. The 1m
: 2m slope is about 58.8dB/decade. The 1m : 3m slope is 55.75dB/decade. Of course if
the range gets to be comparable to the antenna dimensions, that introduces error also.
With our 30cm diameter test loop, we found the 30cm : 45cm slope to be 66.4dB/decade.
The 45cm : 60cm slope was 61.6dB per decade. For our antenna and test lab dimensions,
the sweet spot is from about 75cm to 1.5m, where we get good results without antenna
dimension or room perturbation issues.
2.3 Near-Field Links in Summary
Near-field links have remarkable properties. They exhibit strongly enhanced propagation
characteristics with respect to far-field links. Unlike far-field links which generally
preform worse than free space links in cluttered indoor environments, near-field links are
enhanced by the channeling effects of the indoor propagation environment.
Implementation of a near-field link requires operation at a low frequency whose
wavelength is much longer than typical link range. For instance, our operation around
1MHz with a wavelength of about 300m yields ranges on the order of 20-60m depending
on the specifics of the environment and the level of ambient noise. The long wavelengths
of these signals readily bend and diffract around obstructions. These properties make
near-field wireless systems ideal for robust low-data-rate links in extremely cluttered
indoor environments. Near-field wireless enables reliable, short-range links with virtually
no drops, no deadzones, and no fading.

3 Antennas
This section discusses antenna testing and evaluation. Then, this section reports the result
of measurements on a variety of electrically-small antennas. Finally, this section
discusses limits to antenna gain and a novel formula to predict gain from quality factor
(Q) and electrical size.
3.1 Near-Field Antenna Gain Measurement Techniques
The results of Fig. 6 demonstrate that near-field links approach ideal free-space behavior
at short ranges – ranges comfortably less than half the typical floor-ceiling height in an
indoor environment. Thus, the ideal free-space behavior may be used at relatively short
ranges – on the order of r = 1m – to infer the gain of an electrically small antenna under
test (GAUT) relative to that of a reference antenna (Gref) [23]:


TX
RX
ref
GUT P
P
G
kr
G
6
4

(7)
If a reference antenna with calibrated gain is unavailable, the classic three antenna gain
measurement method [24] may be extrapolated to the near-field case. Given antennas A,
B, and C, with gains GA, GB, and GC, three measurements yield the gain product of each
pair of antennas:

BA
TX
RX
BA P
P
krGG

6
4
,

CB
TX
RX
CB P
P
krGG

6
4
, and

AC
TX
RX
AC P
P
krGG

6
4
. (8a, b, c)
Then:


CB
ACBA
AGG
GGGG
G
,




AC
CBBA
BGG
GGGG
G
, and





BA
ACCB
CGG
GGGG
G
. (9a, b, c)
We applied the near-field three antenna gain technique to three matched Empire (Singer)
LP-105 magnetic loop antennas. Fig. 7a shows the measurement set-up. Fig. 7b shows
the results of the gain measurement 100kHz to 10MHz while Fig. 7c focuses in on results
in the vicinity of 1MHz.

(a) (b)
(c) (d)
Figure 7(a): Near-field antenna gain measurement set-up, (b) gain measurements of three LP-105
antennas 100kHz-10MHz), (c) detailed gain measurement of Empire LP-105 in the vicinity of 1MHz,
and (d) measurement of EMCO 6509 gain compared to manufacturer’s specification
We then used the Empire LP-105 loop antennas as reference antennas to evaluate the gain
of an EMCO 6509 loop and obtained excellent agreement with the manufacturer’s
specifications shown in comparison to our results in Fig. 7d.
We used a similar technique to evaluate the gain of
the electric and magnetic antennas of Fig. 8. Table
2 compares results obtained at Q-Track to those at
the University of Massachusetts, Amherst (UMass).
We found essentially the same gain for the whip
antenna and a 4dB difference for the ferrite-loaded
loopstick.
Q-Track UMass
Whip -64.1 dBi -64.4 dBi
Loop -81.3 dBi -77.1 dBi
Table 2: Comparison of Q-Track to
UMass antenna gain measurements
at 1088kHz.

