Convergence of the Quasi-static Antenna Design Algorithm

Transcription

1 TECHNICAL REPORT 016 Rev. 1 April 013 Covergece of the Quasi-static Atea Desig Algorithm T. O. Joes III Approved for public release. SSC Pacific Sa Diego, CA

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3 TECHNICAL REPORT 016 Rev. 1 April 013 Covergece of the Quasi-static Atea Desig Algorithm T. O. Joes III Approved for public release. SSC Pacific Sa Diego, CA

4 SSC Pacific Sa Diego, Califoria J.J. Beel, CAPT, USN Commadig Officer C. A. Keeey Executive Director ADMINISTRATIVE INFORMATION This report was prepared for the by the ISR/IO Departmet (Code 56), SPAWAR Systems Ceter Pacific, Sa Diego, CA. Released by M. D. Osbur. Head Electromagetics Techology Brach Uder authority of J. McGee, Head SoS & Platform Desig Divisio This is a work of the Uited States Govermet ad therefore is ot copyrighted. This work may be copied ad dissemiated without restrictio. MATLAB is a registered trademark of The Math Works. SB

5 EXECUTIVE SUMMARY The quasi-static atea desig algorithm is a scietific approach to desigig electrically small ateas. Electrically small ateas are small compared to wavelegth. The radiatio resistace is computed from the electrostatic dipole potetial. The capacitace is computed from the electrostatic potetial o the eclosed sphere. The Q is calculated from the radiatio resistace ad capacitace. The geeral thick-disk-cap moopole, eclosed by a sphere, is modeled with electrostatic multipole basis fuctios o the disk. A sequece of solutios coverges i shape ad Q. Computer Simulatio Techology (CST) Microwave Studio is used to compute the impedace of the thick-disk-cap moopole. The impedace is umerically fit to the dipole ad octupole eigemode equivalet circuit. The atea eigemodes are aalogous to radio frequecy (RF) cavity eigemodes. The radiatio resistace is computed from the dipole eigemode. The DC capacitace is accurately computed. The curret sharig betwee the dipole ad octupole eigemodes slightly decreases the radiatio resistace ad slightly icreases the atea Q. The quasi-static atea desig algorithm is a good approximatio of the exact solutio. The figure of merit is the Q-factor ratio, the ratio of the Q to Q Chu, L. J. Chu s [1] theoretical limit for a atea eclosed by a sphere. The thick-disk-cap moopole has the absolute miimum Q- factor ratio of The spherical-cap moopole has a slightly lower Q-factor ratio of At the resoace, the thick-disk-cap moopole s Q is.5. This is much smaller tha the spherical-cap moopole s Q of Above the resoat frequecy, the thick-disk-cap moopole is a superior desig. iii

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7 CONTENTS EXECUTIVE SUMMARY... iii 1. INTRODUCTION CAPACITANCE FOR THE OBLATE SPHEROIDAL ANTENNA GENERAL CALCULATION OF RADIATION RESISTANCE GENERAL OBLATE SPHEROIDAL ANTENNA SECTION CST MODEL OF OBLATE SPHERICAL ANTENNA CST MODEL OF A SPHERICAL-CAP MONOPOLE CONCLUSION REFERENCES Figures 1. First four top-load radial multipole basis fuctios A sequece of atea desigs with oe to five load multipoles The atea shape ear the eclosig surface Electric field calculated as a fuctio of distace from the atea top The three-dimesioal surface with a 45 slice removed The equivalet circuit for the first ad secod eigemode approximatio CST data ad two-eigemode equivalet circuit model Differece betwee the CST data ad the two-eigemode model for ρ < The 4 ad cotributio to resistace Q for the miimum Q desig Q-factor ratio for the miimum Q desig CST impedace ad two-eigemode model Differece betwee the CST data ad the two-eigemode model The ad compoets of resistace Q for spherical-cap moopole Q-factor ratio for spherical-cap moopole... 8 Tables 1. Least square best fit to Stuart's eigemode impedace circuit model Spherical-cap moopole impedace is fit to a two-eigemode model... 3 v

