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Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using High Permeability Shells
 

Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using High Permeability Shells

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A study of the effect of surrounding electrically small, top-loaded, electric-dipole antennas with a thin shell of high-permeability magnetic material. The magnetic polarization currents induced in ...

A study of the effect of surrounding electrically small, top-loaded, electric-dipole antennas with a thin shell of high-permeability magnetic material. The magnetic polarization currents induced in the thin shell of magnetic material reduce the internal stored energy, resulting in a lower Q as compared to conventional designs.

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    Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using High Permeability Shells Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using High Permeability Shells Document Transcript

    • s t r at e g i c w h i t e pa p e rApproaching the Lower Boundson Q for Electrically SmallElectric-Dipole Antennas UsingHigh Permeability ShellsHoward R. Stuart Arthur D. YaghjianMember, IEEE Life Fellow, IEEE Abstract—We study the effect of surrounding electrically small, top-loaded, electric-dipole antennas with a thin shell of high-permeability magnetic material. The magnetic polarization currents induced in the thin shell of magnetic material reduce the internal stored energy, resulting in a lower Q as compared to conventional designs. The simulated Q of thin-magnetic-shell cylindrical and spherical antennas are compared to recently derived lower bounds on Q. The high permeability shells reduce the Q of these antennas to values below the lower bounds for purely global electric current sources. In the case of the spherical electric-dipole antenna, a sufficiently large value of permeability enables the Q to be reduced to a value that is only 1.11 times the Chu lower bound. Index Terms—Dipole antennas, electrically small antennas, lower bounds, quality factor.Manuscript received December 17, 2009; revised March 15, 2010; accepted May 24, 2010. Date of publication September 23, 2010; date of current versionNovember 30, 2010.This work was supported in part by the Defense Advanced Research Projects Agency (DARPA) and in part by the U.S. Air Force Office of Scientific Research (AFOSR).H. R. Stuart is with LGS Innovations, Florham Park, NJ 07932 USA.A. D. Yaghjian is a Research Consultant at Concord, MA 01742 USA.Digital Object Identifier 10.1109/TAP.2010.2078466 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • table of contentS [ 1 ] INTRODUCTION 01 [ 2 ] CY L INDRICA L CAPPED MONOPO L E 02 A. Resonant Modes 02 [ 3 ] S PHERICA L CAPPED MONOPO L E 09 A. Resonant Modes 09 B. Matching to 50 ohms 10 C. Electric Permittivity 11 [4 ] S UMMARY 12 [ 5 ] REFERENCE S 13 [ 6 ] B i o g r a p h i e s 14 iiS t r at e g i c w h i t e p a p e rApproaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • 1. introductionThe lower bounds on the radiation quality factor Q of electrically small antennas were first derived by Chu [1] (see also [2]–[6]) for a spherical surface circumscribing an antenna under the assumption that there is no stored energy inside the sphereexcept possibly for the stored energy needed to tune to zero the reactance produced by the stored energy outside the sphere.It is generally assumed there will be additional stored energy inside an actual antenna that will raise the Q above theChu lower bound. Wheeler [7] was the first to observe that, for the case of a small spherical magnetic-dipole antenna withelectric currents confined to its surface, filling the antenna volume with a material of infinite magnetic permeability wouldreduce the stored magnetic energy inside the sphere to zero, enabling the Chu lower bound to be achieved theoretically.Notwithstanding the practical issues of internal resonances and producing the required low-loss magnetic material,this result provides an approach for reducing the internal energy and lowering the Q of small magnetic-dipole antennasusing permeable materials [8]. Wheeler offered no analogous method for reducing the internal stored electric energy of anelectrically small spherical electric-dipole antenna. (It should be noted that a high-permeability material filling the antennavolume is not equivalent to filling the volume with a perfect magnetic conductor (PMC) because the electric field, unlike themagnetic field, does not approach zero inside a high-permeability core.)Chu’s lower bound calculation was modified by Thal [9] in a manner that accounts for the stored energy inside the boundingsphere when the antenna consists of global1 electric currents confined to the spherical surface. The Thal lower bound fora small electric-dipole antenna is 1.5/(ka)3, a factor of 1.5 above the Chu lower bound for antennas with ka % 1 (k is thewavenumber and a is the radius of the antenna’s circumscribing sphere). This result concurs with the known performanceof several optimized electrically