Near field propagation law & a novel fundamental limit to antenna gain versus size

Abstract

The paper presents a theoretical analysis of the near field channel in free space. The theoretical model is then validated by comparison to data measured in an open field. The results are important for low frequency RF systems, such as those operating at short range in the AM broadcast band (525-1715 kHz). Finally, the paper establishes a novel fundamental limit for antenna gain versus size.

Full text

Preprint – Submitted to IEEE APS Conference July 2005;
© 2005 Dr. Hans Schantz
A Near Field Propagation Law & A Novel Fundamental Limit to Antenna
Gain Versus Size
Hans Gregory Schantz (h.schantz@q-track.com)
The Q-Track Corporation; 515 Sparkman Drive; Huntsville, AL 35816
ABSTRACT
This paper presents a theoretical analysis of the near field channel in free space. Then this paper
validates the theoretical model by comparison to data measured in an open field. The results of this
paper are important for low frequency RF systems, such as those operating at short range in the AM
broadcast band. Finally this paper establishes a novel fundamental limit for antenna gain versus size.
1. INTRODUCTION
The “leading edge” of RF practice moves to increasingly higher and higher frequency in lock step
with advances in electronics technology. The most commercially significant RF systems are those
operating at microwave frequencies and above, such as cellular telephones and wireless data
networks. Microwave frequencies have the advantage of short wavelengths, making antenna
design relatively straightforward, and vast expanses of spectrum, making large bandwidth, high
data rate transmissions possible.
There are many applications, however, that do not require large bandwidths. These include
real time locating systems (RTLS) and low data rate communications systems, such as hands free
wireless mikes or other voice or low data rate telemetry links. For applications like these, lower
frequencies have great utility.
Lower frequencies tend to be more penetrating than higher frequencies.
Lower frequencies tend to diffract around objects that would block higher frequencies.
Lower frequencies are less prone to multipath.
An amazing and often overlooked world of RF phenomena lies within about a half
wavelength of an electrically small antenna. This realm is known as the “near field zone.” The
near field zone is usually neglected by RF scientists and engineers because typical RF links
operate at distances of many wavelengths where near field effects are utterly insignificant. “Near
field” means different things in different contexts. Fortunately there is an excellent article
available that sorts through the various definitions of near field and provides some guidance [1].
The present discussion, takes the near field zone as the region within about a half wavelength of an
electrically small antenna.
The aim of the present paper is to derive a near field propagation equation. This paper
compares the near field propagation equation to data obtained in an open field environment.
Finally, this near field propagation equation is used to derive a fundamental limit for antenna size
versus gain.
2. PATH GAIN IN FAR AND NEAR FIELD
This section will discuss the path gain for traditional far field links and summarize the differences
between far field and near field links. Then, this section will present a recently introduced a near
field link equation that provides path loss for low frequency near field links [2].
The path gain (P) defines the relationship between transmitted power (P
TX
) and received
power (P
RX
) in a far-field RF link. This relation was first given by Harald Friis [3]:
()
() ()
2
2
2
2
1
4
4
,
kd
GG
d
GG
P
P
dfP
RXTXRXTX
TX
RX
=
π
λ
== (1)
In this formula, G
RX
and G
TX
are the receive and transmit antenna gains (respectively), d is the
distance between the antennas, λ is wavelength, and k = 2π/λ is the wave number. The reason for
writing Friis’s law in a non-standard way (using wave number) will become clear momentarily.
The upshot of Friis’s Law is that the far-field power rolls off as the inverse square of the distance

Preprint – Submitted to IEEE APS Conference July 2005;
© 2005 Dr. Hans Schantz
(1/d
2
). Near-field links do not obey this relationship. Near field power rolls off as powers higher
than inverse square, typically inverse fourth (1/d
4
) or higher.
This near field behavior has several important consequences. First, the available power in a
near field link will tend to be much higher than would be predicted from the usual far-field, Friis’s
Law relationship. This means a higher signal-to-noise ratio (SNR) and a better performing link.
Second, because the near-fields have such a rapid roll-off, range tends to be relatively finite and
limited. Thus, a near-field system is less likely to interfere with another RF system outside the
operational range of the near-field system.
Electric and magnetic fields behave differently in the near field, and thus require different link
equations. Reception of an electric field signal requires an electric antenna, like a whip or a dipole.
Reception of a magnetic field signal requires a magnetic antenna, like a loop or a loopstick. The
received signal power from a co-polarized electric antenna is proportional to the time average
value of the incident electric field squared. For the case of a small electric dipole transmit antenna
radiating in the azimuthal plane and being received by a vertically polarized electric antenna, the
received power is:
() () ()








