Determination of Antenna Q from the Reflection-Coefficient Data

Transcription

1 Antnna Dsignr s Notbook Foundd by Hal Shrank Tom Milligan 84 W Polk Pl Littlton CO 8 USA Tl: + () tmilligan@i.org http// Dtrmination of Antnna Q from th Rflction-Cofficint Data Darko Kajfz and Atf Z. Elshrbni Cntr of Applid Elctromagntic Systms Rsarch (CAESR) Elctrical Enginring Dpartmnt Univrsity of Mississippi Univrsity MS USA darko@olmiss.du Elctrical Enginring and Computr Scinc Dpartmnt Colorado School of Mins Goldn CO 84 lshrbni@i.org Abstract Equivalnt circuits containing frquncy-variabl radiation rsistancs ar postulatd for small monopol and small loop antnnas abov a ground plan. Th quivalnt circuits ar valid blow th fi rst natural rsonanc frquncy. By adding an xtrnal sris inductanc or a paralll capacitanc th antnna s Q is valuatd with th us of an invariant form for th rfl ction coffi cint. Th antnna s Q obtaind this way accuratly prdicts th maximum obtainabl bandwidth for ithr a singl- or doubl-tund matchd antnna. At frquncis abov th fi rst rsonanc th Q is valuatd by approximating th impdanc or th rfl ction coffi cint with polynomials within a narrow frquncy rang. Kywords: Low Q; data fi tting; last squars; fi nit diffrncs; small antnnas; bandwidth prdiction; invariant dfi nition of Q 54 IEEE Antnnas and Propagation Magazin Vol. 55 No. 4 August

2 . Introduction Microwav rsonators that ar usd in filtrs oscillators or in th masurmnt of matrial proprtis ar charactrizd by high Q valus oftn xcding valus of or 4. Th rflction cofficints of such rsonators ar charactrizd by prfct circls in th complx plan th so-calld Q circls []. An accurat dtrmination of th Q-factor valu rquirs applying a data-fitting procdur to th Q circl. This was originally manually don on a Smith chart [] and latr don numrically with th us of a computr [ 4]. For considrably smallr valus of th Q factor say blow th valu of th circls on a Smith chart bcom distortd into arcs or loops so that th assumptions usd in dtrmining th high Q factors ar not valid anymor. It thus bcoms ncssary to dvlop mthods of dtrmining th low Q valu from any shap of th rflction cofficint as a func tion of frquncy. Exampls of dvics xhibiting low Q fac tors ar antnnas in particular antnnas that ar small in comparison with a wavlngth [5]. Knowing th Q valu of a particular antnna bfor vn attmpting to match it can b valuabl a priori information in th dsign procss bcaus th valu of Q dtrmins th maximum thortical bandwidth that can b obtaind aftr th match. Rfrnc [6] pointd out simpl quations that prdictd th bandwidths for singl and doubl tuning that can b obtaind for a known valu of th Q factor and for a prscribd standing-wav ratio. A slight improvmnt can b achivd by tripl-tuning [7]. Th mthods to b dscribd hr ar basd on th known bhavior of th rflction cofficint as a function of frquncy. This rflction cofficint can b obtaind ithr by masurmnt using a ntwork analyzr or by computation using lctromagntic-simulation softwar. Two diffrnt quivalnt circuits will b discussd: on for a short dipol (or monopol) antnna and anothr for a short loop antnna. Basd on th quations for th quivalnt circuits a systm of ovrdtrmind linar quations with ral cofficints is dvlopd and solvd by th last-squars procdur. Th cofficints obtaind ar thn usd for computing th Q factor of th antnna. It is wll known that th monopol bcoms rsonant whn its lngth is approximatly on-quartr wavlngth and th loop antnna achivs rsonanc whn its circumfrnc is about on-half of th wavlngth. Th quivalnt circuits accuratly mimic th input rflction-cofficint bhavior starting from th vry low frquncis and up to this first natural rsonanc. At highr frquncis th simpl quivalnt circuits ar not adquat anymor. Howvr it is possibl