1

Optimal Planar Electric Dipole AntennasMiloslav Capek, Senior Member, IEEE, Lukas Jelinek, Kurt Schab, Member, IEEE,

Mats Gustafsson, Senior Member, IEEE, B. L. G. Jonsson, Fabien Ferrero, and Casimir Ehrenborg, StudentMember, IEEE

Abstract—Considerable time is often spent optimizing antennasto meet specific design metrics. Rarely, however, are the resultingantenna designs compared to rigorous physical bounds on thosemetrics. Here we study the performance of optimized planarmeander line antennas with respect to such bounds. Results showthat these simple structures meet the lower bound on radiationQ-factor (maximizing single resonance fractional bandwidth),but are far from reaching the associated physical bounds onefficiency. The relative performance of other canonical antennadesigns is compared in similar ways, and the quantitative resultsare connected to intuitions from small antenna design, physicalbounds, and matching network design.

Index Terms—Antenna theory, optimization methods, numer-ical methods, Q-factor, efficiency.

I. INTRODUCTION

ANTENNA parameters such as gain, Q-factor, and effi-ciency are limited by the geometry made available for a

given design. Given bounds on these parameters under certainconstraints, a designer can rapidly assess the feasibility ofdesign requirements. This feasibility assumes the existence ofan “optimal antenna” design which approaches the bounds oncertain specified parameters. Synthesis of an optimal antennais not a trivial task, and it remains to be demonstrated howan antenna designed to be optimal in one parameter (e.g.,radiation Q-factor) performs relative to bounds on other pa-rameters (e.g., efficiency). The goal of this paper is to discussthe synthesis and analysis of optimal antennas starting fromclassical antenna topologies.

Many strategies have been employed to optimize antennas.Heuristic optimization methods such as genetic algorithms [1],[2] and particle swarm optimization have the advantage ofgenerating design geometries outside of the antenna designer’s

M. Capek and L. Jelinek are with the Department of ElectromagneticField, Faculty of Electrical Engineering, Czech Technical University inPrague, Technicka 2, 16627, Prague, Czech Republic (e-mail: miloslav.capek,lukas.jelinek@fel.cvut.cz).

K. Schab is with the Electrical Engineering Department, Santa ClaraUniversity, 500 El Camino Real, Santa Clara, CA 95053 (e-mail:kschab@scu.edu).

M. Gustafsson and C. Ehrenborg are with the Department of Electricaland Information Technology, Lund University, Box 118, SE-221 00 Lund,Sweden. (Email: mats.gustafsson, casimir.ehrenborg@eit.lth.se).

B. L. G. Jonsson is with the School of Electrical Engineering and ComputerScience, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden.(Email: ljonsson@kth.se).

F. Ferrero is with the Laboratory of Electronics, Antennas and Telecom-munications, CNRS, Universite Cote d’Azur, 06903 Sophia Antipolis, France.(e-mail: fabien.ferrero@unice.fr).

This work was supported by the Swedish Foundation for Strategic Research(SSF), grant no. AM13-0011, by VINNOVA through ChaseOn/iAA and bythe Czech Science Foundation under project No. 19-06049S. The work ofM. Capek was supported by the Ministry of Education, Youth and Sportsthrough the project CZ.02.2.69/0.0/0.0/16 027/0008465.

usual catalog [3]–[5]. Such techniques have been used todesign optimal antennas with radiation Q-factors very close tothe physical bounds [6], though the resulting designs are com-putationally expensive to produce and offer only rough insightinto guidelines for designing optimal antennas in volumeswith arbitrary shapes and electrical size. Conversely, canonicalantenna designs were shown [7], [8] to reach the lower boundon radiation Q-factor, but the question remains whether thesedesigns represent optimal solutions over arbitrary electricalsizes and whether they are optimal in other parameters, e.g.,radiation efficiency and input impedance. The cost of matchingan optimal antenna design to arbitrary impedances is alsounclear, regardless if matching is performed on the antennaitself or through external networks.

In this paper we study whether there exists a simple“recipe” for an optimal planar antenna (Throughout this paper,the term planar means to lie in a plane.) with respect toradiation Q-factor and radiation efficiency. In doing so, weask whether, when prescribed with some form factor andelectrical size, a simple design can be readily employed toachieve an antenna whose properties are sufficiently closeto their bounds. The strategy adopted here is to optimizeparameterizations of canonical antenna geometries known forgood behavior in certain parameters. The examples studiedhere give quantifiable results to this end, i.e., how to designcertain kinds of optimal antennas.

Along the way, we address the crossover of optimalityof antennas across different performance parameters, e.g., dominimum radiation Q-factor antennas have inherently highradiation efficiency? Also discussed are the impacts of certainconstraints, particularly those related to an antenna’s inputimpedance, on optimized parameters.

We stress out that this work differs significantly from otherworks on antenna optimization through parametric, heuristic,or metaheuristic means which typically involves the iterativeevaluation and modification of designs until a local optimumor design goal is reached. Here, instead, we focus on designingantenna performance with respect to physical bounds, whichprovide an absolute measure in judging the quality of thesynthesized design.

