I 8 Sarnoff Symposium A Fundamental Limit on Antenna ain for letrially Small Antennas Andrew J. Compston, James D. Fluhler, and ans. Shantz Abstrat A fundamental limit on an antenna s gain is derived and ompared to measurements taken on a number of different antennas. First, a propagation formula appliable in both the near and far fields is developed, and that result is used to demonstrate that the gain of an antenna is limited by its eletrial size. Index Terms letrially Small Antennas, Antenna ain, Antenna Measurements, Near Field. T I. INTRODUCTION area around antennas is often split into two areas: the near field, whih extends out from the antenna at a distane omparable to a wavelength, and the far field, whih is the area beyond the near field. See [] for an exellent disussion of the properties of the near field, the far field, and exatly where the boundary between the two lies. For most antenna appliations, espeially those at high frequenies, the behavior of the antenna s eletromagneti fields in the near field is often of little onsequene. For example, a typial Wi- Fi signal operating at. z has a wavelength of about.5 m. Beause most Wi-Fi appliations operate at ranges on the order of meters, this system lends itself well to far-field analysis. This is the ase for almost every eletromagneti system in ommon use today. owever, the near field has some interesting properties that some systems exploit. As we will demonstrate, the power transmitted by a near-field link rolls off muh faster than in the far field, whih means that often times signals will not transmit far enough to ause harmful interferene. In turn, short distane, high data-rate ommuniation is possible using a near-field ommuniation link. Another novel use of the near field is in real-time loation systems (RTLS. The Q-Trak Corporation has pioneered a RTLS tehnology known as Near-Field letromagneti Manusript reeived February, 8; revised April, 8. This material is based upon work supported by the National Siene Foundation under Award Number: 339. Any opinions, findings, and onlusions or reommendations expressed in this publiation are those of the authors and do not neessarily reflet the views of the National Siene Foundation. A. J. Compston is a senior eletrial engineering undergraduate student at the eorgia Institute of Tehnology and is with the Q-Trak Corporation, untsville, AL 358 (e-mail: drew.ompston@gateh.edu. J. D. Fluhler is an eletrial engineering undergraduate student at the University of Alabamauntsville and is with the Q-Trak Corporation, untsville, AL 358 (e-mail: j.fluhler@q-trak.om... Shantz is with the Q-Trak Corporationuntsville, AL 358 (e-mail: h.shantz@q-trak.om. Wi-Fi is a registered trademark of the Wi-Fi Alliane. Ranging (NFR that takes advantage of the fat that the phase differene of a transmitting antenna s eletri and magneti fields goes to zero as the distane from the antenna inreases []. Often near-field appliations like these use low frequenies in order to fully realize the advantages of the near field at substantial distanes. Compare the wavelength of a typial Wi-Fi signal (.5 m with the wavelength of a typial Q-Trak NFR signal ( Mz, 3 m. As a onsequene, most antennas used in near-field systems are muh smaller than a wavelength. This paper exploits near-field phenomena in order to derive a fundamental limit on the gain of an antenna versus its eletrial size. First, we derive a propagation formula similar to Friis s formula for the far field. Using this formula, we show a fundamental limit on the gain of an antenna versus its eletrial size, and we ompare the limit to a number of gain measurements taken on atual antennas. II. FRIIS S ROAATION FORMULA AND T FAR FILD arald Friis derived the propagation formula that bears his name in the following form [3] A A. ( ( d is the power reeived by the reeiving antenna, is the power transmitted by the transmitting antenna, A is the effetive area of the reeive antenna, A is the effetive area of the transmit antenna, d is the distane between eah antenna, and is the wavelength of the eletromagneti wave. Note that the antenna effetive area or aperture A is related to the antenna gain by A, ( π so Friis s formula an also be written as A A. (3 πd πd πd This is a very powerful formula, but beause it assumes a plane wave front, it is only appliable in the far field. In his paper, Friis warns that his formula is orret to within a few perent when a d, ( NFR is a registered trademark of the Q-Trak Corporation. Authorized liensed use limited to: rineton University. Downloaded on Otober, 9 at 5:7 from I Xplore. Restritions apply.