to approximat th input impdanc with polynomials within a narrow rang around th frquncy of intrst in ordr to dtrmin th antnna s Q from ths polynomials. A simplr altrnativ procdur will also b dscribd basd on a finit-diffrnc mthod that likwis provids th antnna s Q factor.. Computational Approach Th Q factor is a ral numbr that spcifis how sharp is th rsonanc of a crtain rsonator. Th subjct of this papr is how to dtrmin th unloadd Q factor from a known function of th input rflction cofficint Γ (a complx numbr). In gnral Γ is a rational function of th complx frquncy s σ + jω which can b rprsntd as n n n n... n n... ans + a s + as+ a Γ ( s) n b s + b s + bs+ b Whn th cofficints of th polynomials in th numrator and dnominator ar spcifid th function Γ ( s) is uniquly dfind and can b utilizd to dtrmin th corrsponding Q as will b shown in th papr. Th problm of dtrmining th antnna s Q factor thn rducs to spcifying an appropriat quivalnt circuit which can b dscribd as a rational func tion of th form of Equation () such that th last-squars datafitting procdur can b applid to valuat th coffi cints a n and b n. With th hlp of ths cofficints th func tion Γ ( s) bcoms a smooth and wll-bhavd function amnabl to having drivativs takn. Th valu of th unloadd Q factor can thn b computd from th invariant formula [8] Q dγ f df Γ f f. () () whr f is th rsonant frquncy of th unloadd rsonator. Instad of xprssing th unloadd Q in trms of th rflction cofficint it is possibl to xprss it in trms of th invariant dfinitions of th input impdanc or admittanc [8- ]. W prfr to us th rflction cofficint for two ra sons. First th ntwork analyzr typically provids th output data fil for th ral and imaginary parts of Γ (dnotd S ) valuatd at a numbr of quidistant frquncis. Scond for th numrical data-fitting procdur to run smoothly it is prudnt to avoid data that can tak vry larg valus. Unlik th impdanc or admittanc th rflction cofficint is a boundd function of frquncy th absolut valu of which always rmains smallr than unity (whn only passiv nt works ar involvd). Th dtails of th computational procdur for valuating th Q factors of a small antnna will b xplaind nxt for two IEEE Antnnas and Propagation Magazin Vol. 55 No. 4 August 55

3 practical situations namly for a small dipol and for a small loop. Ths two xampls wr slctd bcaus only a small numbr of ral cofficints is ndd for an accurat rprsntation of th complx function Γ ( f ) a fact that simplifis th prsntation of th computational procdur involvd and bcaus ths antnnas ar dvics whr it is of advantag to possss a low Q factor. Howvr for vry low frquncis th antnnas Q factors again bcom vry larg. Th procdurs to b dscribd hr ar nvrthlss also valid for thos larg Q-factor valus.. Small Dipol Antnna A dipol antnna rachs its lowst natural rsonanc whn its al lngth is qual to on-half of th fr-spac wavlngth. For a monopol antnna th sam is tru whn its lngth is on-quartr of th fr-spac wavlngth. At frquncis blow this rsonanc th input-impdanc quiva lnt circuit consists mainly of th larg capacitiv ractanc and a small radiation rsistanc as shown in Figur a. Th input charactristic impdanc of th ntwork analyzr is typi cally R c 5 Ω. It is thus convnint to normaliz th impd anc valus to R. c Th radiation rsistanc R a of a short dipol antnna is a quadratic function of frquncy [ p. ]. Th capacitiv ractanc X C is dominant bing many tims largr than R a. Th inductiv ractanc X L is small but it nds to b addd for bttr agrmnt with ralistic data. whr Ra d r R d d c f f X + XL xl (5) R XC xc. R c f f c f f Th rflction cofficint bcoms ( ) ( ) ϕ + ϕ ϕ z d j d d Γ u + jv. z + d + + j d d ϕ ϕ ϕ (6) Any Q factor is spcifid at th rsonant frquncy f whr th rflction cofficint crosss th ral axis. At frquncis