+−
642
2
)(
111
~~
kdkdkd
P
ERX
E (2)
Similarly, the received signal power from a co-
polarized magnetic antenna is proportional to the
time average value of the incident magnetic field
squared:
() ()








+
42
2
)(
11
~~
kdkd
P
HRX
H (3)
in (
Thus, the “near field” path gain formulas are:
()
()
() () ()








+−=
=
642
)(
111
4
,
kdkdkd
GG
P
P
fdP
ERXTX
TX
ERX
E
(4)
for the electric field signal, and:
()
() ()








+==
42
)(
11
4
,
kdkd
GG
P
P
fdP
RXTX
TX
HRX
H
(5)
Near Field Channel Model
(Electric TX Antenna)
-30
-20
-10
0
10
20
30
40
50
60
70
0.01 0.1 1
Range (
λ
)
Path Ga dB)
E-Field
H-Field
Far Field
6
0
d
B
/
d
e
c
a
d
e
4
0
d
B
/
d
e
c
a
d
e
2
0
d
B
/
d
e
c
a
d
e
2
0
d
B
/
d
e
c
a
d
e
Figure 1 Path gain of a typical near field
channel
for the magnetic field signal. Equation 4 is the propagation law for like antennas (electric to
electric or magnetic to magnetic) and Equation 5 is the near field propagation law for unlike
antennas (magnetic to electric or electric to magnetic). Typical path gain in a near field channel is
on the order of –6 dB. At very short ranges, path gain may be on the order of +60 dB or more. At
an extreme range of about one wavelength the path gain may be about –18 dB. This behavior is
summarized in Figure 1.
3. PROPAGATION DATA
The Q-Track Corporation has pioneered a novel low frequency tracking technology known as
“Near Field Electromagnetic Ranging,” or NFER
TM
technology. NFER
TM
technology operates in
the AM broadcast band (525-1715 kHz) on an unlicensed basis under the authority of the FCC’s
Part 15 regulations [4]. Q-Track’s 1295 kHz (λ = 213.5 m) prototypes have a demonstrated
accuracy of about 30 cm at ranges of up to 70 m outdoors, and an accuracy of about 4 m at ranges
of up to 70 m indoors. For more information on Q-Track’s NFER
TM
technology, please see the Q-
Track website [5]. Table 1 shows parameters for a prototype NFER
TM
tracking system.

Preprint – Submitted to IEEE APS Conference July 2005;
© 2005 Dr. Hans Schantz
Table 1: Parameters for a
prototype NFER
TM
tracking
system.
(a) (b)
Parameter:
Value:
Transmit Gain
G
TX
= -51 dB
E Receive
Antenna Gain
G
RXE
= -53 dB
H Receive
Antenna Gain
G
RXH
= -71 dB
Transmit
Power P
TX
= +20 dBm
Figure 2 (a) Q-Track’s prototype beacon transmitter with whip antenna.
(b) Q-Track prototype locator receiver with three element array
[both figures courtesy Q-Track; © 2004].
Q-Track’s prototype beacon
transmitter operates at 1295 kHz
with a transmit power at the FCC
limit of 100 mW. The beacon
transmitter uses a 60 cm (2 ft) whip
with a gain of approximately –51
dBi. Figure 2(a) shows Q-Track’s
beacon transmitter with whip
antenna. Q-Track’s prototype
locator receiver uses a three
antenna array to receive both
electric and magnetic field
components. The electric receive
antenna is similar to the electric
transmit antenna and has a gain of
–53 dBi. The magnetic receive
antenna is a box loop with a gain of
about –71 dBi. Figure 2(b) shows
Q-Track’s locator receiver with three element antenna array. Figure 3 compares near field
propagation results for a prototype NFER
TM
tracking system to the theoretical predictions of
Equations 4 and 5. Agreement is generally within a few dB.
Near Field Propagation
1295 kHz Link;
λ
= 231.5 m (760 ft)
-120
-110
-100
-90
-80
-70
-60
0.01 0.1 1Ran
g
e
(
λ
)
Power (dBm) .
PH Data (dBm)
PE Data (dBm)
PH Theory (dBm)
PE Theory (dBm)
Figure 3 Theory vs. experiment for a 1295 kHz link [Courtesy Q-
Track; © 2004].
4. LIMITS TO ANTENNA SIZE AND GAIN
The near field link equations define the path gain as a function of the transmit and receive antenna
gains. Figure 1 appears to indicate that under some circumstances path gain may be greater than
0 dB. This means that the receive power could theoretically be greater than the transmit power.
Since conservation of energy must apply to RF links, antenna gain cannot be arbitrarily large.
There necessarily exists a limit to antenna gain as a function of antenna size.
The treatment of this section borrows on the concept of an antenna “boundary sphere”
introduced by Wheeler and extended upon by Chu [6, 7]. A boundary sphere is the smallest sphere
within which an antenna may be enclosed. Thus the radius of the boundary sphere defines the
characteristic size of an antenna. A matched pair of antennas with boundary spheres of radius R
may be no closer than d = 2 R without overlapping, as shown in Figure 4(a). Taking this as the
limit, one can apply the near field propagation equation for like antennas (Eq. 4) to establish a
limit for antenna gain versus size:

Preprint – Submitted to IEEE APS Conference July 2005;
© 2005 Dr. Hans Schantz
(a)
R R
d = 2R
(b)
Maximum Gain vs Antenna Size
-100
-80
-60
-40
-20
0
20
0.0001 0.001 0.01 0.1 1
R
λ
: Radius (Units of Wavelength)
Maximum Gain (dBi)
Limit
Q-Track
Antennas
Figure 4(a) A matched pair of antennas with boundary spheres of radius R can be no closer than about
d = 2 R without their boundary spheres overlapping. (b) Limit on antenna gain vs. size showing a
variety of Q-Track antennas [Q-Track; ©2004].
()
()()()
()()()
()
()()
()
()()
42
3
42
3
642
642
)(
441
42
221
22
2
1
2
1
2
1
4
2
1
2
1
2
1
4
1,
λλ
λ
π+π−
π
=
+−
=








+−
≤
→








+−≥≤=
RR
R
kRkR
kR
kRkRkR
G
kRkRkR
GG
P
P
fdP
TX
ERX
E
(7)
Figure 4(b) shows the gain limit as a function of boundary sphere radius in units of wavelength.
This figure also shows gain and size of a variety of Q-Track’s electrically small antennas for
comparison.
5. CONCLUSIONS
This paper derived near field propagation relations analogous to Friis’s law and compared theory
to experimental data. This paper further derived a fundamental limit to antenna gain versus size
and compared the result to a variety of antennas designed by the Q-Track Corporation. The often
neglected world of the near field is not only susceptible to mathematical analysis but also yields
lessons applicable to antenna design in general.
6. ACKNOWLEDGEMENTS
The author is grateful to the Q-Track Corporation for support, assistance, and permission to
disclose Q-Track proprietary performance information and results. The author also wishes to thank
Kai Siwiak for his advice and encouragement.
7. REFERENCES
[1] Charles Capps, “Near Field or Far Field,” EDN, August 16, 2001, pp. 95-102. This excellent
article is available online at: http://www.edn.com/contents/images/150828.pdf
[2] Hans Schantz, Near Field Channel Model, IEEE P802.15-04/0417r2, October 27, 2004.
[3] Harald Friis, “A Note on a Simple Transmission Formula,” Proc. IRE, 34, 1946, pp. 254-256.
[4] Section 15.219 authorizes a transmit power of 100 mW (+20 dBm) into an antenna less than
3 m in dimensions on an unlicensed basis in the AM broadcast band.
[5] For more information on NFER
TM
technology see http://www.q-track.com.
[6] Harold A. Wheeler, “Fundamental Limitations of Small Antennas,” Proc. IRE, 35, (1947), pp.
1479-1484.
[7] L.J. Chu, “Physical Limitations of Omni-Directional Antennas,” Journal of Applied Physics,
19, December 1948, pp. 1163-1175