II. MINIMUM RADIATION Q-FACTOR OF PLANAR TMANTENNAS

We begin by studying the synthesis of electricallysmall dipole-like (TM) antennas with minimal radiation Q-factor Qrad (see Box 1 and Box 2). This leads to increasedimpedance bandwidth, however, the lower bound on radiationQ-factor increases rapidly as an antenna design region be-comes smaller (see Box 2). Thus, obtaining low Q-factor Qrad

arX

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9

2

Box 1. Q-factor

The Q-factor of an antenna system tuned to resonance(equality of mean magnetic and mean electric energy) isdefined as [9]

Q =2ωWsto

Prad + Pdiss, (1)

where Wsto represents the cycle mean stored energy,while Prad and Pdiss denote cycle mean radiated and dis-sipated powers, respectively. In single-resonance systems,lower Q-factor implies larger fractional impedance band-width B by an inverse relationship [10]–[12]

B ∼ Q−1. (2)

Evaluated at a single frequency via (1), Q-factor thusbecomes a convenient measure of the frequency selectivityof a system [10]–[22]. Calculation of a system’s Q-factorcan be carried out by a variety of approaches, fromimpedance-based techniques [12] to methods based onthe evaluation of stored energy directly [6], [17]. All ofthese approaches generally agree for electrically small,narrow-band antennas, see [23] for complete discussion andbibliography.

The radiation Q-factor Qrad, in which only radiated poweris considered, can be expressed in terms of Q in (1) andradiation efficiency η (see Box 3, (5)) as

Qrad = Q/η ∼ (Bη)−1. (3)

is a key objective and challenge in the design of electricallysmall antennas.

A. Synthesis of meander line antennas

Drawing from the prevalence of meander line antennas inapplications requiring electrically small planar antennas [24],[40], as well as previous work studying their optimality inradiation Q-factor [8], we focus on determining whether me-ander lines present a consistent, simple solution, to obtainingminimum radiation Q-factor at arbitrary frequencies withinrectangular design regions. Here, and throughout Section III,we specify a rectangular design region of fixed aspect ratio(L/W = 2). The impact of varying aspect ratios is demon-strated and discussed in Section II-B.

From the many possible meander line shapes (for example,rectangular, triangular, sinusoidal [40]) we have chosen thesimple parametrization from Fig. 1. Thin wire versions ofsuch antennas were previously shown to reach the lower boundon radiation Q-factor Qrad for their corresponding rectangulardesign regions with electrical sizes near ka = 0.3 [8]. Here,we use the parameterization in Fig. 1 to optimize the meanderline antenna for resonance by requiring the magnitude ofthe normalized input reactance Xin/Rin to be smaller thana specified tolerance, |Xin/Rin| < 10−3. This procedure isrepeated at many frequencies (electrical sizes, values of ka)to obtain a set of antenna designs, each resonant at a specificfrequency. The Q-factors Qrad of the resulting designs werethen calculated in AToM [41] and compared to the boundsdiscussed in Box 2. The comparison is shown in Fig. 2. Notethat the value of Q-factor Qrad is just weakly dependent on

Box 2. Lower bounds on radiation Q-factor

The approximate inverse proportionality between theQ-factor and the fractional bandwidth (see Box 1)induced considerable effort in lowering the Q-factor forspherical [10], [13], [24]–[27] and arbitrarily shapedantennas [28]–[34]. The sole focus on the radiation Q-factor Qrad in these works avoids the undesired possibilityof reducing Q-factor Q by degrading radiation efficiency.

For electrically small antennas, the lower bound on the Qrad,here denoted as Qlb

rad, is a combination of electric andmagnetic dipoles [10], [26], [32], [35], [36]. In general,it is challenging to excite such a current with a singlefeeding position see [37, Sec. IV] for related discussion. Aconstrained minimization which is more representative forsingle port antennas is to restrict the radiation to TM (electricdipole) modes, yielding the lower bound Qlb,TM

rad , see, e.g.,[38].The importance of Q-factor bounds arises from two keyproperties. First, the Q-factor bound represents the physicallower bound among all possible currents contained withinthe considered region. It thus presents an absolute measureagainst which to compare the performance of differentantenna designs. Practical feasibility of designing antennaswhich reach various bounds remains an open question. Sec-ond, both Qlb

rad and Qlb,TMrad scale approximately as (ka)−3

for electrically small antennas (ka < 1), cf. [39]. Here k isthe free-space wavenumber and a is the radius of the smallestcircumscribing sphere. This (ka)−3 scaling is the root causeof the limited bandwidth in electrically small antennas. Theassociated geometry coefficients for certain shapes are shownin Table I, where η0 denotes free-space impedance and Rs

denotes surface resistance.

dissipation factor (see Box 3) provided that dissipation is notexceedingly high.