Fig. 8(a): Magnetic antenna on 2 in

 0.5 in diam ferrite, and (b) 9in whip and matching circuit.
Figs. 8a and 8b show photographs of the fabricated electric (E) and magnetic (H)
antennas. The E antenna is a 9-inch long whip with a ladder LC matching circuit
optimized for maximum gain around 1080 kHz. The H antenna is a ferrite-loaded loop
antenna with a single turn of 100/46 litz wire as the primary and 50 turns of the same
wire as the secondary. The length and diameter of the Mix-61 NiZn ferrite rod are 2” and
0.5”, respectively. The secondary coil resonates with a parallel capacitor at about 1080
kHz.
3.2 Near-Field Antenna Gain Measurement Results
As we took gain measurements on the ferrite-loaded magnetic antennas of Fig. 9a and the
air loops of Fig. 9b, we noticed an interesting trend. Gain results are proportional to the
loaded quality factor (Q
L
) and to the electrical volume (kR)
3
of the antenna where
k = 2/ is the wave number and R is the radius of the boundary sphere enclosing the
antenna. We used a vector network analyzer to evaluate the S
21
of the link between the
antenna under test and a reference antenna. The center frequency and -3dB bandwidth of
the S
21
yielded the loaded Q. Because the gain of the reference antenna is essentially flat
over the bandwidth of the resonant antennas under test, this is equivalent to the matching
(S
11
) -3dB bandwidth. Table 3 presents results for the ferrite loaded antennas of Fig. 9a
and Table 4 presents results for the unloaded air core loops of Fig. 9b.
(a)
(b)
Fig. 9(a): Four ferrite loaded antennas, from left to right, QT-600 Q antenna, 2 in

0.5in AM band
prototype, 1.14in diam loop prototype, QT640 antenna, and (b) three loop antennas, from left to right,
QT-550 4in loop, Terk 8in loop, and QT-552 2 ¼ in loop.
(a)
(b)

