Fundamental Limit 2008

8/8/2019 Fundamental Limit 2008

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Abstract A fundamental limit on an antennas gain is derivedand compared to measurements taken on a number of differentantennas. First, a propagation formula applicable in both thenear and far fields is developed, and that result is used todemonstrate that the gain of an antenna is limited by its electricalsize.

Index Terms Electrically Small Antennas, Antenna Gain,Antenna Measurements, Near Field.

I.

INTRODUCTION HE area around antennas is often split into two areas: thenear field, which extends out from the antenna at a

distance comparable to a wavelength, and the far field, whichis the area beyond the near field. See [1] for an excellentdiscussion of the properties of the near field, the far field, andexactly where the boundary between the two lies. For mostantenna applications, especially those at high frequencies, thebehavior of the antennas electromagnetic fields in the nearfield is often of little consequence. For example, a typical Wi-Fi1 signal operating at 2.4 GHz has a wavelength of about 12.5cm. Because most Wi-Fi applications operate at ranges on theorder of meters, this system lends itself well to far-fieldanalysis. This is the case for almost every electromagneticsystem in common use today.

However, the near field has some interesting properties thatsome systems exploit. As we will demonstrate, the powertransmitted by a near-field link rolls off much faster than inthe far field, which means that often times signals will nottransmit far enough to cause harmful interference. In turn,short distance, high data-rate communication is possible usinga near-field communication link.

Another novel use of the near field is in real-time locationsystems (RTLS). The Q-Track Corporation has pioneered aRTLS technology known as Near-Field Electromagnetic

Manuscript received February 1, 2008; revised April 2, 2008. Thismaterial is based upon work supported by the National Science Foundationunder Award Number: 0646339. Any opinions, findings, and conclusions orrecommendations expressed in this publication are those of the authors and donot necessarily reflect the views of the National Science Foundation.

A. J. Compston is a senior electrical engineering undergraduate student atthe Georgia Institute of Technology and is with the Q-Track Corporation,Huntsville, AL 35816 (e-mail: [email protected]).

J. D. Fluhler is an electrical engineering undergraduate student at theUniversity of Alabama, Huntsville and is with the Q-Track Corporation,Huntsville, AL 35816 (e-mail: [email protected]).

H. G. Schantz is with the Q-Track Corporation, Huntsville, AL 35816(e-mail: [email protected]).

1 Wi-Fi is a registered trademark of the Wi-Fi Alliance.

Ranging (NFER) 2 that takes advantage of the fact that thephase difference of a transmitting antennas electric andmagnetic fields goes to zero as the distance from the antennaincreases [2].

Often near-field applications like these use low frequenciesin order to fully realize the advantages of the near field atsubstantial distances. Compare the wavelength of a typicalWi-Fi signal (12.5 cm) with the wavelength of a typicalQ-Track NFER signal (1 MHz, 300 m). As a consequence,most antennas used in near-field systems are much smaller

than a wavelength.This paper exploits near-field phenomena in order to derivea fundamental limit on the gain of an antenna versus itselectrical size. First, we derive a propagation formula similarto Friiss formula for the far field. Using this formula, weshow a fundamental limit on the gain of an antenna versus itselectrical size, and we compare the limit to a number of gainmeasurements taken on actual antennas.

II. FRIIS S PROPAGATION FORMULA AND THE FAR FIELD Harald Friis derived the propagation formula that bears his

name in the following form [3]

2)( d A A

PP

TX RX TX

RX = . (1)

P RX is the power received by the receiving antenna, P TX is thepower transmitted by the transmitting antenna, A RX is theeffective area of the receive antenna, ATX is the effective areaof the transmit antenna, d is the distance between eachantenna, and is the wavelength of the electromagnetic wave.Note that the antenna effective area or aperture A is related tothe antenna gain G by

4

2G A = , (2)

so Friiss formula can also be written as

22

2

444 d G A

d AGGG

d PP TX RX TX RX TX RX

TX

RX

==

= . (3)

This is a very powerful formula, but because it assumes aplane wave front, it is only applicable in the far field. In hispaper, Friis warns that his formula is correct to within a fewpercent when

22ad , (4)

2 NFER is a registered trademark of the Q-Track Corporation.

A Fundamental Limit on Antenna Gain forElectrically Small Antennas

Andrew J. Compston, James D. Fluhler, and Hans G. Schantz

T

Reprinted From: 2008 IEEE Sarnoff Symposium,Princeton, NJ, USA; 28-30 April, 2008, Pages:1 - 5.2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material foradvertising or promotional purposes or for creating new collective works for resale or redistribution to servers orlists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

SM


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where a is the largest linear dimension of either of theantennas [3]. Specifically, [t]his criterion has a phase errorof one-sixteenth of a wavelength [1]. Also, Friiss formula isonly strictly valid in free space.