In order to verify the computed data in Fig. 2, an antennadesign sample with ka = 0.42 was scaled to 1.4 GHz andfabricated on a 25µm-thick Polyimide film with 17µm-thickcopper foil bonded by an 25µm-thick acrylic adhesive (εr =4). Thanks to the very thin profile of the substrate as comparedto wavelength, the effect of dielectric can be neglected fora radiation Q-factor evaluation. The input impedance of theprototype was measured using a differential technique [42]and has been used to estimate the Q-factor via the QZ for-mula [12]. Radiation efficiency of the antenna was measuredvia a multiport near-field method [43] and was used to evaluateradiation Q-factor, and its confidence interval of width equalto two times the standard deviation, see triangular marker andcorresponding error bar in Fig. 2.

Figure 2 illustrates that a simple parametrization, such asthe one from Fig. 1, is able to closely approach the radiationQ-factor bound limited to TM radiation Qlb,TM

rad (see Box 2) inthe entire frequency range of electrically small antennas. Fromthis, it is possible to conclude that a complex design (e.g., theparameterizations found in [44]) is not needed to reach thelower bound.

The absolute lower bound for radiation Q-factor Qlbrad is

unreachable by this meander line antenna since its planargeometry and single feed scenario does not allow for anefficient excitation of combined TE and TM radiation. This

3

TABLE ILOWER BOUNDS ON RADIATION Q-FACTORS AND EFFICIENCY IN THELIMIT OF ELECTRICAL SIZE ka→ 0 FOR A SPHERE, A CYLINDRICAL

TUBE, A RECTANGLE, A THIN STRIP DIPOLE AND A SQUARE LOOP.

(ka)3Qlbrad (ka)3Qlb,TM

rad (ka)4η0/Rsδ

a1

3

23

`

`/2π4.5 4.6 59

``/2

4.3 5.2 42

`

`/5016 16 3400

`

0.9`5.3 7.4 130

contrasts to three-dimensional (e.g., spherical) geometries [25],[36], where the dual mode behavior can be realized by a singlefeed network.

Parameters of the self-resonant designs from Fig. 2 areshown in Fig. 3. Design curves are fitted to the optimizedparameters using a polynomial fit with good agreement. Whilesome of the curves from Fig. 3 can be found in [40] for severalparametrization, here all the designing curves are related backto Fig. 2 in which the Q-factor Qrad is minimized. Thepresented data series can therefore be used for designingmeandered dipoles approaching lower bounds on radiation Q-factor for TM antennas. It should, however, be noted thatdesign curves from Fig. 3 depends on the used parametrizationand are valid only for L/W = 2 and w/s ≈ 1.

B. Varying aspect ratios

Meander line antennas, introduced in the previous section,are now studied for various L/W and w/s aspect ratios andcompared against the fundamental bounds calculated for eachform factor.

In all cases, the value of radiation Q-factor Qrad is nor-malized with respect to the minimal TM radiation Q-factor.Generally, Fig. 4 shows that the minimal values can closelybe approached for various L/W aspect ratios. Slightly betterperformance is observed for higher L/W ratios, however, atthe cost of higher absolute bound on radiation Q-factor seetop panel of Fig. 4.

d

w

L

W

s

(a) (b)

Fig. 1. Panel (a) shows a parameterization of a meander line antenna usedin this work. The antenna is fed via delta gap source along a horizontal linecutting the center of the meander line. The feeding region contains a taperbetween the feeding strip of width d and the meander line of width w. Theangle of the taper’s cut is 45. Throughout the paper d = min (w,L/40)in order to keep the feeding region realistically narrow. Panel (b) shows ameander line antenna design “M1” from [8] within the parametrization usedin this paper.

0.2 0.3 0.4 0.5 0.6 0.7 0.8

10

20

50

100

200

500

1000

S

ka

Qra

d

Qlbrad

Qlb,TMrad

Chu boundmeanders

Fig. 2. Radiation Q-factors of self-resonant meander line antennas simulatedas made of PEC (markers) and the lower bound on radiation Q-factor (seeBox 1) corresponding to a rectangular region bounding the meander line.All meander lines are designed using the parameterization in Fig. 1 withw/s = 1 and L/W = 2. The defining parameters of all meandered dipolesare depicted in Fig. 3. A triangular marker with a corresponding error barrepresents measured radiation Q-factor of selected meander line design.

With respect to the varying w/s ratio, slightly better perfor-mance is observed for higher values, i.e., wider metallic strips.The differences become negligible for small values of ka, seeFig. 5. Notice, however, that this behavior is substantiallychanged when ohmic losses are introduced, mainly sincethe spatial proximity of out-of-phased currents degrades theradiation and enhance the ohmic losses [45].

C. The impact of impedance matching on Q-factor Qrad

The designs obtained above are all self resonant (Xin ≈ 0),but no constraint was placed on the value of the inputresistance Rin. In most practical cases, the objective antenna

4

20

40

60

80

N

1

2

3

Leff/λ

0

0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.02

0.04

0.06

ka

Rin/η 0

meandersfit: N = 1.95(ka)−2 − 0.600(ka)−1

meandersfit: Leff/λ0 = 0.548(ka)−1 − 0.0822

meandersfit: Rin/η0 = 0.084(ka)2 + 0.019(ka)3

Fig. 3. Design curves for the meander line antennas from Fig. 1. Thepanels show consecutively total number of meanders N , total length of thestrip Leff normalized to free-space wavelength, and input resistance Rin

normalized to free-space impedance. In all cases w/s = 1, L/W = 2 andPEC are considered. One meander line (N = 1) consists of two horizontalstrips and vertical connections, i.e., meander line antennas in Fig. 1 have 3 and20 meanders, respectively.