small spherical electric-dipole antennas [10]–[13], and is generally thought to represent thebest achievable performance for these electric-dipole antennas.In a recent paper [14], we have derived general expressions for the lower bounds on the Q of electric- and magnetic-dipoleantennas in arbitrarily shaped electrically small volumes. For the case of a spherical electric dipole excited by globalelectric current sources, the lower bound matches Thal’s result of 1.5/(ka)3. For the more general case of both electricand magnetic current sources, the spherical electric-dipole lower bound is reduced to 1/(ka)3, matching the Chu bound.A similar result follows for arbitrary geometries. That is, allowing for the possibility of magnetic current sources yieldsreduced lower bounds as compared to the case of global electric current sources alone. Because magnetic charge does notexist, achieving this general source condition in practice requires the use of magnetic polarization currents, such as thosearising in permeable magnetic materials. These new lower-bound equations therefore imply it is possible to use magneticmaterials to improve the Q of electrically small electric-dipole antennas. The derivation in [14] also suggests the form andphysical mechanism by which these improvements occur. In particular, a suitably driven thin shell of permeable materialsurrounding an electric-dipole antenna can produce magnetic polarization currents that reduce the stored electric energyinside the antenna, thereby improving the Q.In this paper we illustrate antenna designs incorporating thin shells of high relative magnetic permeability to improve theQ of electrically small electric-dipole antennas. We treat top-loaded dipole antennas (or equivalently, monopoles on infiniteground planes) confined to cylindrical (Section II) and spherical (Section III) volumes. The simulated quality factors arecompared to the general lower bounds found in [14] in order to assess the optimality of the designs. For the cylindricalantennas, some basic design equations are provided to estimate the minimum value of relative permeability required to seea notable reduction in Q.1 By “global” electric currents we mean electric currents other than those on small Amperian current loops (or possibly those on slotted electric conductors) that produce effective magnetic current or polarization. 1 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • 2. CYLINDRICAL CAPPED MONOPOLE2 .1 Resonant ModesWe consider the top-loaded monopole structure, shown in cross-section in Figure 1(a), consisting of a short straightconducting wire with a circular conducting cap. When placed over a large ground plane, the structure forms a scatterer thatresonates like a small electric dipole with the lowest resonant frequency occurring at a wavelength that is typically muchlarger than the height of the straight wire because of the capacitive loading of the circular cap. As an example, considera monopole height of 100 mm with a circular cap radius of 150 mm, a center conductor radius of 7.5 mm, and a top platethickness of 2 mm. With no dielectric or permeable material present, the fundamental resonant frequency of the structureshorted to ground is 154 MHz and it has a Q of 14.4, both of which are determined numerically using an eigenmodesimulation with a perfectly matched layer at the outer boundaries [13], [15]. The height of the monopole at this frequency isslightly larger than 1/20 of a wavelength. The electric field of the resonant mode is shown in Figure 2(a). Underneath thecap, the electric field is predominantly z-polarized and uniform along the z-direction, deviating from this behavior mostnotably near the edge of the cap.We next consider how the mode structure is affected by introducing a thin shell with high magnetic permeabilitysurrounding the cylindrical volume formed by the projection of the circular cap. The thin shell sits just under the outeredge of the circular cap [see Figure 1(b)] such that the overall circular cylindrical volume occupied by the open resonator(scatterer) does not protrude beyond the cap. The equations relating the electric and magnetic time-harmonic(exp(–i~t), ~ >0) fields E and H and can be written as (1) (2)where the magnetic induction B is related to H by the magnetic permeability through B = n0nr H, J is the current densityon the conductors, which are assumed to be perfectly electrically conducting, and e0 and n0 are the permittivity andpermeability of free space. Because of the axial symmetry, the fundamental mode oscillates with z-polarized magneticfields (B, H) and with an r- and z-polarized electric field E for cylindrical coordinates (r, z, z), where z points upwards inFigs. 