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9 1. INTRODUCTION Electrically small ateas are commo to portable electroics devices, persoal digital assistats (PDAs), cell phoes, etc. These ateas are small compared to wavelegth, i.e., electrically small ateas. Desigig electrically small ateas is a art; they ormally have a low radiatio resistace, a large reactace, ad a arrow badwidth. Electrically small ateas are used whe reducig the atea size is a critical desig goal. Additioal matchig compoets are required to elimiate the reactace ad icrease the resistace. The objective is to develop a scietific method for desigig electrically small ateas. The radiatio resistace ad capacitace are optimized to give the miimum Q atea. L. J. Chu [1] derived a lower limit for the Q of electrically small ateas. Chu's Q calculatio is based o the radiated eergy ad the stored eergy outside a sphere eclosig the atea; the eergy iside the sphere is assumed to be zero. H. L. Thal [] refied this limit by assumig the atea curret is limited to the surface of the eclosig surface. The folded spherical helix desig by S. R. Best [3] meets Thal's limit. M. Gustafsso, C. Sohl, ad G. Kristesso [4] derived a Q limit based o the optical theorem. A. D. Yaghjia ad H. R. Stuart [5] derived a more restrictive limit o Q. The eergy iside the sphere limits the atea performace. The thi-disk-cap moopole is a leaky capacitor with a large amout of stored eergy iside the sphere. The electric field uder the disk is larger tha the field above the disk. The quasi-static atea desig algorithm [6, 7] aalytically computed a thick-disk-cap moopole. The thick-disk cap reduces the stored eergy ad Q by fillig some of the spherical volume with coductor. The electric field uder the top load is still larger tha the field above the top load. This paper shows how to reduce the electric field uder the top load ad icrease the radiatio resistace of the atea [8]. The atea shape shifts to elimiate the gap (ad the eergy) betwee the top of the atea ad the eclosig sphere. The stored eergy iside the eclosig sphere is limited to the regio betwee the atea ad the groud. Electrically small ateas have electric fields much larger tha the magetic fields below the atea resoace. The Asymptotic Coical Dipole (ACD) [9, 10, ad 11] was the first atea desiged with quasi-static methods. I electrostatics, a perfect coductor is the same as a equipotetial surface. A lie of costat charge o the z-axis, with a image, will geerate the ACD atea desig. Each ACD atea has a differet height. The quasi-static atea desig algorithm [7] fixes the atea height to a costat a ad the legth of the lie charge a is varied; the atea fits withi a eclosig sphere with a radius a. The parameter is a dimesioless. The Q is calculated from 1, (1) Q CRRad where R Rad is radiatio resistace, C is capacitace, ad is agular frequecy. The quatities R Rad ad C are fuctios of. The radiatio resistace is calculated from the effective height. The capacitace is calculated from the charge o the atea arm ad the maximum potetial o the spherical eclosig surface. 1

10 The above equatio for Q is valid below resoace ad it gives oly the first term i Chu's equatio: 1 1 Q, 3 () ka ka where k / ad a is the radius of the eclosig sphere. The ACD desig is exteded by addig a disk-shaped charge distributio as a load o the lie charge [7]. A coductig disk i free space is used as the charge distributio. This disk charge distributio is symmetric. The electric fields betwee the disks ad image will be larger tha the electric fields above the disk. There is o requiremet for the disk charge distributio to be symmetric. Addig a dipole momet moves the charge from the bottom of the disk to the top of the disk. This reduces the electric field betwee the disks. A series of multipole charge distributios ca be added to the disk to model the geeral charge distributio o the disk. Sectios ad 3 are ot eeded to uderstad the results give i Sectios 4 ad 5. Sectio shows how the potetial ad top load capacitace is computed from electrostatic solutios i oblate spheroidal coordiates. Each solutio represets a uique multipole momet with a uique potetial. Oly the rotatioally symmetric top-load multipole modes will be used i this model. The th 1 multipole momet falls off as 1 i the far field. The potetial for the dipole multipole term ad r higher odd multipole momets add i the far field. The moopole ad higher eve multipole momets cacel i the far fields. All of the multipoles make a uique ad dimiishig cotributio to the ear fields. I sectio 3, the effective height is calculated. The effective height calculatio with dipole momet top load is o-trivial. The effective height caot be calculated with the covetioal formula [8]. The effective height is calculated idirectly from the potetial o the eclosig sphere. The perfect coductor boudary coditio, E 0, requires the charge distributio to be eclosed by the atea surface. Oly a subset of the charge distributios satisfies this boudary coditio. The multipole momets have egative potetials, which ca cause the equipotetial surface to termiate o the disk or feed wire. This requires a additio step i the solutio process; the equipotetial surface is sampled to verify that the charge is eclosed by the equipotetial surface. The fial solutio must be verified with a detailed calculatio of the atea shape. I Sectio 4, a sequece of multipole basis fuctios is used to desig miimum Q ateas. The atea desigs appear to coverge i both shape ad Q. The fial atea desig fills the top of the sphere; the area uder the atea is the oly regio with electric fields that cotributes to stored eergy iside the eclosig sphere. The electric field o the surface of the atea is plotted. The electric field uder the atea is reduced by addig the dipole momet. The fial atea has a almost horizotal lower surface. I Sectio 5, Computer Simulatio Techology (CST) Microwave Studio is used to calculate the impedace ad Q for a 1-m high atea with a 1-cm-diameter feed lie. The impedace is calculated with a sequece of eergy-based adaptive iteratios. L. J. Chu [1], H. D. Foltz, J. S. McLea, ad L. Boder [1], ad H. Stuart [13] used a series, L, C, with a resistor, R 0, parallel to the iductor to model both the reactace ad depedece radiatio resistace. The model represets the dipole eigemode. The least squares fit to the CST impedace is a good fit to the reactace. For the resistace, the two curves cross at oly o poit; at other poits the percetage 4 error is large. The radiatio resistace has a ad frequecy-depedet terms; the coefficiets are calculated with a least square fit. The cotributio gives the same effective 4 height as the quasi-static atea desig algorithm. The frequecy-depedet term is the source