III. NEAR -F IELD PROPAGATION FORMULA Whereas the far-field propagation formula developed by

Friis is the same regardless of the type of antenna used, in the

near field one must distinguish between an electric antenna(like a dipole or whip) and its dual: a magnetic antenna (suchas a loop). First the case of an electric transmit antenna isconsidered, followed by the case of a magnetic antenna.

A. Electric Transmit AntennaImagine an infinitesimal current element with length

l


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3

jkr er kr kr

j

kr kr

jkr

Sk I H

++

++

=

)cos()(

1

)(2

)sin()(

1

)(

14

32

32

3

r

. (20)

where S is the surface area of the loop. Using the sameprocedure outlined above for the electric transmitter, thefollowing can be proven for a magnetic transmit antenna

like RX nf H TX

H RX PL AP

P,

,

, = , and unlike RX nf H TX

E RX PL AP

P,

,

, = . (21)

Thus, the near-field propagation formula can besummarized as below

=antennas unlike),,,(

antennas like),,,(),,(

,

,

r PL A

r PL A

Pr P

unlike RX nf

like RX nf

TX

RX , (22)

where like antennas are taken to mean either two electric ortwo magnetic antennas and unlike antennas are taken tomean one electric and one magnetic antenna.

IV. COMPARING THE NEAR -F IELD PROPAGATION FORMULAWITH FRIIS S FORMULA

A. The Special Case of = 90For the special case where both antennas are at = 90

( z = 0, the horizontal plane) relative to each other (as in Fig.2), see in (5) and (6) that the electric and magnetic fields haveno radial components. Therefore, the near-field effective areasof the antennas are equivalent to their far-field effective areas.For an infinitesimal dipole, this is

2

2

,inf ,

2

3

8

3

k

A A A TX TX nf nf

==== , (23)

which further reduces (24) to:

+=

=6422

4

,90 )(

1

)(

1

)(

1

4 kr kr kr A A

k PP

TX RX likeTX

RX

o, and (24)

+=

=422

4

,90 )(

1

)(

1

4 kr kr A A

k PP

TX RX unlikeTX

RX

o. (25)

For large r , the 1/ r 4 and 1/ r 6 terms will go to zero muchfaster than the 1/ r 2 terms, and they can be ignored for allpractical purposes, as demonstrated in Fig. 3. This leaves

2

2

,90,90 )(2 r

A A A A

r k

PP

PP TX RX

TX RX unlikeTX

RX

likeTX

RX

=

==

== oo

. (26)

Thus, the near-field propagation formula converges toFriiss formula in the far field in the horizontal plane.

B. Like Antenna Path-loss function and the Special Case of TX = 0

Fig. 4 shows a plot of PL like as a function of kr for differentvalues of TX . The solid black line represents the special caseof TX = 90 that was previously demonstrated to follow theFriis formula path-loss function for large r . As TX goes to 0 inthe far field, the path-loss function gets smaller but stillfollows the TX = 90 line. This is what one would expect froman ideal dipole with a donut power pattern in the far field.Finally, at TX = 0, the path-loss function is smallest. The TX = 0 line does not follow the TX = 90 line because in thefar field, that corresponds to the null of the antenna, wherethere is ideally no power.

However, the near field tells a different story. The path-lossfunction is largest for small r when TX = 0, which in turn

maximizes the P RX / P TX ratio. For TX = 0, (22) reduces to

+

=

=unlike,0

like,)(

1

)(

123

64,

2

0

kr kr A

k

PP RX nf

TX

RX o

. (27)

Since for TX = 0 the only component present in the fieldequations of an ideal dipole is the radial component, it isconvenient to define a near-field radial component patternfunction of an ideal dipole as

)(cos),( 2, =r nf F . (28)

The near-field radial component directivity can also becalculated

)(cos3

)sin()(cos

)(cos4

)sin(),(

),(4),(

22

0 0

2

2

2

0 0

,

,,

==

=

d d

d d F

F D

r nf

r nf r nf

. (29)

This function is maximum when RX = 0, where thedirectivity is 3. Because an ideal dipole was assumed, the

Fig. 2. Transmit and receive antenna withboth at = 90.