1, 2, and 3.Figure 1Two-dimensional cross-section of the cylindrical capped monopole on an infinite ground plane (a) in free space (b) witha thin cylindrical shell of high magnetic permeability surrounding the cylindrical volume formed by the projection of thecircular cap. 2 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • Figure 2Electric field of the fundamental resonant mode of the capped monopole (a) in free space (b) with a 2 mm thick shell ofmagnetic permeability nr = 472 and (c) with a 2 mm thick shell of magnetic permeability nr = 8000. The thin high-nrlayer in (b) makes the electric field equal to zero near the bottom outside edge and substantially reduces the electricfield everywhere beneath the cap. By further increasing nr, the Ez = 0 point moves to a higher position on the inside ofthe shell layer, and the electric field beneath the cap is further reduced.Figure 3The thin, high-nr layer introduces a discontinuity in the tangential component of the electric field across the boundary.For 9l> 0 and finite nr, Ez is continuous, though rapidly changing, inside the thin layer—the discontinuity referring tothe values of Ez on either side of the layer. In the figure, a positive z-directed magnetic field points into the page.The thin shell is assumed to have a relative magnetic permeability nr& 1. Because the surface layer is everywhere tangentialto the z and z directions and normal to the r direction, a jump in value is introduced in the z-directed electric field acrossthis layer. The size of this jump is determined using (1) and Stokes theorem to relate the line integral of the electric fieldalong the contour shown in Figure 3 to the surface integral of the magnetic induction Bz on the surface enclosed by thecontour. Letting ∆S go the zero causes the Er contributions on the ends of the contour to cancel, and letting ∆l approach thethickness d of the shell yields the following equation for the difference of E z across the shell in terms of Bz or Hz in the shell (3) (4)where h = (n 0 /e 0 )1/2 is the characteristic impedance of free space (120r ohms), k = ~(n 0 /e 0 )1/2 is the free-space wavenumber,and (Bz, Hz) are approximately uniform across the thin shell. In the limit as d becomes vanishingly small but n r is madelarge enough that the quantity n r d is nonzero, (4) illustrates that the high-n r layer behaves like a magnetic surface current,inducing a discontinuity in the tangential component of the electric field across the thin shell. 3 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • We shall now show that it is possible to reduce the E z field under the cap to practically zero and thus reduce the Q of theelectric-dipole radiator to a value close to the smallest possible lower bound for a radiator confined to the given cylindricalvolume. If E z is successfully reduced to a value that is practically zero everywhere inside the cylinder, the magnetic fieldH] inside the cylinder is given from the integral form of (2) as I0/(2rr), where I0 is the electric current along the centerconductor. (The top loading produces a fundamental resonant mode that has an approximately uniform current along z inthe center conductor.) At the outer surface of the cylinder, Hz = I0/(2rrc), where rc is the cylinder radius. The value of theelectric field Ez at the outer surface can be estimated from the static capacitance C DC of the capped monopole structure, Owhere the capacitance can be determined numerically or estimated from known formulas [16]. The resulting electric field Ois Ez = –iI0/(~C DCh), where h is the height of the cylinder above the ground plane. (In practice EO will not be perfectly zuniform but this relationship provides a reasonable estimate.) Using (4), we then determine the following approximatecondition needed to make E z = 0 inside the cylinder (5)Figure 4The magnitude of the electric field versus position along the ground plane for the fundamental resonant mode using a2 mm thick n r = 472 shell.For the dimensions of the structure considered here, the capacitance is determined by numerical simulation to be 18.6 pF.The required n r d depends upon the resonant frequency, which itself will move to lower frequencies due to the presenceof the high-n r shell. For a resonant frequency of about 104 MHz and a shell thickness of d = 2 mm, (5) yields n r = 472. Anumerical eigenmode simulation of the shorted capped monopole with this shell configuration finds a resonant frequency of103.9 MHz and a Q = 34.7. For this case, the electric field indeed goes to zero near the ground plane just inside the cylindersurface [see Figure 2(b)]. This effect is illustrated more clearly in Figure 4. The shell material (positioned at 150 mm) inducesa large change in the electric field such that the field is zero on the ground plane just inside the surface of the material. Itis not zero everywhere underneath the cap, however. This is due, in part, to the fact that the field is only purely z -polarizedat the ground plane; a small r -polarized component becomes more pronounced above the ground. (This component isnormal to the shell surface and unaffected by the high-n r material.) Also, E z increases with height above the ground plane,such that the chosen value n r of does not work ideally at all positions (a larger value is required to zero the field at higherpositions). Nevertheless, there is a pronounced reduction in the size of the electric field throughout the region beneath thecap. The effect becomes more dramatic if we further increase n r [see Figure 2(c)], with larger reductions in stored energyaccompanying larger values of n r . In practice, there will, of course, be a limitation on the maximum obtainable value of n r .Equation (5) is useful for providing a rough estimate for the minimum value of n r d necessary to observe a strong effect. 