11 of the error i the circuit model. The ext eigemode, the octopole, is a circuit coected i parallel with the dipole eigemode model [13]. The octopole (ad higher) equivalet circuits reduce to a capacitor at low frequecies. The radiatio resistace for the octopole eigemode is ot icluded i the simple equivalet circuit. This capacitace combied with dipole eigemode circuit gives a atiresoace. The dipole ad octopole eigemode currets iterfere with each other to create a high impedace at atiresoace. A least square iteratio method was used to calculate the circuit elemets. The improved fit to the 4 radiatio resistace data is attributed to a frequecy-depedet term itroduced by the parallel capacitace. A aalysis of the circuit impedace shows that curret sharig betwee the eigemodes itroduces a small error i the quasi-static atea desig algorithm. The Qad resistace are modified by curret sharig factor. The DC capacitace is accurate. The extra capacitace itroduced by the feed lie is estimated. I Sectio 6, CST is also used to calculate the impedace of the spherical-cap moopole. The same feed wire was used for the 1-m spherical-cap moopole. The data is fit to the same dipole eigemode with a capacitor approximatio of higher eigemodes. The capacitace of the top-load is larger tha the above desig. The effective height is smaller. The Q-factor ratio is about the same as the result obtaied by A. R. Lopez [14]. Electrostatic data is ot available to compare to the circuit results. Sectio 7 is the coclusio. The quasi-static atea desig algorithm gives a solutio very close to the spherical-cap moopole (the top-loaded moopole desig with the lowest Q). The algorithm does ot model all possible atea shapes. The algorithm oly cosiders atea shapes that eclose the source charge o the disk. A thick-spherical-cap moopole could have a lower Q. The quasistatic atea desig algorithm produces a very good atea desig with modest effort. Other desigs ca be computed with a sigificatly higher radiatio resistace ad a slightly larger Q. The algorithm ca be adapted to other eclosig surfaces. 3

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13 . CAPACITANCE FOR THE OBLATE SPHEROIDAL ANTENNA The detailed discussio of the algorithm is give i Referece 7. This discussio is the geeral case. The capacitace of the geeral thick-disk cap is calculated from the potetial o eclosig surface q CV. The potetial is calculated from a aalytic solutio to the electrostatic equatios. The top of the feed lie has a height a ad 1 q charge; the potetial [15, 16] is m R f R t ACD 1 q 1 1 m i l a, (, z) (3) 0 m i a where is for the moopole, is for the image moopole, R t z a is i a R f R b the distace from, z to the top of the moopole, R f z is the distace from, z to the feed poit, ad R b z a is the distace from, z to the bottom of the image moopole. The parameter is the fractio of the total charge q o the atea top load. The superscript idicates a ACD stem. The parameter is dimesioless, 0 1. The stem radius is ot costat; it tapers to a zero as the feed poit is approached The disk also has the height a, a disk radius of a, ad a et charge q where the parameter is dimesioless ad has the rage of values 0 1. G. Arfke [17, pp ad 596] gives the geeral solutio to potetial o the disk, u, v, 0. The potetial o the disk is a m m im liear combiatio of multipole momets, oblate spheroidal coordiate system: K ( u) P cosv e, where v ad u are defied by the z asih( u)cos( v), (4) x a cosh( u)si( v)cos( ), (5) y a cosh( u)si( v)si( ), (6) where a is the disk radius. The atea s desig with coordiate system is called a oblate spheroidal atea (). The atea problem will be simplified by assumig rotatioal symmetry where acosh( u)cos( v) : bm m m im u, v, K ( u) P cosv e. (7) m 0 m0 5

14 The rotatioal symmetry i requires b 0 for m 0 ad the associated Legedre Polyomials 0 reduce to Legedre Polyomials, P cosv P cosv coveiece. The K (u) fuctios are as follows: K m. The u arccot(sih ), 0 u 1 / is added for umerical K (8) K ( u) 1 sih( u) arccot(sih( )), (9) 1 u u (3sih ( u). 1) arccot(sih( u)) 3sih( ) /, (10) u ad K 3( u) sih ( u) sih( u) arccotsih( u) sih ( u). (11) 3 6 The geeral case is K 1 i Q ( i ), (1) where 1 5 Q ( i) ip i arccot P 1i P 3i... (13) 1 3 The fuctio K at large distace is calculated by expadig cot( sh( u)) a ta1/ sih( u) power series of 1 sih( u ). After simplifyig, the leadig term is a i a (14) 1 K1 for u >>, (15) 3 sih ( u) (16) ad (17) 6