I 8 Sarnoff Symposium where a is the largest linear dimension of either of the antennas [3]. Speifially, [t]his riterion has a phase error of one-sixteenth of a wavelength []. Also, Friis s formula is only stritly valid in free spae. III. NAR-FILD ROAATION 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 eletri antenna (like a dipole or whip and its dual: a magneti antenna (suh as a loop. First the ase of an eletri transmit antenna is onsidered, followed by the ase of a magneti antenna. A. letri Transmit Antenna Imagine an infinitesimal urrent element with length l << in free spae. Suppose the urrent element has a uniform urrent distribution I aross its length. This struture, first introdued by einrih ertz [], has sine been analyzed by a number of authors [5-7]. A theoretial idealization, it is an aurate model for the eletrially small antennas being onsidered. The time harmoni eletri field at the point (r, ϕ generated by the infinitesimal dipole is given by r j kr sin( θ ˆ θ πε 3 ( (, (5 r e os( θ ˆ 3 ( ( where k π / is the wave number, ε is the eletri permittivity of free spae, and is the speed of light. The magneti field is r e sin( θ ˆ φ. ( π ( The magnitudes squared of the fields are r sin ( θ πε, and (7 os ( θ sin ( θ π ( ( r. (8 For the sake of ompatness and for reasons that will be learer momentarily, define the path-loss funtions L like and L as L like ( r, φ sin ( θ, and (9 os ( θ L ( r, φ sin ( θ ( (. ( whih means that the eletri and magneti fields an be rewritten as L like r r π, and ( L. ( ε π Define the power densities of the eletri and magneti fields, respetively as W and W, r ( W ε L like, and ( πε W r µ ( µ L. (3 π where µ is the magneti permeability of free spae. Beause the power transmitted by an ideal dipole through a losed sphere around the antenna as pitured in Fig. is given by S r r ( r ds, ( πε the power densities ( and (3 an be rewritten as ( W Llike πε W L like µ ( L µ ε π L, and (5 L. ( The power density at the reeive antenna is equal to the power it reeives divided by its effetive area, so W Llike Anf, Llike. (7 A nf, Fig.. Infinitesimal dipole transmitting with fields at (r, φ. Similarly for a magneti reeive antenna L Anf L. (8 A W, nf, It is important to realize that A nf, is not the effetive area in the traditional far-field sense beause the far-field effetive area assumes a plane wave front. This effetive area must also aount for both the near- and far-field omponents (i.e., not just the θ- and ϕ- but also the r-omponent that is ignored in far-field analyses and will therefore neessarily be different in almost all ases. B. Magneti Transmit Antenna Using the priniple of duality, the eletri and magneti fields of an infinitesimal urrent loop an be written as r 3 I Sk e sin( θ ˆ φ, and (9 πε ( Authorized liensed use limited to: rineton University. Downloaded on Otober, 9 at 5:7 from I Xplore. Restritions apply.
I 8 Sarnoff Symposium 3 Fig.. Transmit and reeive antenna with both at θ 9. r 3 I Sk j kr sin( θ ˆ θ 3 π ( (. ( j r e os( θ ˆ 3 ( ( where S is the surfae area of the loop. Using the same proedure outlined above for the eletri transmitter, the following an be proven for a magneti transmit antenna Anf, Llike, and Anf, L. ( Thus, the near-field propagation formula an be summarized as below (, φ A, L ( r, φ, like antennas r nf like, ( Anf, L ( r, φ, antennas where like antennas are taken to mean either two eletri or two magneti antennas and antennas are taken to mean one eletri and one magneti antenna. IV. COMARIN T NAR-FILD ROAATION FORMULA WIT FRIIS S FORMULA A. The Speial Case of θ 9 For the speial ase where both antennas are at θ 9 (z, the horizontal plane relative to eah other (as in Fig., see in (5 and ( that the eletri and magneti fields have no radial omponents. Therefore, the near-field effetive areas of the antennas are equivalent to their far-field effetive areas. For an infinitesimal dipole, this is 3 3π Anf,inf Anf, A, (3 k whih further redues ( to: k A A, and ( o π ( kr θ 9, like k A A π ( kr o θ 9, Fig. 3. Dependene on the near-field terms of the propagation formula.. (5 For large r, the /r and /r terms will go to zero muh faster than the /r terms, and they an be ignored for all pratial purposes, as demonstrated in Fig. 3. This leaves k A A A A o r ( r 9, like o π θ θ 9,. ( Thus, the near-field propagation formula onverges to Friis s formula in the far field in the horizontal plane. B. Like Antenna ath-loss funtion and the Speial Case of θ Fig. shows a plot of L like as a funtion of kr for different values of θ. The solid blak line represents the speial ase of θ 9 that was previously demonstrated to follow the Friis formula path-loss funtion for