A Performance Comparisonof Fundamental

Small-Antenna Designs

INDIAN INSTITUTE OF TECHNOLOGY KANPUR

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Arun P (12459), Saurabh Kataria (12637),Saurav Prakash (12642), and Vikrant Kumar (12804)

April 13, 2015

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I. ABSTRACT

Small antennas have been an important topic of research for many decades, and interest in the field is increasingwith the development of new systems that require broadband antennas with a small form factor. The analysis ofsmall antennas is generally considered to have begun with the establishment of the theoretical limits that show howelectrical size and bandwidth are related. So Optimization of the performance properties of electrically small antennashas become very important and it has been given considerable attention in recent years.

II. INTRODUCTION

In the design of any small antenna, the performance-optimization goals include achieving an impedance match (lowVSWR), high radiation efficiency, and a low quality factor (Q) or wide operating bandwidth. These are performancematrices that one like to optimize as there are trade -offs between them. Ideally we want to design the antenna in sucha way that VSWR approaches 1:1, and the radiation efficiency approaches 100% over the desired bandwidth. Theobjective is to determine which basic design approach or configuration, if any, offers the best performance in termsof: achieving an impedance match; the radiation-pattern shape; the radiation efficiency; the half-power bandwidth;and the 2:1 VSWR bandwidth.

III. MOTIVATION

Any electrically small antenna can be impedance matched to achieve a low VSWR at any single frequency using anexternal matching network comprised of lossy reactive components. But one challenge in using an external matchingnetwork is that the loss resistances within the matching-network components often exceed the radiation resistance ofthe antenna, resulting in low overall efficiency. So instead of using external matching network we try to modify thestructure of the Antenna itself to achieve the desired performance. The fundamental approaches in modification of theantenna’s structure include the use of capacity or top-hat loading, the use of multiple folded arms in wire monopoleor dipole antennas , inductive loading in wire antennas (an increase in conductor length), and, more recently, the useof metamaterials and multi-arm coupled resonators.

IV. LIMITATIONS ON QUALITY FACTOR AND BANDWIDTH

The most significant challenge in designing a small antenna is optimizing the operating bandwidth, which is oftencharacterized using the antenna’s quality factor (Q). The frequency-dependent impedance of the small antenna isgiven by Z(ω) = R(ω) + jX(ω),where ω is the radian frequency, R(ω) is the antenna’s total feed point resistance(including both radiation and loss terms), and X(ω) is the antenna’s feed-point reactance. The bandwidth and Qof the small antenna are defined at radiation frequency ω0, where the antenna is either self-resonant, or made tobe self-resonant, or tuned to resonance using a lossless series reactance. The small antenna’s bandwidth is oftencharacterized by its Q, because Q and matched VSWR bandwidth have been shown to be inversely related, and alower bound on the minimum achievable Q is defined and well known. If the tuned small antenna exhibits a singleresonance within its defined matched VSWR bandwidth, its Q can be accurately approximated from its impedanceproperties using the following relationship

Q(ω0) ≈ω0

2R(ω0)

√R′(ω0)2 +

[X ′(ω0) +

|X(ω0)|ω0

]2(1)

where R′(ω0) and X ′(ω0) are the frequency derivatives of the antenna’s feed-point resistance and reactance, respec-tively. The lower bound on Q is given by

Qlb = ηr

[1

ka3+

1

ka

](2)

where ηr is the antenna’s frequency-dependent radiation efficiency, k is the wavenumber and a is the radius of animaginary sphere circumscribing the maximum dimension of the antenna.

For the characterization of antennas bandwidth, we have to define bandwidth in such a way that the trade-offbetween VSWR and Q is visible, in other words they have to be inversely related. So we define the bandwidth asfractional matched VSWR bandwidth, FBWV as the ratio of the difference between two frequencies and resonancefrequency.

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The fractional matched VSWR bandwidth and the Q are related by the following equation

Q(ω0) ≈2√β

FBWV (ω0);

√β =

s− 1

2√s≤ 1 (3)

Using (2) and (3), an upper bound on the fractional matched VSWR bandwidth can be written as

FBWV ub =1

ηr

(ka)3

1 + (ka)2s− 1√s

(4)

So we have limitations on both quality factor Q as well as bandwidth.

V. THE FUNDAMENTAL DESIGN APPROACHES

The impedance of small straight-wire dipoles and monopoles is given by ZA(ω) = RA(ω) − jXA(ω) where thevalue of XA(ω) is dominated by the equivalent value of 1/ωC established by the antenna’s structure. So to makethe antenna self-resonant we need to nullify the dominant term by changing the antenna structure. Structure can bemodified to include either a combination of a capacitive top hat or inductive loading (increasing wire length), asillustrated in Fig. 1.