Fig. 3. Dependence on the near-field terms of thepropagation formula.

Fig. 4. Path-loss function for different .


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directivity is equal to the gain. Therefore, the P RX / P TX ratio forthe TX = RX = 0 case can be further defined for the case of ageneral transmitter as

+

=

=unlike,0

like,)(

1

)(

12 64

,,

0

kr kr

GG

PP

TX nf RX nf

TX

RX

o

. (30)

Note that if a non-ideal dipole is assumed, the directivity isnot equal to the gain, but the power transmitted in (14) willalso be multiplied by the antenna efficiency. An analysisaccounting for the antenna efficiency will discover the sameresult of (30).

V. MEASURING FAR -F IELD GAIN OF ELECTRICALLY SMALLANTENNAS IN THE NEAR FIELD

In the case of = 90, the effective areas in the near-fieldpropagation formula are equivalent to the far-field effectiveareas. Therefore, the far-field gain can be measured in the nearfield for this special orientation. Equations (24) and (25) canalso be written in terms of gain as

+=

=642

,90 )(

1

)(

1

)(

14 kr kr kr

GGPP TX RX

likeTX

RX

o . (31)

Assuming that the two antennas are of the same type, (31)can be used to measure gain. Three cases are discussed below.

A. Two Identical AntennasIf the antennas are the same, their gains should be the same.

Therefore, by solving (31) for 2GGG TX RX =

))()()(()(2

)(1

)(1

)(1

4

2464

642

2

kr kr kr PPkr G

kr kr kr P

PG

TX

RX

TX

RX

+=

+

=

. (32)

We used this method to calculate the gain of two Empire(Singer) Model LP-105 loop antennas with known antennafactors. The measured results compared to the expected resultsare shown in Fig. 5. Sources of error in the measurement couldinclude RF coupling through the power (coupling wasnoticeably apparent for electric antennas) and a non free spaceenvironment.

Note in the far-field limit as r gets large, the ( kr )6 willdominate the denominator, and (32) will reduce to

TX

RX

PP

kr G 2= , (33)

which is what Friiss formula also predicts.

B. One Antenna with Unknown GainIf two like antennas are tested and the gain of one is known,

the power transmitted and received can be measured and alsosubstituted into (32). This results in

))()()((

)(4)(

1

)(

1

)(

1

4

246

8

642

kr kr kr GP

r Pkr kr kr

GP

PG

knownTX

RX

knownTX

RX unknown

+=

+

=

. (34)

Again, in the far-field limit as r gets large, the ( kr )6 willdominate the denominator and (34) will reduce to

knownTX

RX unknown GP

Pkr G 2)2(= , (35)

which again is what Friiss formula predicts.

C. Three Like AntennasFor three antennas of the same type (electric or magnetic),

(34) can be used in the same method that Friiss formula isused in the far field to measure antenna gain of three unknownantennas [8]. Measure the power transmitted by each antennaand the power received by each of the other two antennas.This produces three simultaneous equations with threeunknowns (the three gains) that can be solved.

VI. FUNDAMENTAL LIMIT ON ANTENNA GAIN AS A FUNCTIONOF ANTENNA SIZE

To derive the fundamental limit of antenna gain, first theconcept of boundary spheres must be explained. They wereoriginally introduced by Wheeler [9]. One of us [10] hasapplied them to the question of the maximum gain a given

antenna can realize. However, that analysis considered onlythe = 90 case; since the maximum P RX / P TX ratio occurs for = 0, the limit presented in [10] requires revision.

Imagine placing a boundary sphere around an arbitraryantenna with a radius R that is the smallest distance tocompletely enclose the antenna as shown in Fig. 6. Next,imagine a second identical antenna next to the first one alsosurrounded by a boundary sphere. Absent any other sources,the power received by one antenna cannot exceed the powertransmitted by the other in order to comply with the law of conservation of energy. Mathematically,

Fig. 5. Model LP-105 measured gain and expected gain.

-95

-90

-85

-80

-75

600 800 1000 1200 1400

G a

i n ( d B i )

Frequency (kHz)

MeasuredExpected

Fig. 6. Boundary spheres around two arbitrary antennas.