15 a a 3 sice a sih( u) z, the above equatios reduce to,, z 3 z 15 z a 4, ad 35 z a. This is the expected multipole expasio of the field o the z-axis. Figure 1 plots K as a fuctio of sih(u ); the 1 1 plot shows the 1 depedece. The factor is icluded i Equatio (7) to elimiate the α z q depedece i the far field. The value of b is calculated from the solutio of the charged 00 40a q coductig disk i free space V ( u) arccot(sih ) u 40a, where q is the et charge o the disk. The oly restrictio placed o the values of b is the charge distributio must be eclosed withi a m equipotetial surface. Figure 1. First four top-load radial multipole basis fuctios. The combied potetial is ACD TopDisk BotDisk z (, z) u, v u, v,, t t b b (18) where the variables u t, v t, ad u b, v b are defied from z a asih( u t )cos( v ), (19) t a cosh( u t )si( v ), (0) t z a asih( u b )cos( v ), (1) b a cosh( u b )si( v ). () b 7

16 The capacitace is calculated from the maximum value of z, asi : a cos, asi where 0 / o the sphere z a cos ad Max max (3) ad q C. (4) Max I the above coordiate systems, v 0 ad v 0 are defied i the z directio. The charge BotDisk distributio that geerates the term u b,vb TopDisk u,v : t t t b is the mirror image of the charge distributio for b u, v K ( u ) P cosv TopDisk t t t t (5) 0 ad BotDisk u, v 1 K ( u ) P cosv b b b 0 b b. (6) b The sig o K( ub) P cosvb follows from v b The dipole term is the simplest example to uderstad: K ( ut ) P1 cosv t top of the disk ad K u ) P cos ( b 1 v b P cos, beig eve or odd relative to v /. 1 has positive charge o the 1 has positive charge o the top of the image disk (top is +z directio for both disks). Both terms have a egative charge o the bottom of the disk ad image disk. The combied terms K cosv K ( u ) P cosv 1( ut ) P1 t 1 b 1 b (7) are odd about z 0. For the case, b 1 0, the dipole momet of the atea is icreased ad the electric field uder the disk is decreased (with the other parameters uchaged). I the geeral case, whe is odd, the terms icreases the cosv K ( u ) P cosv for odd K ( u ) P (8) t th multipole momet. Whe is eve, the terms t b cosv K ( u ) P cosv for eve b K ( u ) P (9) t t subtract i the far field the first cotributig far-field momet is the 1 th multipole momet. Each term has a uique cotributio to the electric field uder the atea. The ext step is to compute the radiatio resistace. b b 8

17 3. GENERAL CALCULATION OF RADIATION RESISTANCE The effective height for a rotatioally symmetric charge distributio is h Eff 1 za a z zq(, z)ddz. (30) q z0 0 Net From a previous paper [6, 7], this simplifies to h Eff a 1 a. (31) The applicatio of the above effective height calculatio to the dipole term K ( u) P1 cos v obvious. The K1 ( u) P1 cos v term is discotiuous at 0. K 0) P cos v K (0) P cos v 1 is ot u The fuctio is odd about v / or 1 ( The dipole term would give zero for a thi disk with o et charge. To derive the geeral case, the dipole momet used as the startig poit is za a z za 0 p zq(, z)ddz. (3) z The potetial outside (ad o) the sphere ca be represeted as a sum of spherical harmoics: 1 a 0 (33) 1 r r, P cos for r a, where r z p with all of the charge eclosed i a sphere of radius a. The dipole momet is computed from the potetials, 3 p z a a, P1 cos sid. (34) 0 The two equatios yield p z a 10. The effective height is h Eff 1 p z. (35) q Net 9

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19 4. GENERAL OBLATE SPHEROIDAL ANTENNA I the calculatio for the thi-disk-cap moopole, the feed lie is eglected. The capacitace calculatio assumes all of the charge is o the top load disk. The MATLAB calculatio of the oblate spheroidal atea assumes that 99 % of the charge is o the top load ad a toke 1 % of the charge is o the feed lie: ad The 1% charge o the feed lie makes a ½% chage o the radiatio resistace ad about a 1% chage i the capacitace. The feed lie radius tapers to zero at the feed poit. As show i Figure, the atea desigs with four ad five multipoles are almost idetical ad Q-factor ratios have a 0.03% differece. The atea desig appears to coverge i shape ad Q-factor ratio. Radial Distace Figure. A sequece of atea desigs with oe to five load multipoles. Figure 3 shows a detailed plot of the atea surface ear the sphere. Spherical coordiates r z are used for the vertical axis ad arcsi( / r) is used for the horizotal axis. As basis fuctios are added to the atea desig, the shape approaches the surface of the eclosig sphere. Figure 3 shows that addig extra multipole momets has a dimiishig impact o the Q-factor ratio. 1 The multipole momets fall off as 1/ r ; they have dimiishig impact o the electric field outside the sphere ad uder the atea. The electric field is calculated from the potetial ad is plotted i Figure 4. The vertical scale is electric field magitude plotted as a fuctio of distace alog the surface measured from the top of the atea, z 1 ad 0. The electric field uder the atea is reduced by the dipole momet 11