4 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • The measure of whether a high-n r material is successful in significantly reducing the overall stored energy in the modeis the value of Q. The Q increases as a result of adding a high-n r d shell because of the associated lowering of the resonantfrequency. Consequently, for a proper evaluation, the Q must be compared to known lower bounds on Q for the particularelectrical size and shape of the volume to which the antenna is confined. We use the convenient expressions for the lowerbounds on the Q of electrically small dipole antennas derived for the case of solely global electric current excitations [14, Eq.(29b)] (6)and for the case of both electric and magnetic current excitations [14, Eq. (21b)] (7)where the electrostatic polarizability ae in [14] has been expressed as l (the polarizability normalized to the volume) timesthe volume V of the antenna. The parameter l depends solely upon the shape of the antenna volume (l $ 1) and serves as aconvenient parameter for comparing the performance of various structures. (We refer to l as the shape factor, following theterminology of Wheeler, who used a similar shape factor in characterizing the capacitance/inductance of electrically smallcylindrical antennas [17]).For the cylindrical volume considered here, the shape factor l = 2.88 (determined numerically from COMSOL Multi-physics), which corresponds to lower bounds on Q in (6) of 2.71/(ka)3 for global electric current excitations alone, and 1.77/(ka)3 in (7) for both electric and magnetic current excitations. Thus, the Q can be improved, in principal, by a factor ofabout 1.5 by allowing both electric and magnetic currents. From numerical simulation, we determine the resonant frequencyand Q of the fundamental mode of our cylindrical-volume resonator for a range of values of magnetic permeability and shellthickness; the results are shown in Figure 5. For the free-space case (n r = 1), the Q is slightly above the lower bound forglobal electric current excitations [15]. As the shell permeability increases (introducing magnetic currents), the Q crosses theelectric-current lower bound and begins to decrease. After transitioning to lower values, the Q reaches a plateau such thatfurther increases in n r offer little additional reductions in Q. The lowest Q is obtained for the thinnest shell (2 mm) withvery high permeability; in this case we reach a Q = 1.84(ka)3, a value only 4% above the smallest possible lower bound of1.77(ka)3. Increasing the shell thickness d lowers the value of the permeability required to see an effect.However, the minimum achievable Q is higher for thicker shells (which allow more field penetration beneath the cap,thereby increasing the stored energy). Filling the entire cylinder with magnetic material lowers the Q of electrically smallelectric-dipole antennas very little (unlike electrically small magnetic-dipole antennas). The value of n r d required to bringthe n r into the “transition” region of the curve can be estimated using (5) with the frequency set to the resonant frequency(~f s = 154 MHz) of the free-space (n r = 1) antenna. (The values predicted in this way are shown in Figure 5 along the 0x-axis.) This is convenient, as the shift in resonant frequency induced by the introduction of the high-n r shell is nottypically known a priori for a given structure. 5 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • Figure 5The resonant frequency and Q of the fundamental resonant mode of the cylindrical capped monopole [normalized by(ka)3] versus relative permeability of the shell layer for a range of shell thicknesses. The lower bounds for solely globalelectric current excitations, and for both electric and magnetic current excitations, are also shown. By applying (5) fwith the free-space resonants frequency (~ 0 = 154 MHz), we can estimate the permeability value required to reach thetransition portion of the curve for each case.2 .2 The Corresponding Cylindrical AntennaThe capped monopole open resonator (scatterer) discussed in the previous section can be converted into an electric-dipoleantenna by driving the base of the center conductor through the ground plane. For a d = 2 mm shell of n r = 100, theeigenmode analysis predicts a resonant frequency of 80.8 MHz and a Q = 69.4(Q(ka)3 = 1.97). The time-harmonic simulationof the driven antenna produces the impedance shown in Figure 6. The Q of the antenna is estimated from the frequencyderivative of the impedance using the formula derived by Yaghjian and Best [18, Eq. (96)] (8)where the reactance is assumed to be tuned to zero at each frequency ~ and R(~) and X(~) are the resistance and reactanceof the untuned antenna. The resulting Q z near the resonance is shown in Figure 7. These results confirm that the high-n rlayer produces an antenna exceeding the performance