Ferrite Loaded Antennas Gain (dBi)
Antenna f(kHz) |S21|(dB) QL R(m) Gref (dBi) kR Theory Data Delta
QT600/11 turns 897.0 -71.0 17.6 0.0229 -87.7 4.30E-04 -88.6 -83.2 5.4
QT600/11 turns 994.0 -71.3 16.7 0.0229 -86.0 4.76E-04 -87.4 -82.5 4.9
QT600/11 turns 1050.0 -71.6 15.7 0.0229 -85.0 5.03E-04 -87.0 -82.3 4.7
QT600/11 turns 1206.0 -70.7 15.0 0.0229 -82.7 5.78E-04 -85.4 -80.2 5.2
QT600/2 turns 1203.0 -67.7 21.3 0.0229 -82.7 5.76E-04 -83.9 -77.2 6.7
QT600/2 turns 1056.0 -68.8 21.8 0.0229 -84.9 5.06E-04 -85.5 -79.5 6.0
QT600/2 turns 896.0 -68.5 23.0 0.0229 -87.7 4.29E-04 -87.4 -80.7 6.7
2inAM 2013.0 -68.9 22.3 0.0262 -74.0 1.10E-03 -75.2 -73.7 1.6
2inAM 859.0 -70.5 25.1 0.0262 -88.4 4.71E-04 -85.8 -83.1 2.7
2inAM 1071.0 -70.5 24.3 0.0262 -84.7 5.88E-04 -83.1 -81.0 2.0
2inAM 1412.0 -69.0 28.7 0.0262 -80.0 7.75E-04 -78.7 -77.0 1.7
FT114-61 1026.0 -76.5 20.8 0.0150 -85.4 3.23E-04 -91.5 -87.4 4.1
FT114-61 1686.0 -74.3 19.5 0.0150 -77.0 5.31E-04 -85.4 -80.7 4.7
QT640 1 turn 726.5 -76.7 25.1 0.0132 -91.3 2.01E-04 -96.9 -90.8 6.1
QT640 1 turn 1038.1 -75.6 24.8 0.0132 -85.2 2.87E-04 -92.3 -86.4 5.9
QT640 1 turn 1922.0 -73.6 23.7 0.0132 -74.8 5.32E-04 -84.5 -78.8 5.7
UMass 1 turn 1077.7 65.3 0.0262 5.91E-04 -78.7 -81.3 -2.6
Table 3: Gain measurements for ferrite loaded antennas compared to a prediction of G = QL (kR)3.
Air Core Antennas Gain (dBi)
Antenna f(kHz) |S21|(dB) QL R(m) Gref (dBi) kR Theory Data Delta
QT550 (1:1) 803.0 -62.2 5.7 0.060 -89.6 1.02E-03 -82.2 -75.4 6.8
QT550 (1:1) 1109.0 -62.9 5.0 0.060 -84.1 1.40E-03 -78.6 -73.1 5.5
QT550 (1:1) 1531.0 -63.1 4.8 0.060 -78.7 1.94E-03 -74.6 -70.4 4.2
QT550 (4:1) 609.5 -58.6 16.9 0.060 -94.2 7.72E-04 -81.1 -74.3 6.8
QT550 (4:1) 994.0 -59.1 12.8 0.060 -86.0 1.26E-03 -75.9 -70.3 5.6
QT550 (4:1) 1531.0 -59.4 9.9 0.060 -78.7 1.94E-03 -71.4 -66.7 4.8
Terk(1:1) 511.4 -50.7 20.6 0.108 -97.2 1.16E-03 -75.0 -68.0 6.9
Terk(1:1) 716.1 -50.4 18.7 0.108 -91.5 1.62E-03 -71.0 -64.6 6.4
Terk(1:1) 1068.0 -51.3 15.1 0.108 -84.8 2.42E-03 -66.7 -61.9 4.8
Terk(1:1) 1392.0 -51.3 13.3 0.108 -80.3 3.15E-03 -63.8 -59.4 4.4
Terk(1:1) 1867.0 -52.8 11.4 0.108 -75.3 4.23E-03 -60.7 -58.3 2.4
Terk(4:1) 511.0 -53.5 25.0 0.108 -97.2 1.16E-03 -74.1 -70.8 3.3
Terk(4:1) 716.0 -51.6 24.8 0.108 -91.5 1.62E-03 -69.8 -65.8 3.9
Terk(4:1) 1056.0 -52.3 23.5 0.108 -84.9 2.39E-03 -64.9 -63.0 2.0
Terk(4:1) 1424.0 -50.2 19.9 0.108 -79.9 3.22E-03 -61.8 -58.1 3.6
Terk(4:1) 1845.0 -51.0 20.7 0.108 -75.5 4.18E-03 -58.2 -56.6 1.7
QT552 (4:1) 668.0 -64.8 21.2 0.060 -92.7 8.46E-04 -78.9 -79.7 -0.7
QT552 (4:1) 863.5 -64.2 19.8 0.060 -88.3 1.09E-03 -75.9 -76.7 -0.8
QT552 (4:1) 1056.0 -64.7 18.2 0.060 -84.9 1.34E-03 -73.6 -75.4 -1.8
QT552 (4:1) 1462.0 -64.2 16.4 0.060 -79.4 1.85E-03 -69.8 -71.9 -2.1
QT552 (4:1) 1718.0 -64.6 15.2 0.060 -76.7 2.18E-03 -68.1 -70.8 -2.8
QT552 (4:1) 2942.0 -66.9 11.8 0.060 -67.6 3.72E-03 -62.1 -68.2 -6.0
Table 4: Gain measurements for air core antennas compared to a prediction of G = QL (kR)3.

3.3 Gain Limits
The ideal or radiation quality factor (Qideal) for a small magnetic antenna is [25, 26]:

kR
kR
Qideal
11
3 . (10)
where R is the radius bounding the antenna. However, the Chu/McLean analysis assumes
no energy is stored within the sphere of radius R. For practical antennas, this is not the
case. For instance, for a small air core (

r = 1) loop antenna instantiating a TE1 mode [27,
28, 29]:

rad
idealrad R
L
kR
kR
QQ

 33
33. (11)
where the angular frequency is

= 2f, L is the antenna inductance, and Rrad is the
radiation resistance. Practical antennas are also characterized by loss resistance (Rloss) that
further decreases even their unloaded QU:
lossrad
URR
L
Q


. (12)
The efficiency (

) of a small loop antenna may be written in terms of the quality factor
[30, 31]:






33
2
33 2
3
2
3





kR
kRQ
kR
kRQ
Q
Q
RR
RL
U
rad
U
lossrad
rad

, (13)
where for a well-matched antenna the loaded Q (QL) is half the unloaded QU. The gain of
a small loop antenna is related to the efficiency and directivity (D):