20 term. The spike i the electric field is at the trasitio from almost spherical surface to the atea uderside. For the solutio with five multipoles, this trasitio is very sharp. Figure 4 shows that the positio of the spike moves to the right as multipoles are added. The added multipoles are icreasig the surface area of the top of the atea ad the total charge o the top of the atea. Figure 3. The atea shape ear the eclosig surface. Figure 4. Electric field calculated as a fuctio of distace from the atea top. 1

21 5. SECTION CST MODEL OF OBLATE MONOPOLE ANTENNA Computer Simulatio Techology (CST) Microwave Studio T-solver was used to model the atea; Figure 5 is a plot of the atea surface with a 45 slice removed to highlight the edge. The tapered feed lie i the atea caot be modeled with Microwave Studio. To simplify the CST model, the tapered feed lie is replaced with a 1-cm-diameter hollow feed lie. The hollow feed lie was used to reduce the capacitace at the feed poit. The small capacitace itroduced by the feed lie is estimated i this sectio. A accurate model for the eergy is critical for Q calculatios; CST icludes a eergy-based adaptive meshig that icreases the mesh resolutio i areas of high-eergy desity. A sequece of adaptive meshig models was ru to refie the startig mesh for the ext model 1. A 1-cm diameter, 4.3-ohm cylidrical edge port was used at the feed poit. The curret flows o the cylidrical edge port surface. A lie source would have a much larger iductace. The excitatio frequecy rage was from 15 to 5 MHz; this reduces the eergy reflected at the iput to the atea. The hollow feed lie was modeled idepedetly to determie the required meshig. The CST calculatio used the exact atea geometry. The covergece i the solutios is evaluated by lookig at the largest differece betwee previous ad curret reflectio coefficiet S 11. I the fial ru, this differece i S 11 is 3.4e-4. This process was time cosumig, but it elimiated the risk of usig too fie a mesh ad itroducig umerical errors i the solutio. The atea capacitace, iductace, Q, ad effective height ca be extracted from the CST impedace data with a MATLAB program. The data with a reflectio coefficiet, 0. 9, is used i the aalysis of the impedace. The MATLAB program calculated the least square best fit to Stuart's eigemode impedace circuit model, Figure 6 ad Table 1. The series C ad L with a parallel resistace R 0 represets the dipole eigemode of the atea. The fit to the reactace was very good but the radiatio resistace has a large error. A additioal octopole eigemode ca be modeled at low frequecies as a parallel capacitor C, where the subscript idicates octopole plus OP higher order eigemodes (the iductors short the other circuit elemets at low frequecies). This approximatio is very accurate; the octopole eigemode 3 is resoat at 4. MHz with oly 3.17 Ω radiatio resistace. The additio of the COP elimiates the error i the umerical fit to the radiatio resistace. The impact of COP o the reactace is ot sigificat. The superscript idicates the atea type as a (oblate spheroidal atea). The circuit parameters are computed with a iteratio process. A two-step iteratio process is used to miimize the error i the radiatio resistace. The dipole eigemode L ad C values are a good approximatio to the reactace; they are used as fixed iputs to a least square fit for R 0 ad C. The ew R 0 ad C values are used as fixed iputs to OP a least square fit for improved L ad C values. This iteratio process was repeated util a self-cosistet umerical solutio was obtaied. Figure 7 shows the CST impedace ad the two OP 1 Shrikrisha Hegde at Soet Software provided isight ito eergy adaptatio ad meshig. James Whillhite at Soet Software suggested matchig the source resistace to the atea resistace ad usig a arrow frequecy rage to maximize the eergy delivered to the atea. 3 The atea was excited by a short voltage pulse. After the iitial excitatio, V 0 at the feed poit, the atea is shorted to the groud. 13

22 eigemode circuit approximatio of the impedace. The differece is so small it is plotted i Figure 8. The errors itroduced by the CST umerical computatio ad the eigemode circuit model approximatio are idepedet. The small differece betwee the two models idicates that both are accurate. Figure 5. The three-dimesioal surface with a 45 slice removed. Figure 6. The equivalet circuit for the first ad secod eigemode approximatio. The DC capacitace is C C C (36) DC CST OP The impact of the 1-cm-diameter hollow feed lie ca be estimated. The curret o the feed lie is almost costat; almost all of the charge flows oto the top load. A reasoable approximatio for the charge distributio o the feed lie is a triagular distributio. The et charge o the feed lie is oehalf the moopole charge distributio (costat charge distributio for electrically small moopole). The capacitace of the m high moopole is 6. pf. The feed-lie capacitace is about 3.1 pf. The quasi-static atea desig algorithm assumes 1% of the total charge is o the tapered feed Stem lie; the capacitace is 0.9 pf pf. This differece is C. pf. 14