1 2 3 5 10 15 205

10

15

6.145.12 5.15 5.66 6.29

L/W

(ka)3

Qlb,T

Mra

d

0.2 0.3 0.4 0.5 0.6 0.7 0.81

1.05

1.1

1.15

1.2

1.25

ka

Qra

d/Q

lb,T

Mra

d

ka = 0.25

L/W

11.7234

Fig. 4. Radiation Q-factor performance of PEC self-resonant meandereddipoles from Fig. 2 for various L/W aspect ratios. The radiation Q-factor isnormalized to fundamental bound for a rectangular region which is shownin the top panel as a function of the aspect ratio. The minimum of thefundamental bound is found around ratio L/W ≈ 5/3. Generally, the higherthe L/W ratio, the closer the meander lines are to the bound, however, atthe cost of increasing absolute value of radiation Q-factor.

0.2 0.3 0.4 0.5 0.6 0.7 0.81.05

1.1

1.15

1.2

ka

Qra

d/Q

lb,T

Mra

d

0.2 0.25

1.06

1.07

1.08

1.09

w/s

1/23/413/22

Fig. 5. The same study as in Fig. 4 done for various ratios of meanderline width w to spacing s.

TABLE IITHREE IMPEDANCES OF PRACTICAL SIGNIFICANCE FOR ANTENNA

SYSTEM DESIGN.

System Input impedance Z0

Power amplifier (PA) 15 + j50 ΩRFID chip (passive RX) 20− j200 Ω

Low-noise amplifier (LNA) 50 Ω

input resistance is not driven by any antenna considerationbut is set by the radio frequency electronic equipment tobe interfaced with a particular antenna. Transmission linesand active receivers based on Low Noise Amplifiers (LNA)often require matching to 50 Ω. However, where deviceswith complex impedances are used, antenna resonance maynot be ideal for conjugate matching and maximum powertransfer. For example, a typical Power Amplifier (PA) outputimpedance is complex [46], with an input resistance lowerthan 50 Ω and an inductive (positive) reactive component.Similarly, passive RFID receivers based on Schottky dioderectifiers typically exhibit input resistances lower than 50 Ωand strong capacitive (negative) reactance [47]. Examples ofnominal impedances Z0 for these systems are listed in Table II.

Antennas may be designed to have input impedances whichconjugate match a desired load. However, any of the designsshown in Fig. 2 can be conjugate matched to an arbitrarycomplex impedance Z0 through an L-network consisting oftwo reactive components [48]. In many instances, the storedenergies within these reactances will raise the radiation Q-factor of the system. To assess the cost of this form of simplematching, we select the design in Fig. 2 corresponding to selfresonance at ka = 0.479. A set of lossless networks wasgenerated to conjugate match the antenna to arbitrary compleximpedances and the matched radiation Q-factors were calcu-lated. A typical frequency dependence of this cost is depictedin Fig. 6 while the dependence on matching impedance isdepicted in Fig. 7. We observe that it is generally possible totransform the resonant antenna impedance to an arbitrary realvalue with minimal increase in radiation Q-factor, except whensmall resistance and high reactance is required. As expected,adding a reactive component to the real-valued (resonant)

5

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

6

7

8

9

10

ka

(ka)3

Qra

d

bare meanderself-resonance of meanderPA: 15 + j50 ΩRFID: 20 − j200 ΩLNA: 50 Ω

Fig. 6. Matched radiation Q-factor of a selected meander line antenna forseveral impedance matching scenarios listed in Table II. Matching to eachimpedance is accomplished via a two-element reactive L-network, see insetin Fig. 7 for a schematic. In cases when several L-networks exist for a givenmatching impedance, the network with the lowest stored energy has beenused. Abrupt jumps of the matched radiation Q-factor curves result from non-existence of matching by two inductances in certain frequency ranges. Thisdouble inductance matching is the most favorable scenario for a capacitiveantenna.

10 20 30 40 50

−200

−100

0

100

200

R0 [Ω]

X0

[Ω]

PA: 15 + j50 ΩRFID: 20 − j200 ΩLNA: 50 Ω

1

1.1

1.2

1.3

Fig. 7. Matched radiation Q-factor of a selected meander line antenna forvarying complex matching impedance normalized to radiation Q-factor of bareantenna. For each complex impedance, the meander line is conjugate matchedat its self-resonance (circle mark in Fig. 6, ka ≈ 0.479) using the lossless L-network matching circuit with lowest Q-factor. The markers denote the threeimpedances from Fig. 6 and Table II.

antenna impedance necessarily increases radiation Q-factor,though this increase is on the order of 30% for the mostextreme of the three test impedances (RFID) examined here.Additionally, Fig. 6 shows that it is often possible to moveslightly away from the self-resonant frequency and lower theoverall radiation Q-factor by a small amount. Nonetheless, theminimum radiation Q-factor of the matched antenna is, forpractical values of the matching impedance, within the vicinityof the self-resonance of the antenna.