large r. As θ goes to in the far field, the path-loss funtion gets smaller but still follows the θ 9 line. This is what one would expet from an ideal dipole with a donut power pattern in the far field. Finally, at θ, the path-loss funtion is smallest. The θ line does not follow the θ 9 line beause in the far field, that orresponds to the null of the antenna, where there is ideally no power. owever, the near field tells a different story. The path-loss funtion is largest for small r when θ, whih in turn maximizes the / ratio. For θ, ( redues to A, like nf, π. (7 o θ, Sine for θ the only omponent present in the field equations of an ideal dipole is the radial omponent, it is onvenient to define a near-field radial omponent pattern funtion of an ideal dipole as θ F ( θ, φ os (. (8 The near-field radial omponent diretivity an also be alulated Fnf, r ( φ D ( φ π π Fig.. ath-loss funtion for different θ. π π π π F ( θ, φsin( θ dθdφ os ( θ os ( θ sin( θ dθdφ 3os ( θ. (9 This funtion is maximum when θ, where the diretivity is 3. Beause an ideal dipole was assumed, the Authorized liensed use limited to: rineton University. Downloaded on Otober, 9 at 5:7 from I Xplore. Restritions apply.
I 8 Sarnoff Symposium -75-8 ain (dbi -85-9 -95 diretivity is equal to the gain. Therefore, the / ratio for the θ θ ase an be further defined for the ase of a general transmitter as nf, nf,, like. (3 o θ, Note that if a non-ideal dipole is assumed, the diretivity is not equal to the gain, but the power transmitted in ( will also be multiplied by the antenna effiieny. An analysis aounting for the antenna effiieny will disover the same result of (3. V. MASURIN FAR-FILD AIN OF LCTRICALLY SMALL ANTNNAS IN T NAR FILD In the ase of θ 9, the effetive areas in the near-field propagation formula are equivalent to the far-field effetive areas. Therefore, the far-field gain an be measured in the near field for this speial orientation. quations ( and (5 an also be written in terms of gain as. (3 o ( kr θ 9, like Assuming that the two antennas are of the same type, (3 an be used to measure gain. Three ases are disussed below. A. Two Idential Antennas If the antennas are the same, their gains should be the same. Therefore, by solving (3 for. (3 ( We used this method to alulate the gain of two mpire (Singer Model L-5 loop antennas with known antenna fators. The measured results ompared to the expeted results are shown in Fig. 5. Soures of error in the measurement ould inlude RF oupling through the power (oupling was notieably apparent for eletri antennas and a non free spae environment. Note in the far-field limit as r gets large, the (kr will dominate the denominator, and (3 will redue to Measured xpeted 8 Frequeny (kz Fig. 5. Model L-5 measured gain and expeted gain. Fig.. Boundary spheres around two arbitrary antennas. kr, (33 whih is what Friis s formula also predits. B. One Antenna with Unknown ain If two like antennas are tested and the gain of one is known, the power transmitted and reeived an be measured and also substituted into (3. This results in unknown known. (3 known 8 ( βr ( Again, in the far-field limit as r gets large, the (kr will dominate the denominator and (3 will redue to unknown (kr, (35 known whih again is what Friis s formula predits. C. Three Like Antennas For three antennas of the same type (eletri or magneti, (3 an be used in the same method that Friis s formula is used in the far field to measure antenna gain of three unknown antennas [8]. Measure the power transmitted by eah antenna and the power reeived by eah of the other two antennas. This produes three simultaneous equations with three unknowns (the three gains that an be solved. VI. FUNDAMNTAL LIMIT ON ANTNNA AIN AS A FUNCTION OF ANTNNA SIZ To derive the fundamental limit of antenna gain, first the onept of boundary spheres must be explained. They were originally introdued by Wheeler [9]. One of us [] has applied them to the question of the maximum gain a given antenna an realize. owever, that analysis onsidered only the θ 9 ase; sine the maximum / ratio ours for θ, the limit presented in [] requires revision. Imagine plaing a boundary sphere around an arbitrary antenna with a radius R that is the smallest distane to ompletely enlose the antenna as shown in Fig.. Next, imagine a seond idential antenna next to the first one also surrounded by a boundary sphere. Absent any other soures, the power reeived by one antenna annot exeed the power transmitted by the other in order to omply with the law of onservation of energy. Mathematially, Authorized liensed use limited to: rineton University. Downloaded on Otober, 9 at 5:7 from I Xplore. Restritions apply.