 
1
133
23
2
3
2
3
2
3









 kRkRQ
kR
kRQ
kR
kRQ
DG L
LL

, (14)
A second line of investigation leads to a similar prediction. A recent paper presents a link
budget between resonant loops [32]:
1
6









kRQQ
r
RR
P
P
RXTX
RXTX
TX
RX , (15)
where RTX and RRX are the radii of the transmit and receive loops, respectively, and QTX
and QRX are the loaded quality factors of the transmit and receive loops, respectively.
Assume a matched pair link for (3), oriented coaxially (as in Fig. 1d) where kr << 1 so:

6
kr
GG
P
PRXTX
TX
RX . (16)

The matched-pair link means that RTX = RRX = R, and QTX = QRX = QL. Equating (15) and
(16) yields the same expression for gain as a function of loaded quality factor and
electrical volume:


3
kRQG L
. (17)
This relation is of great utility in electrically-small antenna design for near-field links.
The loaded quality factor QL is related to the fractional bandwidth of an antenna:


LU
LU
LU
LU
LU
C
Lff
ff
ff
ff
ff
f
bw
Q





 2
1
1 (18)
where the upper and lower frequencies (fU and fL, respectively) follow from the -3dB
points, and the center frequency fC is the geometric average of the upper and lower
frequencies. Note in the narrowband limit, the geometric average becomes equivalent to
the arithmetic average.
The gain relation of (17) yields an easy and reliable prediction of antenna performance
given the desired bandwidth and size of the antenna. Gain measurements of electrically
small antennas are often subtle and challenging. A long, low-Q cable may possibly have a
higher effective gain than a small high Q antenna under test. The gain relation (17) allows
an evaluation of antenna gain using only a matching measurement (to yield QL) and a
ruler. This circumvents many difficulties in electrically small antenna gain measurement.
One interesting consequence is that for ideal free-space links with comparable antennas
of the same size R and the same QL, performance is independent of frequency, as
demonstrated by Azad et al [32] and captured in (15). Choice of a frequency of operation
follows from other factors such as regulatory constraints, the frequency dependence of
ambient noise, and the desired near-field range characterized by a few times the
radiansphere distance at a particular frequency.
4 Fields Around Small Antennas
In small antenna practice, we characterize the transmission of an antenna according to its
Equivalent Isotropic Radiated Power (EIRP = PTX GTX). This approach works well for a
system analysis of a link. In theory, however, we often model a small loop antenna as a
magnetic dipole characterized by a dipole moment m = N I A n, where m = |m|, N is the
number of turns, the current I, the area A, and the normal vector n to the area of the loop.
This theoretical approach is helpful for characterizing the fields around a small antenna.
However, connecting the theory of fields around a dipole to the performance of a small
loop antenna requires a relation between EIRP and dipole moment. This section aims to
derive such a connection.

Direct measurement of the current in a resonant, high-Q parallel LC circuit is difficult
without perturbing the resonance and obtaining an erroneous measurement. One useful
technique follows from application of the law of conservation of energy. For a high Q
resonant antenna:
2
2
1
2
2
1LICV  (19)
so:
L
C
VI . (20)
In (20) the voltage (V) and current (I) may be rms or peak so long as both are similarly
defined. Even a megaohm impedance voltage probe risks perturbing the impedance of a
high Q resonant circuit, however, so an alternate approach to relating current and dipole
moment to transmit power are needed.
The antenna factor relates electric field intensity to received voltage [30, see p. 287]:
V
E
GZ
Z
AZ
Z
AF
load
S
load
S
RX  2
4


(21)
where Zs =376.7 is the free space impedance, (2) relates aperture (A) to gain (G), and
Zload is the receive impedance. Solving (21) for gain:
RX
S
load
S
load
SP
E
Z
Z
V
E
Z
AFZ
Z
G2
2
2
2
2
22
444






 (22)
where the received power is PRX = V2/Zload. Substituting (22) into the link law between
unlike antennas (3):
 







 242
2
114
4krkr
P
E
ZGP
PRX
S
TXTX
RX


, (23)
and solving for the electric field intensity yields:
 







 24
2
211
krkr
Z
GPE S
TXTX


. (24)
Electric field intensity also follows from the theoretical model of a magnetic dipole:
 







 24
82
0
2
6
0
211
32 krkr
c
m
E


. (25)

Noting that:
c
ZS
00
01


 , 22
2
24
1
c



, and 3
0
2
24c
ZS




, (26a, b, c)
and equating (24) and (25) yields an expression for EIRP in terms of dipole moment:
5
0
42
0
3
0
2
282
0
2
62
0
8
432
c
m
GP
c
GP
Z
GP
c
m
TXTX
TXTX
S
TXTX