of intrst th small dipol is not yt rsonant bcaus its input impdanc is dominatd by capacitiv rac tanc X. Th radr may b asking How com th paprs on small antnnas display th antnna s Q factor as a continu ous function of frquncy? Sinc soonr or latr th input impdanc of th small antnna will hav to b matchd to bcom mainly ral Collin [] proposd to dfi n th antnna s Q by adding an idal losslss ractiv lmnt so that th impdanc bcoms purly ral at th frquncy of intrst. This is illustratd in Figur b whr an xtrnal inductanc X is addd to th circuit so that th al input impdanc bcoms ral at th prscribd frquncy f. C Figur a. Th quivalnt circuit of th unloadd mono pol antnna. For convninc of th data-fitting numrical procdur w introduc th normalizd frquncy ϕ : f ϕ () f so that th normalizd input impdanc z bcoms ϕ ϕ ϕ z d + j d d (4) Figur b. Th antnna augmntd with an xtrnal sris inductanc. 56 IEEE Antnnas and Propagation Magazin Vol. 55 No. 4 August

4 Dividing Equation (6) into its ral and imaginary parts w obtain two quations: ϕ ϕ ϕ d u + d v d v + u (7) ( ) ( ) ϕ ϕ ϕ d v+ d u d u v. (8) For ach frquncy thr ar two quations lik th abov. W thus hav a systm of linar quations of th typ di i h. (9) i Th partitiond vctors apparing in Equation (9) ar dfind as follows: ( ϕ ϕ v 4. Small Loop Antnna Figur a shows th quivalnt circuit of a small loop antnna. Th impdanc bhavior is dominatd by th loop s inductanc X L and th radiation rsistanc R a. For a small loop antnna th radiation rsistanc is proportional to th fourth powr of frquncy [ p. 5]. Th prsnc of any stord lctric nrgy must b modld by a paralll capacitiv ractanc X C bcaus th input impdanc at vry low frquncis must in th limit approach a short circuit so X C cannot b in sris with X L as in Figur a. To mak th small loop antnna rsonant at frquncy f an xtrnal loss lss capacitiv ractanc X should b addd to th circuit as shown in Figur b. Th radiation rsistanc and th rac tancs ar normalizd in a similar way as in th prvious sc tion. Th ovrall rflction cofficint bcoms 4 5 ϕ + ϕ + ϕ 4ϕ d d j d d Γ d d j d d 4 5 ϕ + ϕ + ϕ+ 4ϕ. (4) h ϕv ϕ ( ϕ v ϕ ( + u. v () To solv th ovrdtrmind systm of Equation () on dfins matrix D as follows: ( ) D () so that th solution in th last-squars sns is Figur a. Th quivalnt circuit of th unloadd loop antnna. d d d ( DD ' ) h. () d Onc w know th cofficints d to d w can prform th drivativ rquird in Equation () valuatd at ϕ : Γ d+ j d + d ϕ ϕ + + d d d j d d. () Th absolut valu of th abov is substitutd into Equa tion () and thn dividd by Γ to gt th numrical valu of Q valid at th frquncy f. Figur b. Th antnna augmntd by an xtrnal paralll capacitanc. IEEE Antnnas and Propagation Magazin Vol. 55 No. 4 August 57

5 By dividing Equation (4) into its ral and imaginary parts w formulat again a linar systm of th typ in Equation (9). Th partitiond vctors apparing in th quation ar th fol lowing: 4 ( ϕ 4 vϕ ( + ϕ vϕ vϕ (5) ( ϕ involvd in obtaining th masurd or simulatd data th rsults of Equation (7) ar not as accurat as th rsults obtaind by th last-squars procdur. Howvr whn th procdur basd on Equation (7) is rpatd for svral nighboring valus of th indx i th rsultant valu of Q is not supposd to apprciably chang. W found that if th finitdiffrnc procdur is applid to say fiv nighboring points th man valu of ths fiv rsults yilds Q within a fw prcnt of th valu obtaind by th analytic diffrntia tion dscribd in th prvious sctions. Thrfor it is con vnint to valuat Q by both mthods to chck whthr thy provid similar rsults. Anothr advantag of using th finit-diffrnc procdur is that w can intrprt th standard dviation of ths fiv points to indicat th standard rror of Q albit with a pssimistic stimat. 