The importance of Q-factor is its relation to fractionalbandwidth which is predicated on simple, single resonance

0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51−12

−10

−8

−6

−4

−2

0

Matched4.49 %

LNA4.30 %

RFID3.65 %

PA4.24 %

B−3 dB

ka

τ[dB]

Matched: 9 ΩPA: 15 + j50 ΩRFID: 20 − j200 ΩLNA: 50 Ω

Fig. 8. Frequency dependence of normalized power delivered Pdel/Pcm tothe meander line studied in Figs. 6 and 7. The separate curves correspond todevices exhibiting the three practically relevant impedances from Table II andto a device with impedance corresponding to that of the meander line at itsself-resonance (“Matched”). In each case, the antenna is conjugate matchedusing an L-network at its self resonant frequency, ka ≈ 0.479. The −3 dBbandwidths 50% power delivered bandwidths B−3 dB for each scenario arealso listed.

behavior [12]. We demonstrate that the low variance in Q-factor corresponds to consistent realized bandwidth whenL-networks are used to conjugate match an antenna to anarbitrary impedance. Figure 8 shows the power delivered Pdel

to the meander line antenna studied above using a matchedsource (Z0 = Rin) as well as with L-networks designedto match the antenna to the three complex impedances ofpractical interest in Table II. In each case, a network tunesthe antenna to the desired (possibly complex) impedance atits natural resonant frequency. The frequency profile of themismatch factor [48], [49]

τ =Pdel

Pcm=

4RminR0

|Zmin + Z0|2

(4)

is nearly identical in all four cases, in agreement with thepredictions based on the relatively invariant Q-factor acrossthese cases. Here, Zm

in is the antenna impedance including thetuning network, Pdel is the power delivered to the antenna,and Pcm is the power delivered under a conjugate matchcondition. It is necessary to point out that we have assumednon-dispersive matching impedances, i.e., ∂Z0/∂ω = 0. Inpractice, the matching impedance may be dispersive withinthe band of interest, in which case the relation between Q-factor and bandwidth described in [12] ceases to be valid.However, inclusion of a dispersive load impedance may notnecessarily cause major changes to the realized bandwidthdue to the already heavily frequency-dependent nature of theimpedance of high Q-factor antennas. Despite this simplifica-tion, when generating Fig. 8, lumped inductors and capacitorsin each tuning network are modeled as frequency dependentimpedances.

The results in Figs. 7 and 8 numerically suggest that thereis little cost in bandwidth to match a self-resonant antennato arbitrary impedances. However, further considerations re-veal why it is of practical importance to design an antennawith a given impedance, rather than relying on this form

6

of matching. First, the use of lumped components increasescomplexity and cost of an antenna system and the requiredcomponent values for the L-networks described in this sectionmay not be realizable. Second, lumped components made ofany practical, lossy material (e.g., metallic inductors) increasethe net loss in an antenna system while not adding anypotential radiation mechanism. This guarantees a decrease inoverall efficiency, particularly in high Q-factor antennas [50].Additionally, tunability or the use of broadband multipleresonance matching may benefit from the design of an antennawith specific impedance characteristics, e.g., to increase theradiation resistance [8], [51].

III. RADIATION EFFICIENCY OF Q-OPTIMAL ANTENNAS

The previous section demonstrated that meander line an-tennas are nearly optimal with respect to radiation Q-factor,including the cases when matching to realistic compleximpedances is desired. This section studies how these antennasperform with respect to another critical antenna metric: radi-ation efficiency (see Box 3). Specifically, we examine theirperformance with respect to radiation efficiency bounds (seeBox 4).

Before presenting the radiation efficiency of matched me-ander line antennas it is necessary to deal with losses in thematching circuit since, similarly to the case of Q-factor, anymatching circuit with finite losses will worsen the overallefficiency of the antenna system. Throughout this sectionwe will assume that all matching networks are composedof lossless capacitors and lossy inductors1. The inductorsare further assumed to be planar, made of the same ma-terial (metallic sheet, surface resistivity Rs) as the antennaitself. Under such restrictions it is possible to estimate theloss added by a matching network quite precisely usingdata from Fig. 9, which shows the normalized reactance,(103/kaL

)(Rs/η0) (XL/RL), of several spiral inductors as a

function of their electrical size. Here η0 denotes the free spaceimpedance. The normalized reactance in Fig. 9 is independenton surface resistance Rs and, at small electrical size, justweakly dependent on number of turns and frequency, consis-tent with classical relations for helical air-core inductors [52].A conservative value

(103/kaL

)(Rs/η0) (XL/RL) = 66 will

be used in this section to determine losses of all inductorswithin the L-matching network, assuming further that induc-tors are always ten times smaller in electrical size than theantenna, i.e., aL = a/10. This last assumption enforces theuse of an electrically small, approximately lumped element,matching network.