(27)
Similarly, the magnetic dipole moment may be expressed in terms of EIRP:
TXTX
S
TXTX
S
TXTX
S
SS
TXTX
GP
Z
AP
kZ
GP
kZ
m
m
Z
m
kZ
GPEIRP
3
4
24
2
0
2
0
4
3
2
0
4
2
88
2
8







(28)
The fields may be written in terms of EIRP:
 







 242
211
krkr
Z
GPE S
TXTX


E, and (29)
  







 246
2
2111
krkrkr
GP
Z
HTXTX
S


H. (30)
These expressions describe the field intensities of the peak transverse near fields around a
small loop antenna. As previously noted, the longitudinal magnetic field component will
be 6dB (a factor of two) stronger.

(a) (b)
Fig. 10(a): NEC model, and (b) FEKO model.
5 Numerical Validation
This section presents results from two numerical analysis packages: NEC [33] and FEKO
[34]. We implemented models of a 1m diameter loop comprising six turns of AWG #10
(wire radius 1.294mm) copper wire, as shown in Figure 10. NEC modeling employed
both “EZ-NEC” [35] and “4NEC2” [36].
5.1 Validation of Gain Relation
We numerically evaluated the SWR of each antenna and tuned to achieve a resonance in
the vicinity of 1.1MHz. We assumed a matched load and evaluated the -3dB bandwidth
to determine the simulated loaded Q
L
. We compared the models’ predicted gains to the
gains predicted using the relation G = Q
L
(kR)
3
. In each case the numerical model yielded
a predicted gain not quite 3dB below the theoretical prediction. Table 5 shows details.
f
C
(kHz) f
L
(kHz) f
U
(kHz) Q
L
kR
Gain
Theory
(dBi)
Model Error Z (ohm) C(pF)
NEC 1099.9 1097.85 1101.97 267 0.012138 -33.21 -36.17 2.96 0.6842 402.5
FEKO 1102.5 1100.47 1104.53 271 0.012158 -33.12 -35.7 2.58 0.7511 367.0
Table 5: Gain predictions and other characteristics of NEC and FEKO models.
5.2 Validation of Link Relation
For each model, we implemented a duplicate small loop antenna and replaced the source
in the duplicate with a matched load. Then we arranged the matched pair of antennas to
assess co-POL and L-POL link relations. We assumed 1W transmit power, and evaluated
the received power in the matched load of the other antenna at various distances. We
compared the simulated received power to what would be predicted by the link relations
(3) and (4) assuming the gain predicted by the numerical model. Figure 11 shows the
excellent agreement between the numerical models and the link law predictions. The only
significant deviation occurs at short ranges where the weak coupling assumption breaks
down due to the close proximity of the receive and transmit antennas.

(a)
(b)
Fig. 10(a): NEC link model, and (b) FEKO link model. Note that conservation of energy prohibits received
power greater than the 1W transmit power or S21 > 0dB. In this limit a strong coupling model like that of
Azad et al [32] becomes relevant.
10
0
10
1
10
2
10
3
−140
−120
−100
−80
−60
−40
−20
0
r (m)
|S21| (dB)
Co−POL FEKO
Co−POL Theory (w/ FEKO gain)
L−POL FEKO
L−POL THeory (w/ FEKO gain)

5.3 Validation of Field Relations
Finally, we implemented another simple
model to assess the validity of the field
relations of (29) and (30). A 1m per side
1cm-wire-diameter copper “wire” square
loop driven at 1MHz has gain -37.64dBi
according to a NEC simulation. When
driven with 10W, we obtain a near perfect
agreement except in the immediate
vicinity of the antenna. For instance, at
30m in the plane of the loop antenna,
NEC predicts 14.2196mV/m and the
theory predicts 14.227mV/m, agreement
within 0.05%. The figure below shows the
agreement in this case. If we tell NEC to
assume lossless conductors, the gain is
+1.76dBi as expected for a dipole. Then,
NEC predicts E = 1.327V/m at 30m range, and theory predicts E = 1.328V/m - again,
excellent agreement. The magnetic field results show similar agreement: within a fraction
of a percent, except at very short ranges where the weak coupling assumption breaks
down. Fig. 12 displays the results.
Fig. 12(a): Comparison of theoretical and modeled magnetic field, and (b) comparison of theoretical and
modeled electric field.
Fig. 11: NEC model of a loop to evaluate field
predictions.