4 5 vϕ 5 ( + ϕ 6. Exampls 6. Sphrical-Cap Dipol + u h. v Not that thr ar now four cofficints although thr ar only thr ral circuit unknowns: R a X L and X C. Ths cofficints provid a good dscription of Γ ( f ) and w s no problm in using thm to obtain th valu of Q by th invariant formula of Equation (). It will b rcalld that th drivativ of a rational function is a ba ab. b b (6) Thortically dipol antnnas can achiv a valu of Q lowr than []. Lopz [6] dscribd a small dipol loadd with a sphrical cap which h matchd (ithr singl-tund or doubl-tund) to a bandwidth corrsponding to Q 9. For th frquncy rang 86.5 MHz to.5 MHz th input rflction cofficint of this antnna was plottd in Lopz s Figur b. W procssd ths rflction-cofficint data two tims: first for th quivalnt circuit in Figur a and thn again for th quivalnt circuit shown in Figur b. Whn th coffi cints d i ( i to ) wr valuatd for th first tim th rflction cofficint Γ was valuatd by Equation (6). Figur shows a Although th numrator and th dnominator hav to b diffrntiatd sparatly and thn ths factors combind according to Equation (6) all ths stps ar asily pro grammd on a computr. 5. Finit-Diffrnc Procdur Th input data fil that contains valus of th rflction cofficint as a function of frquncy is usually stord with data at quidistant frquncy intrvals f. Suppos that th ith point of th data fil dnots th dsird frquncy f. Th valu of Q is thn obtaind by th scond-ordr finitdiffrnc approximation as follows: ( i ) ( i ) Γ + Γ f Γ i Q i f i. (7) Bcaus of th limitd numbr of digits with which th data ar xprssd as wll bcaus of any othr approxima tions Figur. Th rflction cofficints of th unloadd and augmntd sphrical-cap antnna. 58 IEEE Antnnas and Propagation Magazin Vol. 55 No. 4 August

6 Lopz s original points shown as circls and th rcomputd points obtaind from Equation (6) shown as + symbols. At first glanc th two data sts agr wll with ach othr. Th actual distancs btwn th data ar plottd in Figur 4. It can b sn that th ovrall agrmnt with th modl from our Figur a was about on-half prcnt. An advantag of our rprsntation in Equation (6) is that it is continuous. It is thrfor possibl to accuratly prdict th valu of th xtrnal ractanc X at th frquncy of intrst vn if that frquncy dos not coincid with on of th data points. For a targt frquncy f MHz th nor malizd cofficint was found to b d x.74. According to Figur b this ractanc was addd to th antnna s ractanc. Th normalizd impdanc was obtaind from th rflction-cofficint data as z a +Γ. (8) Γ Th impdanc was thn augmntd by th xtrnal induc tanc as follows: a z ϕ z ϕ + jx ϕ (9) and th rsult was transformd back into th rflction cofficint z Γ z +. () Th data-fitting procdur was rpatd for th aug mntd rflction cofficint using Equation () and th nw cofficints d i ( i to ) wr found. Th valus of Γ wr now cntrd on th ral axis. As sn in Figur 4 th data obtaind by Equation () still agrd with th data obtaind from th nw cofficints d i ( i to ) substitutd into Equation (6) but not as prfctly as bfor. Most actual distancs shown in Figur 4 wr now smallr than % of th Smith-chart radius. Our valus of th antnna Q factor wr first as valuatd by data fitting Q 7.9 ; and scond as valuatd by finit diffrnc Q 8. ±.. It is customary to spcify th largst mismatch that can b tolratd within th prscribd bandwidth in trms of th rturn loss RL in dcibls. Th rturn loss is rlatd to th absolut valu of th rflction cofficint Γ as follows: RL. ρ Γ () Th thortical rlativ bandwidth that can b achivd for a givn Q and a givn Γ was found by Whlr (rfrnc [] in th Lopz papr). For a singl-tund matching th rlativ bandwidth is Figur 4. Th distancs btwn th data fittd points and th original data. BW Q ρ ρ () whras for th doubl-tund matching th thortically possibl rlativ bandwidth is ρ BW Q ( ρ ). () By substituting Q into Equations () and () it was possi bl to prdict that th sphrical-cap antnna could b singl-tund to a -db rlativ bandwidth of.7% or doubl-tund to a rlativ bandwidth of 9.