Lossy elements with the above mentioned specifications areused to match the meander studied in Figs. 6–8 to impedanceZ0 = 50 Ω over a band of interest near the meander line’sself-resonant frequency. The resulting radiation Q-factor andefficiency (here presented in the form of dissipation factor, δ)are depicted in Fig. 10 as functions of frequency (scaledas electrical size ka). The figure reiterates the previously-observed near-optimal performance of meander line antennas

1Q-factors of lossy capacitors are typically much higher than those of lossyinductors.

10−3 10−2 10−140

45

50

55

60

65

70

75

kaL

103

kaL

(R

s

η 0

)(X

L

RL

)

2 turns3 turns4 turns

Fig. 9. Normalized reactances of selected rectangular spiral inductors. Thequantities XL, RL, and Rs denote input reactance, input resistance, andsurface resistance, respectively. The radius aL defines the smallest spherecircumscribing the inductor.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.81

2

3

5

10

20

30

50

ka

δ/δ lb,

Qra

d/Q

lb,T

Mra

d

Qrad/Qlb,TMrad

δ/δlb

minimumresonance

Fig. 10. Normalized radiation Q-factor and normalized dissipation factorof a selected meander line antenna matched to 50 Ω over a range offrequencies (scaled here as electrical size ka). The self-resonance of theantenna (ka = 0.47) is denoted on each trace with a circular marker whilethe minimum of each trace is also marked. The same data are also plotted asa curve parameterized by frequency in Fig. 11.

with respect to radiation Q-factor, but, surprisingly, shows arather poor performance with respect to radiation efficiency.This metric is, at the self-resonance frequency of the antenna,almost one order of magnitude worse than the value of thephysical bound (see Box 4). Similarly to radiation Q-factor,dissipation factor reaches its minimum in the vicinity of theresonance frequency, at least in the case of realistic values ofmatching impedances used here.

Within the used normalization of dissipation factor andradiation Q-factor, it is reasonable to represent the data fromFig. 10 as a two dimensional curve (radiation Q-factor vs.dissipation factor) parametrized by frequency, see Fig. 11. Thefigure also shows the Pareto front (represented by the blackline) evaluated by the method from [53], which demonstratesthe optimal trade-off between radiation Q-factor and dissipa-tion factor for the given design geometry and frequency. ThePareto front has been evaluated at ka = 0.5, but, due to theused normalization, it is almost independent of electrical size.

7

0 2 4 6 8 10 12 14 16 18 20

1

1.5

2

0.32

0.37

0.45

0.55

0.62min δminQrad

resonance

Pareto front

simulated antennas

δ/δlb

Qra

d/Q

lb,T

Mra

d

Fig. 11. A two-dimensional view of the lower bounds on dissipation factorand radiation Q-factor. A black solid curve denotes the Pareto front describingthe trade-off between these quantities and the corresponding feasible region(shaded). The data from Fig. 10 are drawn as a curve parameterized byfrequency, with the relevant points in Fig. 10 being similarly marked. Thecircle markers along the curve representing the simulated antenna show theelectrical size ka.

The Pareto front was evaluated for a combination of TM andTE modes which, as normalized to the TM bound Qlb,TM

rad ,gives values lower than one. The reason for this particularnormalization is that TM bounds represent meaningful limitof one-port planar antennas.

The two-dimensional plot in Fig. 11 represents a com-plete comparison of various antenna designs with respectto matched efficiency and matched radiation Q-factor. Anexample of such comparison is shown in Fig. 12, wherethe normalized and frequency-parameterized Q–δ curves aredrawn for several small antenna designs within the samedesign specifications2. Figure 12 clearly presents the superiorperformance in efficiency and Q-factor of simple meanderline antennas shown in Fig. 1 with respect to other designs. Italso shows that although there exist other meander lines whichperform slightly better in radiation efficiency (Palmier pastrytype, [54]) this improvement costs much in the radiation Q-factor. In conclusion, simple meander line antennas present thebest trade-off between radiation Q-factor and dissipation factorfrom the depicted antennas when matching to real impedancesis demanded. As in the previous section, we note that the use ofmore advanced matching topology (e.g., folding or impedancetransformer) may benefit from alternative antenna designs.

Figures 11 and 12 show that the considered antenna struc-tures, which are close to optimal in radiation Q-factor, arefar away from the efficiency bounds. This is puzzling sinceresonant modes optimal in radiation Q-factor and efficiencyare similar in nature. However, there are important differences.Radiation Q-factor restricted to TM modes is minimized byseparation of charges and inducing dipole like currents [38].These modes can be tuned to resonance by inducing edgeloops along the structure. TM efficiency, on the other hand,is minimized by inducing homogeneous currents [55]. Theseare similar in nature to the dipole like currents minimizing Q-

2The bounding geometry, material parameters, and restrictions on matchingnetwork topology and losses are all kept constant across each design.