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5

1TX

RX

PP

. (36)

Furthermore, the minimum separation between the antennassuch that neither boundary sphere intersects is

Rr 2= . (37)Take the maximum P RX / P TX ratio as (30). Combining (30),

(36), and (37) results in:

2

6

64

,,

64,,

0

)2(1

)2(2

)2(1

)2(1

2

1)2(

1)2(

12

kR

kR

kRkR

GG

kRkRGG

PP

TX nf RX nf

TX nf RX nf

TX

RX

+=

+

+=

= o

. (38)

Because both antennas are assumed to be identical, their gainsare also identical, so

23

2

6

2,,

)4(1

2)4(

)2(1

)2(2

R

RkR

kRG

GGG

nf

nf TX nf RX nf

+=

+

=

, (39)

where R is taken to mean R in units of wavelength (so R / ).Finally, note that the near-field gain accounts for all

components of the fields, whereas the far-field gain onlyaccounts for two components. Therefore, one would expectthat the near-field gain must be at least greater than or equal tothe far-field gain. Therefore, in terms of the far-field gain

23

2

6

)4(1

2)4(

)2(1

)2(2

R R

kR

kRGG nf ff

+=

+ . (40)

To check the limit suggested above, we measured the gainsof a number of different magnetic antennas and comparedthem against their theoretical limit for their size (see Fig. 7).

For the antennas with data taken at multiple frequencies, wemeasured the power received by an Empire LP-105 loopantenna with a known gain and changed the transmit antenna.We measured the gain of EMCO Model 6509 antenna usingthis method and compared it to its expected values based on itsantenna factor. For the antennas with data at a singlefrequency, we first measured the power transmitted out to aknown distance by the EMCO Model 6509 loop antenna witha known gain. We then transmitted the exact same power foreach antenna at the same distance, and the relative differenceof the power received by this new antenna was added to thegain of the EMCO.

Some antennas seem to perform better than the theoretical

limit would predict. However, sources of error in themeasurements, including RF coupling and non-free-spaceconditions, can account for this discrepancy. Recall that theEmpire LP-105 antennas measured gain was as high as 5 dBoff of the expected value. Accounting for a measurement errorof 5 dB, it is entirely plausible that all of the data fall belowthe expected limit.

VII. CONCLUSION The near fields are an often overlooked aspect of antenna

analysis that can often yield interesting results. For example,the gain limit derived above has profound implications onelectrically small antennas, which are becoming more andmore common as near-field applications are increasing inpopularity. However, we suspect that we are only beginning toscratch the surface on this fascinating topic.

REFERENCES [1] C. Capps, Near Field or Far Field, EDN, August 16, 2001, pp. 95-102.

Available: http://www.edn.com/contents/images/150828.pdf. [AccessedFeb. 1, 2008].

[2] H. Schantz, A Real-Time Location System Using Near-FieldElectromagnetic Ranging, IEEE Antenna and Propagation SocietyInternational Symposium, 2007. Available:http://www.q-track.com/Technology.aspx?ID=25. [Accessed Feb. 1,2008]. See http://www.q-track.com/ for more on NFER.

[3] H. Friis, A Note on a Simple Transmission Formula, Proc. IRE, 34,1946, pp. 254-256.

[4] H. Hertz, Electric Waves , English ed. New York: Dover Publications,1962, pp. 141-151.

[5] C. A. Balanis, Antenna Theory: Analysis and Design , 2nd ed. UnitedStates: John Wiley & Sons, 1997, pp. 133-143.

[6] J. D. Kraus, Antennas , 2nd ed. United States: McGraw-Hill, 1988, pp.201-213.

[7] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design , 2nd ed.United States: John Wiley & Sons, 1998, pp. 20-24.

[8] IEEE Standard Test Procedures for Antennas , IEEE Std 149-1979, NewYork: IEEE, p. 96.

[9] H. A. Wheeler, Fundamental Limitations of Small Antennas, Proc.IRE, 35, 1947, pp. 1479-1484.

[10] H. G. Schantz, A Near-Field Propagation Law & A Novel FundamentalLimit to Antenna Gain Versus Size, IEEE Antenna and Propagation

Society International Symposium, 2005, Vol. 3A, pp. 237-240.Available: http://www.q-track.com/Technology.aspx?ID=25. [AccessedFeb. 1, 2008].

Fig. 7. Measured antenna gains compared to the theoretical limit.

-105

-100

-95

-90

-85

-80

-75

-70

-65

-60

0.0001 0.001

G a i n

( d B i )

R

Theoretical LimitEMCOEMCO ExpectedEmpire LoopEmpire Loop ExpectedTerk AM AdvantageMultiple FrequenciesSingle Frequency

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Fundamental Limit 2008