% vn bfor th actual match was attmptd. Ths valus agrd wll with Lopz s rsults. 6. Loop Antnna Abov th Ground Plan Figur 5 shows a printd-circuit loop antnna on a ground plan. Th loop consistd of a mm-wid strip with outsid dimnsions of 4 mm 4 mm. Th substrat was.5 mm thick and its rlativ dilctric constant was ε r.5. As this antnna could b containd within a hmisphr of radius a 4 47 mm it was considrd small for ka < i.. up to a frquncy of 5 MHz. Th input impdanc of th loop antnna in a printd form ovr a dilctric substrat was simulatd using th FDTD mthod basd on [4]. For computational simplicity th ground plan was rmovd and th input impdanc was computd for a loop twic as big in fr spac. Th computd impdanc was dividd by two to rprsnt th impdanc of a loop on an infinit ground plan. Th computd valus ar shown in Figur 6 for th frquncy rang MHz to 5 MHz. IEEE Antnnas and Propagation Magazin Vol. 55 No. 4 August 59

7 Thortically at zro frquncy th input impdanc was a short circuit which is th lft-most point on th ral axis of th Smith chart in Figur 6. As frquncy grw th points movd on th priphry of th Smith chart in th clockwis dirction. Th impdanc at first was purly inductiv. Th lowst computd point shown in Figur 6 was at MHz. As frquncy furthr grw th points movd in a clockwis mannr and at 48 MHz thy crossd th ral axis and bcom capacitiv. This was th first natural rsonant frquncy of th antnna. Although th impdanc was ral th valu was vry high (about 9 kω ) and difficult to match to 5 Ω. Aftr that th points bcam capacitiv in natur and movd inward on th Smith chart. Figur 5. Th printd-circuit loop antnna mountd on a ground plan (not in scal). Figur 6. Th input rflction cofficint of th loop antnna in th frquncy rang of MHz to 5 MHz. Th quivalnt circuit from Figur a was valid for th uppr half of th Smith chart. Figur 7 shows th diffrnc btwn th rflction cofficint computd by th circuit in Figur a and th input rflction cofficint obtaind by FDTD. Ovr th ntir rang of MHz to 5 MHz th diffrnc was lss than % of th radius of th Smith chart. At any frquncy of intrst within this rang on could find th rquird valu of th paralll suscptanc that would bring th antnna to rsonanc at this frquncy. For instanc at f MHz an xtrnal paralll capacitanc of C.5 pf would cancl th antnna s suscptanc so that th Q factor obtaind from Equation () was found to b Q.. This was alrady a disappointingly high antnna Q but th valu of th input rsistanc obtaind by this xtr nal matching capacitanc was vn mor disappointing: R kω. Abov th first natural rsonanc frquncy th quivalnt circuit in Figur a is no longr appropriat for accuratly simulating th input impdanc of th unloadd loop antnna. W usd polynomial approximations of ordr thr to sparatly rprsnt th ral and th imaginary parts of th rflc tion cofficint and to find th analytical valu of Q with th us of Equation (). This works only in a rlativly narrow rang in th vicinity of frquncy f. As long as th rsults agr with th finit-diffrnc rsults from Equation (7) w fl confidnt in th valus obtaind. in Tabl summarizs th rsults of dtrmining Q for th small loop antnna. BW and BW ar th singl-tund and doubl-tund db bandwidths computd by using Equations () and () using Q as listd in th scond column of th tabl. R in is th input rsistanc of th antnna aug mntd by a sris inductanc that mad th antnna rsonat at f. At frquncis blow th first natural rsonanc (48 MHz) th bandwidths wr too narrow and th matchd input rsistanc R was too high for practical applications. in Figur 7. Th data fitting rror for th quivalnt circuit from Figur a was smallr than % ovr a wid rang of frquncis blow th first rsonanc. At f 5 MHz th loop antnna is still considrd small and at this frquncy th antnna s Q was rlativly low namly Q 7.5 whil R in was clos to 5 Ω. At f 5 MHz R in bcam practically qual to 5 Ω. Th rquird xtrnal sris inductanc at this frquncy was L 48 nh. As can b sn from Figur 8 th input impd- 6 IEEE Antnnas and Propagation Magazin Vol. 55 No. 4 August