23 The differece betwee the two calculatios is about 3.3 pf or about 3.6%: Stem C C C (37) DC QS QS Table 1. Least square best fit to Stuart's eigemode impedace circuit model. Parameters Eigemode 1 Eigemode 1 & QSADA C DC pf C pf pf N/A L uh uh N/A R K K N/A f MHz MHz N/A h Eff N/A C OP N/A pf N/A Q-factor ratio at resoace N/A Q-factor ratio a / 40 N/A 1.8 The impedace of the circuit is rewritte to give isight ito the Figure 6 curret flow. The quasistatic atea desig algorithm calculates the DC capacitace. The fractio of curret flowig ito the dipole eigemode is C / C COP I additio to curret sharig, there is a circulatig curret flowig betwee the two eigemodes. The L C icreased the radiatio OP resistace ad iductace. The value L COP is ear the resoace ad 1 at the atiresoace. The term L / R is also small ear resoace. I the iteratio process, chages i the value of R 0 ad C make a isigificat cotributio to the reactace: X Total j C R Total OP 1 j L 1 L COP C OP 1 L COP L / R0 L R0. R 1 L C L 0 The term L C makes a larger cotributio to the resistace ear resoace, 1%. OP C L / R ad OP 0 ; the values of L ad without restrictig the accuracy of the least squares fit to the resistace. The resistace depeds o the ratio OP I the limit as 0, the radiatio resistace ad Q reduces to R Total L, R 0, C ca be fixed 15

24 where Q Total C The above term ca be simplified by itroducig the idicates the dipole eigemode Q where The [18]: Dipole EM DipoleEM 1 C 1, C R R Dipole EM OP R Total. Q Dipole EM, where the subscript DipoleEM 0 L. R Q of the dipole eigemode is icreased by Dipole EM 1 /, the same result cited by H. R. Stuart Q Total QDipole EM /. The radiatio resistace, Figure 9, has two compoets R b 4 / b /, 1 r r 4 ad terms, where r is the agular resoat frequecy, b ad b The effective height i Table 1 is calculated from the term i the radiatio resistace. The quasi-static atea desig algorithm is close to the value i Table 1; but, it does ot iclude the curret sharig factor i the radiatio resistace. The curret sharig factor icreases the Q by 1/ The octopole Dipole EM eigemode icreases the radiatio resistace at higher frequecies. The Q, show i Figure 10 is umerically calculated from the fit data usig the A. D. Yaghijia ad S. R. Best [19] result: Figure 11 shows the 3 dr X dx Q. (39) R d d Q ka Q-factor ratio for the atea that was calculated for a exteded frequecy rage. The miimum Q-factor ratio is The Q at resoace is

25 Figure 7. CST data ad two-eigemode equivalet circuit model. 17

26 Figure 8. Differece betwee the CST data ad the two-eigemode model for ρ <

27 Figure 9. The 4 ad cotributio to resistace. 19

28 Figure 10. Q for the miimum Q desig. 0

29 Figure 11. Q-factor ratio for the miimum Q desig. 1

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31 6. CST MODEL OF A SPHERICAL-CAP MONOPOLE The same method was used to model a spherical-cap moopole. The edge of the spherical-cap moopole requires a higher resolutio CST model. If the edge thickess was set too small, CST will ot accurately model the coductor ad the electric field. A accurate model of the electric field is critical to the calculatio of the Q. The shell thickess is 0.83 cm; the horizotal edge is 1 cm ad the height of the edge is m. A. R. Lopez [14] published a Q-factor ratio of Q ka , where a The same MATLAB program was used to umerically fit the data to a two-eigemode model, Table ad Figure 1. The superscript SP is for spherical-cap moopole. Figure 13 shows the magitude of the resistace ad reactace error is ad 0.00., respectively. The spherical-cap moopole has a larger surface area ad a smaller charge desity larger tha the above atea. The potetial o the spherical-cap moopole will be smaller ad the DC capacitace is 8% larger tha the above atea. The effective height is 5% smaller tha the above atea. The iductace of the spherical-cap moopole is 5% larger tha the oblate spheroidal atea. A thier spherical shell will have less charge o the edge ad a higher charge desity o the sphere. The higher charge desity icreases the potetial o the sphere, which decreases the capacitace. Movig the edge charge to a higher positio icreases the effective height. H. R. Stuart ad A. D. Yaghjia [0] poited out that a thicker shell has a higher Q-factor ratio. The feed-lie aalysis is the same as above. A 1-m moopole has 9.54-pF capacitace. The feed lie would have a capacitace aroud 5 pf. The CST model is reasoably accurate. The higher iductace reduces the resoace to MHz. The Q at resoace is 46.1, much higher tha the oblate spheroidal atea. 4 The radiatio resistace, show i Figure 14, has two compoets, ad : R SP 4 / s /, s (40) 1 r r where r is the agular resoat frequecy, s ad s The Q is plotted i Figure 15 ad the Q-factor ratio is plotted i Figure 16. Table. Spherical-cap moopole impedace fit to a two-eigemode model. Parameters Eigemode 1 Eigemode 1 & C SP pf pf L SP µh µh SP R SP f MHz MHz h Eff N/A C SP OP N/A pf Q Factor N/A