bounds for electric-dipole antennas with solely global electric currentexcitations and achieves a Q that is only about 11% higher than the smallest possible lower bound for antennas with bothelectric and magnetic current excitations. Figs. 6 and 7 illustrate the accuracy of the eigenmode simulation in predictingthe Q of the driven antenna at resonance. Similar results are observed for the other resonator configurations studied in theeigenmode analysis.As discussed in [15], the impedance properties of a single resonance top-loaded monopole antenna are well described overa wide frequency range using the equivalent circuit model shown in Figure 8. The lumped element values are determinedfrom four parameters: the resonant frequency ~ 0 , the Q of the fundamental resonant mode, the radiation resistance R 0 ofthe fundamental mode at its resonant frequency, and the static capacitance C D C of the antenna, through the equationsC=1/~ 0 QR 0 , L=QR 0 /~ 0 , R=1/[R 0 (~ 0 C)2] and C res=C D C − C . The three parameters ~ 0 , Q, and C D C were determinednumerically in the previous section and we shall next derive an equation for determining R 0 . 6 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • Figure 6 Figure 7Impedance versus frequency for the cylindrical capped Qz versus frequency for the cylindrical capped monopolemonopole antenna with a 2 mm shell of n = 1000. The with a 2 mm shell of n = 1000.parameters of the lumped element model are derivedfrom the resonant frequency, the Q, the resonantradiation resistance of the fundamental mode, and thestatic capacitance.For excitations with global electric currents alone, the radiation resistance at resonance can be determined from the well-known equation for a small monopole of uniform current and height h over an infinite ground plane [19] (9)where m 0 is the resonant wavelength. Equation (9) was shown in [15] to provide a highly accurate value for the radiationresistance of the fundamental mode of the capped monopole antenna (with no dielectric or permeable materials). Equation(9) is no longer accurate once the high-n r shell is added to the structure. This is because the magnetic currents induced inthe high-n r shell, in addition to greatly reducing the electric fields interior to the shell, also provide a contribution to thetotal dipole moment of the resonance beyond that of the electric currents alone.To account for this effect, we first write the electric-current contribution to the electric-dipole moment ( ) as [20] (10)where the factor of 2 accounts for the presence of the ground plane. In order to determine the magnetic-current contributionto the electric dipole, we first rewrite (1) in the form (11)where the magnetization M is related to the magnetic induction B through the constitutive relation B = n 0 (H+M). Themagnetization M = M ] ] is nonzero only inside the high-n r material. For a very thin shell, M ] is approximately uniformacross the shell and equals (n r -1 ) H ] , where H ] is the value of the approximately uniform magnetic field in the shell. Theright side of (11) can be interpreted as a magnetic-current source Jm analogous to the electric-current source J in (2), whereJm= - i~n 0 M . Taking the magnetic-current analog of the more familiar relationship in which a circulating electric currentproduces a magnetic-dipole moment, we write the magnetic-current contribution to the electric-dipole moment as ( ) as (12) 7 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • where A is the cross-sectional area of the circular cylinder (rrc2) and Im is the magnetic current in the high-n r shell. Thetotal magnetic current circulating inside the shell material can be expressed as (13)where we have assumed a uniform H ] along the entire height of the cylinder with the factor of 2 again accounting for theground plane. Because Im is a magnetic current, it has units of volts (not amperes), such that the units in (12) are that of anelectric-dipole moment (ampere-second-meters), that is, identical to the units in (10). The two electric-dipole moments add inphase to give the total dipole moment, which produces a total radiated power of [21, p. 437] (14)where the factor of 1/2 accounts for the infinite ground plane. Combining (10) with (12)–(14) and using the relationships andH ] = I0/(2rrc) and R0= 2P/|I0 | 2 yields the following expression for the resonant radiation resistance (15)where we have let (n r -1 ) . n r because n r & 1 . The second term in the last factor of (15) accounts for the increase in theradiation resistance due to the presence of the high-n r shell, and illustrates how the magnetic currents directly assist inpolarizing the radiating dipole (this term, without the approximation, goes to zero when n r =1 ). It is important to recall :that m0 is itself a function of n r (m0 increases as n r in the limit of high n r) such that the resonant radiation resistance andthe ratio of the dipole moments (pmc/ pec ) approach limiting values as n r becomes very large. This is implied by Q reaching e ea plateau for large values of n r .Equation (15) is used in combination with the modal analysis and the static