8 Tabl. Th loop antnna s Q and associatd bandwidths. f (MHz) Q using () Q using (7) BW (MHz) BW (MHz) R in ( Ω ) 5 8 4±7.6.4 k. 7±9.. k ± k ± ± ± anc in this cas passd through th cntr of th Smith chart and no furthr matching was ncssary if on was satisfid with a singl-tund bandwidth of BW 4 MHz. Tabl also shows that th antnna Q of th loop antnna from Figur 5 could b as low as Q 8.8 at th somwhat highr frquncy of f 4 MHz. Th magnitud of th rflction cofficint basd on th data computd by th FDTD is shown in Figur Conclusions Simpl quivalnt circuits hav bn vrifid to fit th rflction-cofficint data obtaind by lctromagntic-simulation softwar wll. An xampl of th sphrical-cap dipol was simulatd in th litratur by th Momnt Mthod and th printd-circuit loop antnna was simulatd hr by th FDTD mthod. In both cass it was found that th input impdancs basd on th quivalnt circuits agrd wll with thos valus from th softwar simulations starting from vry low frquncis up to th first natural rsonanc of th unloadd antnna. Figur 8. At f 5 MHz th sris inductanc matchd th antnna to 5 Ω. Th symbols dnot th computd input impdanc and th + symbols dnot th polyno mial approximation. At frquncis highr than th first rsonanc th simpl quivalnt circuits wr not adquat. W usd polynomial approximations for th ral and imaginary parts of th rflction cofficint. Onc th rflction cofficint was xprssd with continuous functions th invariant forms of th unloadd Q factor providd good stimats of th antnna s Q. Although th invariant forms ar also known for th impd anc and th admittanc w prfrrd to us th invariant form for th rflction cofficint. 8. Rfrncs. W. Altar Q Circls A Mans of Analysis of Rsonant Microwav Systms Procdings of th IRE 5 (Part I) April 947 pp. 55-6; (Part II) May 947 pp E. L. Ginzton Microwav Masurmnt Nw York McGraw Hill D. Kajfz Q Factor Oxford MS Vctor Filds D. Kajfz Q Factor Masurmnts Using MATLAB Norwood MA Artch Hous. 5. S. R. Bst A Discussion on th Quality Factor of Impd anc Matchd Elctrically Small Wir Antnnas IEEE Transactions Antnnas and Propagation January 5 pp Figur 9. Th magnitud of th rflction cofficint basd on th data computd by th FDTD. 6. A. R. Lopz Doubl-Tund Impdanc Matching IEEE Antnnas and Propagation Magazin 54 April pp IEEE Antnnas and Propagation Magazin Vol. 55 No. 4 August 6

9 7. T. Simpson Tripl-Tuning Small Antnnas for Incrasd Bandwidth IEEE Antnnas and Propagation Magazin 54 4 August pp D. Kajfz and W. P. Whlss Invariant Dfinitions of th Unloadd Q Factor IEEE Transactions Microwav Thory Tchniqus MTT-4 7 July 986 pp W. Gyi P. Jarmuszwski and Y. Qi Th Fostr Rac tanc Thorm for Antnnas and Radiation Q IEEE Transactions Antnnas and Propagation AP-48 March pp A. D. Yaghjian and S. R. Bst Impdanc Bandwidth and Q of Antnnas IEEE Intrnational Symposium on Antnnas and Propagation Jun pp J. D. Kraus Antnnas Scond Edition Nw York McGraw Hill R. E. Collin and S. Rothschild Evaluation of Antnna Q IEEE Transactions on Antnnas and Propagation AP- 964 pp G. A. E. Vandnbosch Simpl Procdur to Driv Lowr Bounds for Radiation Q of Elctrically Small Dvics of Arbitrary Topology IEEE Transactions on Antnnas and Propagation 59 6 Jun pp A. Elshrbni and V. Dmir Th Finit Diffrnc Tim Domain Mthod for Elctromagntics with MATLAB Simulations Raligh NC SciTch Publishing 9. 6 IEEE Antnnas and Propagation Magazin Vol. 55 No. 4 August