32 Figure 1. CST impedace ad two-eigemode model. 4

33 Figure 13: Differece betwee the CST data ad the two-eigemode model. 5

34 4 Figure 14. The ad compoets of resistace. 6

35 Figure 15. Q for spherical-cap moopole. 7

36 Figure 16. Q-factor ratio for spherical-cap moopole. 8

37 7. CONCLUSION The quasi-static atea desig algorithm modeled a thick-disk cap as a sum of multipole momets. The algorithm appears to coverge i Q ad shape. I the case of a spherical eclosure, the Q is oly slightly larger tha the miimum Q spherical-cap moopole. Above the resoat frequecy of the oblate spheroidal atea, the Q is oe-half of the spherical-cap moopole. This is caused by the reduced iductace of the feed lie. This icreases the resoat frequecy ad greatly reduces the Q. This algorithm oly icluded solutios with the (charge distributio) disk eclosed iside the atea surface. The allowed solutios are limited to the thick-disk-cap dipole. A thick-spherical-cap moopole could be thick at the ceter ad taper to a thi edge. The thick ceter could reduce the stored eergy uder the spherical cap. The dipole momet ad radiatio resistace would be reduced. The thick-spherical cap could reduce Q-factor ratio. The electromagetic properties of a atea ca be described as a liear combiatio of eigemodes. The quasi-static atea desig algorithm is a method for calculatig the radiatio resistace of the dipole eigemode (the ω term) ad the DC capacitace of the atea. The threeelemet circuit model of the dipole eigemode impedace does ot accurately explai the impedace ad Q. A accurate model for the impedace requires a cotributio from the octopole ad higher eigemodes. The octopole ad higher eigemode ca be approximated with a parallel capacitor. The iteractio of the dipole ad octopole eigemode itroduces the ω 4 term required i the radiatio resistace. The quasi-static atea desig algorithm igores the octopole ad higher order eigemodes of the atea. The curret sharig betwee the dipole eigemode ad higher eigemodes is also igored. This itroduces a small error i the Q ad dipole eigemode radiatio resistace. The circulatig curret i the circuit model icreases both the effective radiatio resistace ad the effective iductace. The quasi-static atea desig algorithm is a excellet approximatio of electrically small ateas. I most cases, the error caused by the curret sharig factor ratio should be isigificat. 9

38

39 8. REFERENCES 1. L. J. Chu Physical Limitatios of Omidirectioal Ateas, Joural of Applied Physics, vol. 19, o. 1 (December), pp H. L. Thal New Radiatio Q Limits for Spherical Wire Ateas, IEEE Trasitios. Ateas ad Propagatio, vol. 54, o. 10 (October), pp S. R. Best The Radiatio Properties of Electrical Small Folder Spherical Helix Ateas, IEEE Trasactios Ateas ad Propagatio, vol. 5 (April), pp M. Gustafsso, C. Sohl, ad G. Kristesso Illustratios of New Physical Bouds o Liearly Polarized Ateas, IEEE Trasactios o Ateas ad Propagatio, vol. AP-57, o. 5 (May), pp A. D. Yaghjia ad H. R. Stuart Lower Bods o the Q of Electrically Small Dipole Ateas, IEEE Trasactios o Ateas ad Propagatio, vol. 58, o. 10 (October). 6. T. O. Joes III. 01. Quasi-Static Desig Approach for Miimizig the Quality Factor ' Q ' for Electrical Small Ateas, U.S. Patet 8,11,81 B1 (February 1). 7. T. O. Joes III A Quasi-Static Atea Desig Approach for Miimum-Q Ateas, IEEE Ateas ad Propagatio Magazie, vol. 53, o. 3 (Jue), pp T. O. Joes III Dipole Momet Term for a Electrically Small Atea, U.S. Patet B1 (February 5). 9. S. A. Schelkuoff ad H. T. Friis Ateas, Theory ad Practice, p. 318, Figure 10.9, Joh Wiley & Sos, Ic., New York, NY. 10. N. C. De, T. K. Ghosh, D. R. Poddar, ad S. K. Chowdhury Desig ad Experimetal Ivestigatio of the Asymptotic Coical Dipole Atea, IEEE Trasactios o Electromagetic Compatibility, vol. 37, o. (May), pp T. Simpso The Scheikuaff-Friis Dipole: the Simplest Atea of All, IEEE Ateas ad Propagatio Magazie, vol. 48, o. 4 (August), pp H. D. Foltz, J. S. McLea, ad L. Boder. 00. Closed-form Lumped Elemet Model for Folded, Disk Loaded Moopoles. Proceedigs of IEEE Ateas ad Propagatio Society Iteratioal Symposium (pp ) Jue, Sa Atoio, TX. 13. H. R. Stuart Eigemode Aalysis of a Two Elemet Segmeted Capped Moopole Atea, IEEE Trasactios o Ateas ad Propagatio, vol. 57, o. 10 (October), pp A. R. Lopez Fudametal Limits of Small Ateas: Validatio of Wheeler's Formula, IEEE Atea ad Propagatio Magazie, vol. 48, o. 4 (August), pp The author derived this result from [16]; he is ot first to calculate this result. 16. I. S. Gradshtey ad I. M. Ryzhik Table of Itegrals, Series, ad Products, Corrected ad Elarged Editio, Fourth Editio. Academic Press, New York, NY. 17. G. Arfke Mathematical Methods for Physicists, Secod Editio. Academic Press, New York, NY. 31