capacitance to determine values for thelumped element parameters in Figure 8. The resulting model produces the solid lines shown in Figure 6. Although thelumped element model overestimates the radiation resistance at resonance by approximately 8% (not surprising giventhe approximation used to derive (15) that H ] is uniform along z), the model nonetheless closely matches the simulatedimpedance over two octaves around the resonance. A similarly close correspondence between the lumped element modeland the simulated impedance is seen for other values of n r , with the correspondence becoming slightly worse for smallern r . This degradation is caused by the greater inaccuracy of (15) when larger electric fields exist beneath the cap (as happens mcfor both smaller n r and thicker shells). Likewise, (12) overestimates p e for thick shells, implying that (15) is most accuratefor very thin shells. Nonetheless, for a 50 mm shell with n r =100 (15) only overestimates R0 by 16%, close enough that theimpedance curves of the model match the time harmonic simulation reasonably well at and below resonance. However,when the entire cylindrical volume is filled with the high- n r material, the discrepancy between (15) and the observed R0becomes too large (greater than a factor of two) to provide a useful estimate.FIGURE 8Lumped element model for short top-loaded monopoles. 8 S T R AT E G I C W H I T E P A P E R APPROACHING THE LOWER BOUNDS ON Q FOR ELECTRICALLY SMALL ELECTRIC-DIPOLE ANTENNAS USING HIGH PERMEABILITY SHELLS
    • 3. SPHERICAL CAPPED MONOPOLE3.1 Resonant ModesAlthough spherical antennas may be less practical than cylindrical antennas, they are useful in determining how closelythe original Chu lower bound can be approached. In this section we study the spherical capped monopole shown in cross-section in Figure 9(a) (the spherical analog of the cylindrical antenna discussed in the previous section). The antennaconsidered here has a radius of 100 mm with a center conductor radius of 7.5 mm. The shell thickness will be varied, butin each case the thickness of the electrically conducting portion of the shell (upper region) is made equal to the thicknessof the high-n r shell (lower region), forming a solid hemispherical shell of uniform thickness. The conducting portion of theshell extends from the top of the sphere down to an angle of 50 degrees from the top (this is roughly the optimal angle forminimizing the Q at resonance relative to lower bounds for the free space antenna). The entire shell conforms to a sphericalbounding surface in order to minimize the Q with respect to the original Chu lower bound.With no dielectric or permeable material present in the lower portion of the shell, the antenna shorted to ground has aresonant frequency of 249.3 MHz (ka = 0.522) and a Q of 11.9 for a conducting shell thickness of 1 mm. This is a factor of1.34 above the Chu lower bound2 [1/(ka)3 + 1(ka)], and a factor of 1.008 above the lower bound 1.33[1/(ka)3 + 1(ka)] of Thal [9,Table I] for spherical antennas with global electric currents confined to the surface of the sphere. Increasing the thickness ofthe conducting shell raises the resonant frequency (253.6 MHz for a 10 mm thickness), and also raises the Q relative to thelower bounds very slightly (1.35 times Chu lower bound and 1.024 times Thal lower bound at a 10 mm thickness).We next introduce magnetic material into the lower portion of the shell. Figure 10 illustrates the effect of increasing relativepermeability values on the resonant frequency and Q of the fundamental resonant mode. In the limit of a thin shell withvery high permeability, the fundamental mode has a Q that is only a factor of 1.11 above the Chu lower bound.The physical mechanism enabling the Q to closely approach the lower bound is identical to that seen previously in thecylindrical antenna. A magnetic polarization current is induced in the high-n r shell by the magnetic field of the resonantmode. This polarization acts as a magnetic surface current that reduces the magnitude of the electric field inside the sphereand thus the stored electric energy inside the sphere. For very high values of relative permeability, a small amount ofelectric field energy is still present inside the sphere (concentrated predominantly near the junction between the high- andconducting regions of the shell), such that the Q does not reach the Chu lower bound but achieves a value that is 11% abovethe Chu lower bound.Figure 9(a) Cross-section of the spherical capped monopole antenna on a large ground plane. A thin, high-n r shell is used to com-plete the sphere, in a geometry analogous to the cylindrical antenna. The center conductor of the antenna is fed throughthe ground plane. (b) Splitting the feed into two asymmetric posts and grounding one allows for matching to 50 ohms.2 Because ka = 0.522 is not < 1, we include the 1/(ka) term in addition to the 1/(ka)3 term in the Chu lower bound. 