Fig. 1. Depictions of the capacitive top-hat-loaded monopole and the inductively loaded monopole.Both antennas had the same overallheight(8.48 cm) and the same resonant frequency (≈ 300 MHz).

From the perspective of comparing the advantages of one design approach relative to the other, we need tocompare the relative performance of the antennas as a function of ka, or, more appropriately, how close the antenna’sQ approaches the lower bound, Qlb. The relative comparison as a function of Q/Qlb indicated that the top-hatmonopole performed substantially better than the meander-line antenna. This is the general perception in the designof small antennas: top-loaded antennas perform better then inductively loaded antennas. But we note that the boththe designs have not efficiently utilized the full spherical volume that is available based on the definition of ka. Inthe following sections we propose structural changes in the design of antenna keeping the same in focus that is wewould attempt to utilize the entire spherical volume to optimize the performance properties specially Q/Qlb ratio.

VI. SMALL ANTENNA DESIGNS

Steven Best and Drayton L. Hanna [1] have done a comprehensive comparison of the performances of some of thefundamental antenna designs. We studied four of them: the folded spherical helix, the folded cylindrical helix, thedisk-loaded dipole, and the spherical-cap dipole. The various properties being considered are antenna’s impedance,radiation efficiency, pattern characteristics, matched VSWR bandwidth, and quality factor Q.

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Fig. 2. One-turn, four-arm folded spherical helix antenna.

A. The Folded Spherical HelixWith the inductively loaded antennas, achieving the minimum possible Q required that the wire comprising the

antenna structure be wound on the outside of the available volume. For a spherical-shaped antenna, this required thewire to be wound on the outside of the required ka sphere. Achieving a high radiation efficiency and a lower Qrequired that the wire diameter be sufficiently large enough so that the radiation resistance was substantially largerthan the loss resistance. At the same time, use of a larger wire diameter usually results in a higher resonant frequency,and it will impose physical constraints as to how much wire length can fit within a given volume or surface area.

With a single-arm dipole winding on the outside of the spherical volume defined by ka, the antenna could bemade self- resonant with an appropriate choice of wire length. However, the value of the resonant resistance wouldbe much less than 50 Ω, as was the case with the meander-line antenna discussed in the previous section. To increasethe radiation resistance of the antenna, three additional folded arms were added to the structure, as described in [2],[3] (see Fig. 2).

The four-arm folded spherical helix was specifically designed to have a low VSWR in a 50 Ω system, high radiationefficiency, and a low Q. It exhibited a single resonance at Q that was within 1.5 times the lower bound at a value ofka ≈ 0.263 . The folded spherical helix was considered the baseline antenna for this study. The antennas comparedto the folded spherical helix were designed to have the same length, the same value of ka, or to operate at (or verynear) the same frequency. The design objective was to operate all of the antennas as close to 300 MHz as possible.

The folded spherical helix (wire-grid version), its impedance, matched VSWR (ZCH = 47.5Ω), and resonantradiation patterns are presented in Fig. 3(a), 3(b), 3(c). The value of ZCH was chosen to match the antenna’sresonant resistance.

B. The Folded Cylindrical HelixThe design concept of the folded spherical helix can be applied to any volumetric shape: a sphere, cube, cylinder,

etc. In this section, we consider a multiple folded-arm dipole design that fits within a cylindrical volume. A wiregrid version of the spherical cap dipole type antenna is illustrated in Fig. 4(a).

For the cylindrical folded helix, the maximum diameter and length were equal to the diameter of the folded sphericalhelix considered in the previous section. The folded cylindrical helix had the same overall length and diameter asthe folded spherical helix, 8.36 cm. The conductor diameter was the same , at 2.6 mm. The total wire length in eachdipole arm was 60.2 cm, just slightly less than that of the spherical helix. The cylindrical helix occupied a largerphysical volume, and therefore had a larger value of ka than the spherical helix, it was expected to have greaterbandwidth and lower Q. However, it occupied less of the spherical volume defined by its value of ka, therefore itsQ would not approach its associated lower bound as closely as that of the spherical helix. The cylindrical foldedhelix was self-resonant at 301.1 MHz, with a value of ka = 0.373. The resonant resistance of the folded cylindrical

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(a) The four-arm folded spherical helix. The folded spherical helix had anoverall diameter of 8.36 cm and was comprised of four one-and-one-half-turn arms, each having a length of 63.6 cm. The folded spherical helixwas resonant at 299.8 MHz.

(b) The impedance of the four-arm folded spherical helix over a frequencyrange of 270 to 330 MHz.

(c) The matched VSWR of the four-arm folded spherical helix.

Fig. 3.