6 Conclusions
We presented simple formulas to describe near-field links, antennas, and fields. Friis’ law
of propagation may be extended to the case of electrically-small antennas operating at
near-field ranges in the weak coupling limit. Friis’ Law bifurcates into two laws - one for
“like” antennas (both electric or both magnetic), and one for unlike antennas (one electric
and one magnetic):
  
 










unlike
krkr
like
krkrkr
GG
P
PRXTX
TX
RX
42
642
11
111
4
(3)
Near-fields propagation also exhibits an additional longitudinal polarization or “L-POL”
component not present in far-field signals. L-POL links are about 6dB stronger than
comparable near-field links, but do not extend into the far-field:
 







 64
11
krkr
GG
P
P
RXTX
TX
RX
. (4)
These near-field link laws assume small loop or dipole antennas characterized by TM1 or
TE1 fields, respectively. More complicated multipole configurations will necessarily
exhibit different link behavior characterized by correspondingly higher power
enhancements in the deep near-field. We found in practice that indoor near-field links are
enhanced by the indoor propagation channel to yield power law dependence on the order
of inverse r3 or so.
There exists a general formula to describe gain for small resonant antennas of the form:


3
kRQG L
. (17)
This paper derived the gain relation (17) by two separate approaches. Experimental
measurements of air core loops tend to indicate a relation more on the order of
G = 2 QL (kR)3. Ferrite-loaded loops appear enhanced further, perhaps because of a
change in the ideal Q, or alternatively because the effective electrical size R of the ferrite-
loaded loop is enhanced. Numerical modeling tends to indicate the relation might be
closer to G = ½ QL (kR)3. It is entirely possible (17) is off by a small constant factor and
may need to be modified to capture the impact of ferrite loading, shape factors, and other
key characteristics. Preliminary investigation shows that loops with a large length-to-
diameter have lower gain than those with a small length relative to diameter.
This gain relation (17) has great practical value for predicting the performance of
electrically small antennas. In addition, the gain relation enables a better validation of Q
limits. Since a small TM1 (or TE1 antenna for that matter) has an ideal gain of 3/2, in the
limit as Q approaches ideal values, gain as a function of Q must approach 3/2. Although
the present work focuses almost entirely on small magnetic loop antennas, preliminary
investigation indicates that similar relations holds for small electric dipole antennas as

well. In short, we find in the small antenna limit a general relationship for antenna gain of
the form:


3
kRCQG L
 (31)
where C is a constant dependent upon shape factors, loading, and the electric or magnetic
nature of the antenna.
The intensity of the transverse fields around a small antenna may be written in terms of
EIRP:
 







 242
211
krkr
Z
GPE S
TXTX


E, and (29)
  







 246
2
2111
krkrkr
GP
Z
HTXTX
S


H. (30)
These relations are helpful for predicting emission in support of regulatory testing of
near-field wireless devices as they describe the behavior of the peak transverse
component of each field.
The progression of RF technology is a story of ever increasing frequency: LF-band spark
gaps yielding to MF-band broadcasts, and HF-band shortwave signals, yielding in turn to
VHF- and UHF-band TV, microwave radars, cellular communications, and now
commercial millimeter-wave wireless. High-data-rate high-capacity links do call out for
high-bandwidth signals. This necessitates operation at relatively high frequencies. But a
higher frequency is not necessarily a better frequency for all applications.
Architectural pioneer Louis Sullivan established the principle that the shape of a building
or an object should be primarily based on its intended function or purpose: “form follows
function” [37]. The corresponding rule for RF design should be “frequency follows
function.” The demands of an RF application lead to an appropriate choice of operating
frequency.
Increasingly, RF scientists and engineers are recognizing that low-frequency and near-
field wireless links can offer substantial advantages for high-performance, short-range
wireless systems, particularly those operating in difficult or cluttered environments. We
offer these simple formulas as a starting point for those interested in exploring how near-
field wireless can solve real-world problems.
7 Acknowledgements
This material is based upon work supported by the National Science Foundation under
Grant No. 1217524. Any opinions, findings, and conclusions or recommendations
expressed in this publication are those of the author(s) and do not necessarily reflect the
views of the National Science Foundation.” We would also like to acknowledge helpful
discussions with Kai Siwiak, Steve Werner, and Eric Richards.

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