9 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • Figure 10The resonant frequency and Q-factor of the fundamental resonant mode of the shorted spherical capped monopoleantenna versus the relative permeability of the shell layer, normalized to the Chu lower bound [1/(ka)3 + 1/(ka)] . Forvery thin, very high permeability shells, the Q is 1.11 times the Chu lower bound3 . 2 M at c h i n g t o 5 0 o h m sThe shorted structure of Figure 9(a) can be made into an antenna by driving the conducting post through the ground plane.Numerical simulations confirm that the resulting antennas have resonant frequencies and Q’s corresponding to thosepredicted by the eigenmode simulations. As in the case of the cylindrical antennas, the radiation resistance is typicallysmall at resonance (< 10 ohms). However, there are well known feeding techniques that can be applied to matching theseantennas (both the spherical and cylindrical versions) to 50 ohms. One example is shown in Figure 9(b); the feed postis divided asymmetrically into two arms, with the wider arm shorted to ground and the narrow arm fed through theunderlying ground plane. Simulations of this geometry confirm that the spherical antenna can be matched to a 50 ohmimpedance (see Figure 11). In this case, the sphere radius is 100 mm, and the two posts are composed of flat plates withwidths of 2.5 mm (feed post) and 7.5 mm (shorted post), separated by 2.5 mm. The resulting antenna is matched to 50ohms at 163 MHz. The magnetic shell is 5 mm thick with n r = 100; in this configuration Figure 10 predicts a Q at 1.15times the Chu lower bound, or equivalently, a -3 dB matched fractional bandwidth of 6.2% (10 MHz) for ka = 0.341. Thisperformance is observed in the simulated antenna (see inset of Figure 11). 10 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • Figure 11The simulated impedance (solid line: resistance, dashed line: reactance) for the spherical antenna structure shown inFigure 9(b), with a 5 mm magnetic shell of n r = 100. The antenna is matched to 50 ohms with bandwidth performancecorresponding to a Q at 1.15 times the Chu lower bound.3 . 3 El e c t r i c P e r m i t t i v i t yIn all of the cases discussed so far, the relative electric permittivity in the shell material has been assumed to be 1. Becausethis is unlikely to occur in real materials, Figure 12 illustrates the effect of setting the electric permittivity of the magneticmaterial to values greater than 1 for one particular structure, namely the spherical antenna with a 2.5 mm shell thickness.The baseline structure in air achieves a Q just slightly above the Thal lower bound, and increasing the electric permittivityof the shell increases the Q relative to this bound. A correspondingly higher value of n r is then required to reduce the Q tovalues appreciably below this bound. Figure 12 makes clear the importance of minimizing the electric permittivity of themagnetic material in order to obtain the optimal performance.It is important to note that the simultaneous requirement of high relative permeability and low relative permittivity donot apply along the same axes. For the antennas considered here, the high-n r is required only in the azimuthal direction,whereas the low-e r is required only in the r- and z-directions. The use of anisotropic magnetic materials may enable somemitigation of the effects seen in Figure 12.Figure 12The Q-factor versus shell permeability of the fundamental resonant mode normalized to the Thal lower bound for thespherical capped monopole with a 2.5 mm shell. Increasing the relative electric permittivity er above 1 degrades theperformance. 11 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • 4. SUMMARYWe have illustrated how thin shells of high magnetic permeability can be utilized to reduce the radiation Q of electricallysmall, top-loaded, electric-dipole antennas. The Q’s of these antennas approach the recently derived general lower boundson Q for antennas with both electric and magnetic current excitations [14], and for the case of spherical antennas, the Q’sapproach the Chu lower bound. A relative permeability of n r = 100 would enable the realization of antennas with notablyimproved performance, provided the electric permittivity could be kept relatively small. The practical challenge of realizingthese designs is in the development of low-loss materials with high magnetic permeability and relatively low electricpermittivity. Finally, we assumed in all simulations that no frequency dispersion is present in the magnetic material,as there is no fundamental limitation preventing negligible dispersion for high permeability materials over the antennabandwidths. However, any frequency dispersion that may be present could reduce the achievable bandwidth. 12 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • 5. REFERENCES[1] L. J. Chu, “Physical limitations on omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948.[2] R. F. Harrington, “Effect of antenna size on gain, bandwidth, and efficiency,” J. Res. Nat. Bureau Stand., vol. 64D, p. 112, Jan. 1960.[3] R. E. Collin and S. Rothschild, “Evaluation of antenna Q,” IEEE Trans. Antennas Propag., vol. 17, pp. 23–27, Jan. 1964.[4] R. L. Fante, “Quality factor of general ideal antennas,” IEEE Trans. Antennas Propag., vol. 17, pp. 151–155, Mar. 1969.