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(a) The four-arm folded cylindrical helix. The folded cylindrical helix hadan overall diameter of 8.36 cm and was comprised of four one-turn arms,each having a length of 60.2 cm. The cylindrical folded helix was resonantat 301.1 MHz.

(b) The impedance of the four-arm folded cylindrical helix over afrequency range of 270 to 330 MHz.

(c) The matched VSWR of the four-arm folded cylindrical helix.

Fig. 4.

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helix (83.6 Ω) was greater than that of the spherical helix. The Q of the folded spherical helix was 53.5, a value lessthan that of the spherical helix, as expected. However, it was approximately 2.5 times the lower bound of 21.35. Itsradiation efficiency was 98.3%. Its half-power and 2:1 VSWR bandwidths were 3.75% and 1.33%, respectively, bothgreater than those of the spherical helix.

C. The Disk-Loaded DipoleFor the folded-helix antennas (including both spherical and cylindrical), self-resonance was achieved by increasing

the total wire length in each arm. For antennas of disk-loaded dipole type, lowest Q is achieved when conductoris wound on the outermost surface of available volume, and the impedance match is achieved through the use ofmultiple folded arms. The self-resonance in the reduced-length dipole is achieved by capacitive or disk loading atthe end of the conductors [4]. End-loading the dipole increased the capacitance between the upper and lower dipolearms, resulting in a decrease in the magnitude of the feed-point capacitive reactance, −j/ωC. With an appropriatevalue of capacitance determined by the disk diameter, the small dipole could be made self-resonant.

The solid top-disk version of the antenna (as shown in Fig. 5) has less total feed-point capacitive reactance, sincethere was more capacitance between the upper and lower disks. Without additional tuning, neither antenna wasresonant near 300 MHz, since the limited diameter of the disk did not provide sufficient capacitance to achieve self-resonance at this frequency. The feed point impedance of wire-grid loaded-dipole antenna and sold-disk loaded-dipoleantenna is shown in Fig. 6(a) and 6(b) respectively.

Fig. 5. The solid-disk loaded dipole antenna.

D. The Spherical-Cap DipoleSpherical-cap dipole, like the disk-loaded dipole antenna, has a top-loaded structure. A wire grid version of the

spherical cap dipole type antenna is illustrated in Fig. 7. The spherical cap of this type of antenna is a part of theimaginary sphere that encloses the maximum dimension of the antenna i.e. within the spherical volume defined bythe value of ka. In [5], Wheeler described the spherical-cap dipole in these terms: ”the best simple utilization of aspherical volume is achieved by spherical caps covering about one-half the surface, tuned by a distributed inductor”.The distributed inductor Wheeler referred to is the helical-coil dipole that was used to tune the disk-loaded dipole,as described before.

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(a) The impedance of the wire-grid loaded-dipole antenna. The an-tennas was not self-resonant at 300 MHz because the disk diameterwas not large enough to provide sufficient parallel capacitance tomake the antenna self-resonant at this frequency.

(b) The impedance of the solid disk-loaded loaded dipole antenna.The antennas was not self-resonant at 300 MHz because thedisk diameter was not large enough to provide sufficient parallelcapacitance to make the antenna self-resonant at this frequency.

Fig. 6.

Fig. 7. The spherical cap dipole. Fig. 8. Impedance of the spherical cap dipole over a frequency rangeof 270 to 330 MHz.

In [6], Lopez modeled an untuned spherical-cap dipole and achieved a Q that was within 1.75 times the lowerbound. Lopez concluded that this was the lowest achievable bound for electric antennas. However, the folded sphericalhelix more closely approaches the lower bound, being within approximately 1.5 times its value. The black circle inFig. 8 shows the impedance of the spherical cap dipole over a frequency range of 270 to 330 MHz.

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REFERENCES

[1] S. R. Best and D. L. Hanna, “A performance comparison of fundamental small-antenna designs,” Antennas and Propagation Magazine,IEEE, vol. 52, no. 1, pp. 47–70, 2010.

[2] S. R. Best, “The radiation properties of electrically small folded spherical helix antennas,” Antennas and Propagation, IEEE Transactionson, vol. 52, no. 4, pp. 953–960, 2004.

[3] B. Steven R, “Low Q electrically small linear and elliptical polarized spherical dipole antennas,” Antennas and Propagation, IEEETransactions on, vol. 53, no. 3, pp. 1047–1053, 2005.

[4] K. Fujimoto, Small antennas. Wiley Online Library, 1987.[5] H. Wheeler, “Antenna topics in my experience,” Antennas and Propagation, IEEE Transactions on, vol. 33, no. 2, pp. 144–151, 1985.[6] A. R. Lopez, “Fundamental hmrations of small antennas: validation of wheeler’s formulas,” Antennas and Propagation Magazine, IEEE,

vol. 48, no. 4, pp. 28–36, 2006.