40 18. H. R. Stuart. 01. Uderstadig the Badwidth Limitatios of Small Ateas: Wheeler, Chu ad Today, Ateas ad Propagatio Society Iteratioal Symposium (APSURSI) (pp. 1 ) July, Chicago, IL. Istitute of Electrical ad Electroics Egieers (IEEE). 19. A. D. Yaghjia ad S. R. Best Impedace, Badwidth ad Q of Atea, IEEE Trasactios o Ateas ad Propagatio, vol. 53, o. 4 (April), pp H. R. Stuart ad A. D. Yaghjia Approachig the Lower Bouds o Q for Electrically Small Electric-Dipole Ateas Usig High Permeability Shells, IEEE Trasactios o Ateas ad Propagatio, vol. 58, o. 1 (December), pp

41 REPORT DOCUMENTATION PAGE Form Approved OMB No The public reportig burde for this collectio of iformatio is estimated to average 1 hour per respose, icludig the time for reviewig istructios, searchig existig data sources, gatherig ad maitaiig the data eeded, ad completig ad reviewig the collectio of iformatio. Sed commets regardig this burde estimate or ay other aspect of this collectio of iformatio, icludig suggestios for reducig the burde to Departmet of Defese, Washigto Headquarters Services Directorate for Iformatio Operatios ad Reports ( ), 115 Jefferso Davis Highway, Suite 104, Arligto VA Respodets should be aware that otwithstadig ay other provisio of law, o perso shall be subject to ay pealty for failig to comply with a collectio of iformatio if it does ot display a curretly valid OMB cotrol umber. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY). REPORT TYPE 3. DATES COVERED (From - To) April 013 Fial 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Covergece of the Quasi-static Atea Desig Algorithm 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHORS T. O. Joes III 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER SSC Pacific 56 Hull Street Sa Diego, CA SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) Noe TR 016 Rev SPONSOR/MONITOR S ACRONYM(S) 11. SPONSOR/MONITOR S REPORT NUMBER(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release. 13. SUPPLEMENTARY NOTES This is work of the Uited States Govermet ad therefore is ot copyrighted. This work may be copied ad dissemiated without restrictio. 14. ABSTRACT The quasi-static atea-desig algorithm uses multipole basis fuctio to model the geeral thick top load. A sequece of solutios coverge i shape ad Q. The absolute miimum Q-factor, 1,85, is obtaied for a thick disk top load eclosed by a sphere. This is sigificatly smaller tha the thi disk top load Q-factor.349 ad previously derived thick-disk Q-factor.078. A aalytic potetial is derived for each multipole basis fuctio. The capacitace ad effective height is calculated from the potetials o the eclosig sphere. The impedace is computed with Computer Simulatio Techology (CST) Microwave Studio. The impedace data is umerically fit to a dipole eigemode equivalet circuit. The radiatio resistace does ot fit the expected frequecy depedece (effective height). 4 The error is a term that is explaied by a capacitor approximated for the octupole eigemode equivalet circuit. The quasistatic atea desig algorithm predicts the DC capacitace ad the dipole eigemode effective height. The octupole eigemode icreases the radiatio resistace. The Q-factor, 1.77, is lower tha expected. These results are compared to the spherical cap top 4 load. The existece of a limits the accuracy of the theoretical limits i the Q values for ateas. 15. SUBJECT TERMS oblate spheroidal atea Q-factor multipole basis fuctio radiatio resistace quasi-static atea equivalet circuit model top load dipole eigemode two-eigemode circuit model 16. SECURITY CLASSIFICATION OF: a. REPORT b. ABSTRACT c. THIS PAGE 17. LIMITATION OF ABSTRACT 18. NUMBER OF PAGES 19a. NAME OF RESPONSIBLE PERSON T. O. Joes III U U U U 46 (619) B. TELEPHONE NUMBER (Iclude area code) Stadard Form 98 (Rev. 8/98) Prescribed by ANSI Std. Z39.18

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