[5] J. S. Mclean, “A re-examination of the fundamental limits on the radiation-Q of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, pp. 672–676, May 1996.[6] A. D. Yaghjian, “Internal energy, Q-energy, Poyntings theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antennas Propag., vol. 55, pp. 1495–1505, Jun. 2007.[7] H. A. Wheeler, “The spherical coil as an inductor, shield, or antenna,” Proc. IRE, vol. 46, pp. 1595–1602, Sep. 1958.[8] O. S. Kim, O. Breinbjerg, and A. D. Yaghjian, “Electrically small magnetic dipole antennas with quality factors approaching the Chu lower bound,” IEEE Trans. Antennas Propag., to be published.[9] H. L. Thal, “New radiation Q limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2757–2763, Oct. 2006.[10] S. R. Best, “The radiation properties of electrically small folded spherical helix antennas,” IEEE Trans. Antennas Propag., vol. 52, pp. 953–960, Apr. 2004.[11] H. R. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antennas Propag., vol. 54, pp. 1644–1653, Jun. 2006.[12] H. R. Stuart and S. R. Best, “A small wideband multimode antenna,” presented at the IEEE Antennas Propag. Soc. Int. Symp., San Diego, CA, 2008.[13] H. R. Stuart, “Eigenmode analysis of small multielement spherical antennas,” IEEE Trans. Antennas Propag., vol. 56, pp. 2841–2851, Sep. 2008.[14] A. D. Yaghjian and H. R. Stuart, “Lower bounds on the Q of electrically small dipole antennas,” IEEE Trans. Antennas Propag., vol. 58, pp. 3114– 3121, Oct. 2010.[15] H. R. Stuart, “Eigenmode analysis of a two element segmented capped monopole antenna,” IEEE Trans. Antennas Propag., vol. 57, pp. 2980–2988, Oct. 2009.[16] H. A. Wheeler, “A simple formula for the capacitance of a disc on dielectric on a plane,” IEEE Trans. Antennas Propag., vol. 30, pp. 2050–2054, Nov. 1982.[17] H. A. Wheeler, “Fundamental limitations of small antennas,” Proc. IRE, vol. 35, pp. 1479–1484, Dec. 1947.[18] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag.,vol. 53, pp. 1298–1324, Apr. 2005.[19] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997.[20] H. A. Wheeler, “A helical antenna for circular polarization,” Proc. IRE, vol. 35, pp. 1484–1488, Dec. 1947.[21] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. 13 S t r at e g i c w h i t e p a p e r Approaching the Lower Bounds on Q for Electrically Small Electric-Dipole Antennas Using HigH Permeability Shells
    • 6. BiographiesHoward R. Stuart (M’98) received the S.B. and S.M. degrees in electrical engineering from the Massachusetts Institute ofTechnology, Cambridge, in 1988 and 1990, respectively, and the Ph.D. degree in optics from the University of Rochester,Rochester, NY, in 1998.From 1990 to 1993, he worked as a Research Scientist for the Polaroid Corporation, Cambridge, MA. In 1998, he joined BellLaboratories, Lucent Technologies as a Member of Technical Staff in the Advanced Photonics Research Department,Holmdel, NJ. Since 2003, he has worked in the Bell Labs Government Communications Laboratory, which became part ofLGS Innovations in 2007. He has published papers on a variety of research topics, including small resonant antennas, metalnanoparticle enhanced photodetection, multimode optical fiber transmission, optical waveguide interactions and devices,optical MEMS, and optical performance monitoring.Dr. Stuart served as the Integrated Optics Topical Editor for the OSA journal Applied Optics from 2002–2008.Arthur D. Yaghjian (S’68–M’69–SM’84–F’93–LF’09) received the B.S., M.S., and Ph.D. degrees in electrical engineering fromBrown University, Providence, RI, in 1964, 1966, and 1969.During the spring semester of 1967, he taught mathematics at Tougaloo College, MS. After receiving the Ph.D. degree hetaught mathematics and physics for a year at Hampton University, VA, and in 1971 he joined the research staff of theElectromagnetics Division of the National Institute of Standards and Technology (NIST), Boulder, CO. He transferredin 1983 to the Electromagnetics Directorate of the Air Force Research Laboratory (AFRL), Hanscom AFB, MA, wherehe was employed as a Research Scientist until 1996. In 1989, he took an eight-month leave of absence to accept avisiting professorship in the Electromagnetics Institute of the Technical University of Denmark. He presently worksas an Independent Consultant in electromagnetics. His research in electromagnetics has led to the determination ofelectromagnetic fields in materials and “metamaterials,” the development of exact, numerical, and high-frequency methodsfor predicting and measuring the near and far fields of antennas and scatterers, the design of electrically small supergainarrays, and the reformulation of the classical equations of motion of charged particles.Dr. Yaghjian is a Life Fellow of the IEEE, has served as an Associate Editor for the IEEE and URSI, and is a member of SigmaXi. He has received best paper awards from the IEEE, NIST, and AFRL. 14 © 2 0 1 2 – L G S I n n o v at i o n s L L C - All R i g h t s R e s e r v e d L G S , L G S I n n o v at i o n s , a n d t h e L G S I n n o v at i o n s l o g o a r e t r a d e m a r k s o f L G S I n n o v at i o n s L L C .