0 2 4 6 8 10 12 14 16 18 20

1

2

3

meander (A)folded meander (B)palmier (C)loop antenna (D)bowtie antenna (E)

δ/δlb

Qra

d/Q

lb,T

Mra

d

Pareto front

A B C D E

Fig. 12. Two dimensional frequency parametrized plot representing thephysical bound and several antennas. The matching impedance equals to 50 Ω.The tip of each curve lies in the vicinity of resonance or anti-resonance of theantenna. The electrical sizes at the resonance or anti-resonance frequency are:A (ka = 0.48, res.), B (ka = 0.85, res.), C (ka = 0.50, res.), D (ka = 0.59,anti-res.), E (no resonance).

factor, but the loop currents which minimize TE Q-factor andmaximize TE efficiency are fundamentally different. Wherelow Q-factor loops tend to be confined towards the edges ofthe structure, high efficiency loops are spread across the wholearea [53, Fig. 4]. Such loop currents are naturally restricted asan original simply connected object fully filling a prescribedbounding box is perforated, forcing the current distributioninto more inhomogeneous forms. Thus, low Q-factor loopsare tolerant of alterations to a structure whereas high efficiencyloops are harshly disrupted.

In Fig. 13, the optimal resonant Q-factor and dissipationfactor are plotted normalized to the corresponding boundsof a rectangular plate. Data for different shapes made byremoving portions of the plate are shown. The currents onthe structures in Fig. 13 have been calculated with currentoptimization without physical feeding. It is clear that removingmetal does not greatly affect the achievable radiation Q-factor, at worst reducing it to the TM-only bound. However,when metal is removed from the plate the loss factor issignificantly increased, especially for small electrical sizes.Thus, while optimal radiation Q-factor and radiation efficiencymodes are fairly similar, removing design space has a muchgreater effect on the loss factor than the Q-factor in relationto the physical bounds. This can be seen in Fig. 13 wherethe loss factor of the optimal resonant currents is very highfor the structures with slots in them. Consider the meanderline antenna which has significantly higher loss factor atelectrical sizes ka < 0.4, here the loop modes are extremelydisrupted, however, the Q-factor is hardly affected. The sharpchange in the meander line’s loss at around ka = 0.6 is due toits resonance, where it is possible to induce a resonant dipole

8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.81

2

3

45

10

15

20

A

B

C

D

ka

δ/δ lb,

Qra

d/Q

lb rad

Qrad/Qlbrad

δ/δlb

A B C D

Fig. 13. Illustration of how much different metrics deteriorate when a solidplate is permuted. The figure shows four generic shapes, a single slot, threeslots, a meander line, and a loop. The ratio between optimal loss factor fordifferent geometries and the optimal loss factor for the full plate are shown insolid lines. The ratio between the optimal mixed mode Q-factor for the samegeometries and the optimal mixed mode Q-factor of the full plate are shownin dashed lines.

Box 3. Radiation efficiency

The radiation efficiency of an antenna is defined as

η =Prad

Prad + Pdiss=

1

1 + δ, (5)

where, as in (1), Prad and Pdiss are radiated and ohmicdissipated power, respectively, and δ = Pdiss/Prad is thedissipation factor [56]. Along with bandwidth radiationefficiency is a key antenna performance parameter,particularly in electrically small systems where it is knownto decrease rapidly with antenna size.

For objects with homogeneous loss properties, e.g., uniformsurface resistance or conductivity, the dissipation factor δis a linear function of those properties. As such, values ofdissipation factor can be normalized by surface resistivityfor ease of comparison.

mode on the structure. This example illustrates a fundamentalchallenge in designing efficient small resonant antennas: manyof the strategies normally utilized to induce resonance, suchas meandering, harshly limit the achievable efficiency.

IV. ANTENNAS OPTIMAL IN OTHER PARAMETERS

Determining the best possible Q-factor can be formulatedas a minimization problem. Therefore it is possible to adddifferent or additional constraints to such an optimization.So far, in this paper, we have considered the constraints ofefficiency and impedance matching. Another type of con-straints are different kinds of field-shaping requirements ofnear and/or far-fields [62], [63]. For small antennas it is wellknown that the radiated far-field tends to resemble a dipole

Box 4. Lower bounds to dissipation factor

Two different paradigms for minimization of dissipationfactor exist. The first assumes that tuning or generalimpedance matching of the antenna can be performedin a lossless manner. Under this assumption, the optimalcurrent density minimizing dissipation factor is the resultof a generalized eigenvalue problem [55]–[58]. Such lowerbounds were shown to scale with electrical size as (ka)−2

and are straightforward to calculate. Their major drawback,however, is that, by neglecting matching network losses,the resulting dissipation factors are overly optimistic andunachievable by realistic designs where some form ofmatching is required [50].

One solution to the aforementioned drawback is a paradigmin which the optimal currents are calculated while takinginto account the dissipation cost of achieving resonance orgeneral matching [50], [59]. Dissipation factors coming fromthis second paradigm are generally closer to realistic designsand scale with electrical size as (ka)−4 [53], [59]–[61].Lower bounds to tuned dissipation factor for several selectedshapes are shown in Table I.

pattern, meander line antenna treated in this paper being noexception, see Fig. 14. However, with these types of Q-factoroptimization procedures it is possible to determine the Q-factorcost, to have the antenna radiating with a certain front-to-back ratio or (super-) directivity in a given direction. Theseclasses of bounds indicate that for a limited bandwidth cost itis possible to extend, e.g., the directivity beyond the traditionaldipole pattern, see [62]–[69].