I 8 Sarnoff Symposium 5. (3 Furthermore, the minimum separation between the antennas suh that neither boundary sphere intersets is r R. (37 Take the maximum / ratio as (3. Combining (3, (3, and (37 results in: nf, nf, (kr (kr θ o (kr. (38 nf, nf, (kr (kr (kr Beause both antennas are assumed to be idential, their gains are also idential, so nf, nf nf, nf (kr (kr (πr 3 (πr, (39 where R is taken to mean R in units of wavelength (so R/. Finally, note that the near-field gain aounts for all omponents of the fields, whereas the far-field gain only aounts for two omponents. Therefore, one would expet that the near-field gain must be at least greater than or equal to the far-field gain. Therefore, in terms of the far-field gain ff (kr 3 nf (πr. ( (kr (πr To hek the limit suggested above, we measured the gains of a number of different magneti antennas and ompared them against their theoretial limit for their size (see Fig. 7. For the antennas with data taken at multiple frequenies, we measured the power reeived by an mpire L-5 loop antenna with a known gain and hanged the transmit antenna. We measured the gain of MCO Model 59 antenna using this method and ompared it to its expeted values based on its antenna fator. For the antennas with data at a single frequeny, we first measured the power transmitted out to a known distane by the MCO Model 59 loop antenna with a known gain. We then transmitted the exat same power for eah antenna at the same distane, and the relative differene of the power reeived by this new antenna was added to the gain of the MCO. Some antennas seem to perform better than the theoretial limit would predit. owever, soures of error in the measurements, inluding RF oupling and non-free-spae onditions, an aount for this disrepany. Reall that the mpire L-5 antenna s measured gain was as high as 5 db off of the expeted value. Aounting for a measurement error of 5 db, it is entirely plausible that all of the data fall below the expeted limit. ain (dbi - -5-7 -75-8 -85-9 -95 - -5 Fig. 7. Measured antenna gains ompared to the theoretial limit. analysis that an often yield interesting results. For example, the gain limit derived above has profound impliations on eletrially small antennas, whih are beoming more and more ommon as near-field appliations are inreasing in popularity. owever, we suspet that we are only beginning to srath the surfae on this fasinating topi. RFRNCS Theoretial Limit MCO MCO xpeted mpire Loop mpire Loop xpeted Terk AM Advantage Multiple Frequenies Single Frequeny. R. [] C. Capps, Near Field or Far Field, DN, August,, pp. 95-. Available: http://www.edn.om/ontents/images/588.pdf. [Aessed Feb., 8]. []. Shantz, A Real-Time Loation System Using Near-Field letromagneti Ranging, I Antenna and ropagation Soiety International Symposium, 7. Available: http://www.q-trak.om/tehnology.aspx?id5. [Aessed Feb., 8]. See http://www.q-trak.om/ for more on NFR. [3]. Friis, A Note on a Simple Transmission Formula, ro. IR, 3, 9, pp. 5-5. []. ertzletri Wavesnglish ed. New York: Dover ubliations, 9, pp. -5. [5] C. A. Balanis, Antenna Theory: Analysis and Design, nd ed. United States: John Wiley & Sons, 997, pp. 33-3. [] J. D. Kraus, Antennas, nd ed. United States: Mraw-ill, 988, pp. -3. [7] W. L. Stutzman and. A. Thiele, Antenna Theory and Design, nd ed. United States: John Wiley & Sons, 998, pp. -. [8] I Standard Test roedures for Antennas, I Std 9-979, New York: I, p. 9. [9]. A. Wheeler, Fundamental Limitations of Small Antennas, ro. IR, 35, 97, pp. 79-8. [].. Shantz, A Near-Field ropagation Law & A Novel Fundamental Limit to Antenna ain Versus Size, I Antenna and ropagation Soiety International Symposium, 5, Vol. 3A, pp. 37-. Available: http://www.q-trak.om/tehnology.aspx?id5. [Aessed Feb., 8]. VII. CONCLUSION The near fields are an often overlooked aspet of antenna Authorized liensed use limited to: rineton University. Downloaded on Otober, 9 at 5:7 from I Xplore. Restritions apply.