030

60

90

120

150180

210

240

270

300

330

−60

−40

−20

0

x-z cut, e = ϑ

y-z cut, e = ϕa)

030

60

90

120

150180

210

240

270

300

330

−60

−40

−20

0

y-z cut, e = ϑ

x-z cut, e = ϕb)

Fig. 14. Directivity (in dB scale) with respect to an isotropic radiator of ameander line antenna depicted in Fig. 2 as the very right inset. Antenna isplaced in x-z plane, with a longer side aligned with z-direction.

To illustrate bounds on superdirectivity, Q-factor optimiza-tion for a given directivity described in [62], [63], was solvedfor a small antenna with length to width ratio of 2:1, infinites-imal thickness, and electrical sizes ka ∈ 0.2, 0.5, 0.8. Thebounds for low directivities are identical to the lower boundon the Q-factor, where the radiation pattern changes from thatof an elliptically polarized dipole with D ≈ 1.5 to that of aHuygens source with directivity just below D = 3 and themain beam pointing in the direction of the longest side [21].Higher directivities require quadrupole and higher order modeswhich increases the Q-factor rapidly [10]. The direction of themain beam changes from the longest side of the antenna to an

9

2 3 4 5 6

100

101

D

Qra

d/Q

lb rad

ka = 0.2ka = 0.5ka = 0.8

meanders

1.5 1.54 1.581

1.2

1.4

Fig. 15. The cost in Q-factor for a desired directivity in a lossless 2:1 shapedantenna. The three line colors represent electrical sizes ka = 0.2, 0.5, 0.8and the radiation patterns correspond to the ka = 0.5 case for the optimalcurrent. All Q- vs D-values for the meander lines of Fig. 2 are representedin purple, and the relevant region is zoomed in the right bottom inset. Dashedcurves show the results for an array composed of a meander line antennaelement with one feed and a loop antenna element with two feeds.

endfire pattern along the shortest side at Qrad/Qlb,TMrad ≈ 3 as

indicated by circles in Fig. 15.Much like bounds on other parameters (e.g., efficiency), it is

an open problem if the directivity-constrained limits are reach-able for all sizes and desired directivity, even under idealizedlossless conditions. As a demonstration of one possible highdirectivity, low Q-factor design, a three port array composedof a meander line and a loop structure with optimized feedingis presented in relation to the bounds, see Fig. 15. However,high directivity for single port antennas remains, as of yet,far from the bound and new designs ideas that allow a highdirectivity with larger bandwidth are desired.

V. CONCLUSION

The possibilities how to approach the fundamental boundson selected antenna metrics were investigated. A planar regionof rectangular form factor was considered. It was observedthat the lower bound on Q-factor with radiation restrictedto TM modes only is closely approached by a meanderline antenna for a broad range of electrical sizes. The optimaldesign parameters were depicted and various aspect ratios ofthe bounding rectangle were studied together with selectedratios of the strip and slot widths. The simulated resultswere verified by a measurement of a fabricated prototype.The impedance matching and its impact on the Q-factor ofthe antenna was studied, concluding that the effect of theimpedance matching on radiation Q-factor is minor and, insome cases, that matching the antenna slightly away fromits self-resonance can even decrease its Q-factor. Radiationefficiency of the meander line antennas optimal in Q-factorwas evaluated, taking into account ohmic losses dissipatedin the matching circuit. It was observed that the radiationefficiency of the studied meander line antennas is far froman upper bound of a rectangular patch. Several other planarantennas were similarly evaluated against fundamental boundsyielding consistent conclusions: synthesizing antenna designs

which approach the upper bound on radiation efficiency ismore difficult than designing those which reach the lowerbound on Q-factor. The reason was identified in the highsensitivity of radiation efficiency to the perturbation of idealconstant current density. Namely, when an initial structurefully filling the prescribed bounding box is perforated (as isdone in a practical synthesis procedure), the performance ofmaximum efficiency current distributions drops much fasterthan that of a minimum Q-factor distribution. Finally, aPareto-type bound between Q-factor and directivity has beencalculated and compared to meander line antennas. An attempthas been made to find an antenna with reasonably low Q-factor and directivity higher than that of an electric dipoletype antenna. Nevertheless, no planar antenna with one feedfulfilling these contradictory constraints was found. This taskand its feasibility remains as a subject for ongoing research.

The fundamental bounds, i.e., the lower bounds on Q-factor,the upper bounds on radiation efficiency, the Pareto-optimalitybetween Q-factor and efficiency, or Q-factor and directivity,were demonstrated to be powerful tools for judging theperformance of the radiating devices. If the realistic designsare compared to the fundamental bounds, designer can assesshow far from the optima the design is, therefore, if furtherimprovement is needed. Furthermore, incremental progress indesign improvement can be put into context by considering theremaining distance between an antenna’s realized performanceand the fundamental bounds. It is the normalized ratio of theactual device’s performance to the fundamental bounds whatreveals the real quality of the design.

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