A HIGH-RESOLUTION ANTENNA DIAGNOSTICS TECHNIQUE FOR SPHERICAL NEAR-FIELD MEASUREMENTS

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1 A HIGH-RESOLUTION ANTENNA DIAGNOSTICS TECHNIQUE FOR SPHERICAL NEAR-FIELD MEASUREMENTS C. Cappelli 1-, O. Beibjeg 1, A. Fadse 1 Østed DTU, Techical Uivesit f Dea, DK-800 Kgs. Lgb, Dea, Eail: cca@ested.dtu.d, b@ested.dtu.d TICRA, Lædestæde 3, DK-101 Cpehage K, Dea, Eail: af@tica.c ABSTRACT A ew diagstics techique f spheical ea-field atea easueets, that ca pvide a high spatial esluti f the ecstucted apetue field, is peseted. This techique is ealied b tasfig the spheical wave epasi SWE f the adiated field it the cespdig plae wave epasi PWE, ecveig a sigificat pat f the ivisible egi f the plae wave spectu. Thugh the ivese Fuie tasf IFT, the field a plae utside as well as iside the atea iiu sphee is ecstucted, with a esluti that eceeds the liit f e half a wavelegth pvided b the taditial IFT f the fa-field. 1. INTRODUCTION Atea diagstics is a techique t detect ad idetif electical ad echaical es i a atea thugh a ispecti f its adiated ea-field. S fa, seveal techiques, cl based fa-field, plaa spheical ea-field easueets, have bee develped, all pesetig liitatis i thei pactical ad theetical ealiati [1]-[]. We ppse a ew atea diagstics techique f spheical ea-field easueets t be ipleeted at the DTU-ESA Spheical Nea-Field Atea Test Facilit lcated at the Techical Uivesit f Dea [3]. The easueets caied ut i the DTU-ESA Facilit ae based the spheical wave epasi SWE f the field adiated b the atea. This field epasi is atheaticall valid i a suce-fee egi f space utside the s-called iiu sphee f the atea, the sallest sphee, ceted at the igi f the cdiate sste, which cpletel eclses the atea. B wig the field the easueet sphee, the field ca be evaluated a ew ad salle sphee, lage tha the iiu sphee. This is the liit f the st c diagstics techiques used f spheical ea-field easueets. We ppse a wa t eceed this liit. The idea is t deive f the SWE f the adiated field the plae wave spectu f the sae field. Oce the plae wave spectu a give plae utside iside the iiu sphee is calculated, the plae wave epasi PWE f the field that plae is w. This allws the apetue field t be cputed as the ivese Fuie tasf IFT f the spectu. B evaluatig the plae wave spectu als i pat f the ivisible egi, the achieved spatial esluti the plae ca eceed the taditial value f half a wavelegth, pvided b the taditial techiques. The fist step csists f btaiig the SWE cefficiets f a spheical ea-field easueet; secd, the plae wave spectu i the visible, as well as i pat f the ivisible egi, is calculated. The ivese Fuie tasf is late applied t btai the field the desied plae clse t the atea. I this auscipt the the behid the ew techique will be descibed ad aaltical calculatis as well as ueical siulatis will be shw. Ivestigatis the ube f spheical des ecessa f the PWE cvegece, ad the cespdig saplig desit the easueet sphee, will be peseted. The ifluece f tucatis i the ivese Fuie tasf will be fiall studied. All theetical esults ae epessed i the S.I. atialied sste with e -jωt tie depedece.. THEORY We stat b itducig the SWE f the electic field E adiated b a geeal atea cicuscibed b a iiu sphee f adius. I a suce-fee egi > the field ca be epessed as a weighted su f spheical waves [], E η 1 Q1 F1 + Q F 1 whee ad ae the epasi cefficiets, that Q 1 Q ca be btaied f a spheical ea-field easueet, ad F1 ad F ae the pwealied spheical vect wave fuctis. The ediu itisic adittace is deted b η, is the wave ube, ad is the psiti vect epessed as a fucti f the taditial spheical cdiates, θ,

2 φ. I pactice, the -suati f the SWE is tucated at N, N beig usuall equal t N +10. The PWE f the sae electic field i the spectal -dai valid f >, with beig the lagest - cdiate f the suce egi, is give b [5] 1 E,, whee,, T, e j wave ppagati vect with e j + d d ae the catesia cpets f the. I pactice, the,, itegals ae tucated at a fiite value ± a ad ± a. The plae wave spectu f j a give -cdiate is thus T, e, ad it ca be deived b the ivese f Eq., i tes f the cuet suce, b [5] 1 T, J e πη V j + + dv It will w be shw hw the SWE f Eq. 1 ca be tasfed it the PWE f Eq.. F this pupse we itduce the PWE valid f > 0, i the spectal α,β- dai f the spheical vect wave fuctis as give b [6]-[7] F 1 j + 1 F j j + 1 π πc+ π πc+ Y ŝ Y α, β e jŝ α, β e siα dα dβ jŝ siα dα dβ 5 with ŝ si αcs β ˆ + siαsi β ŷ + csα ẑ, β [-π, π] ad α C+, see Fig. 1. I {α} 0 π/ Re {α} The fucti Y α, β is the vect spheical haics defied b 1 Y α, β j P csα e α 1 jβ P α je ˆ cs α siα jβ ˆ β + with P cs α beig the alied assciated Legede fuctis as defied b [], ˆ α csα csβ ˆ + csα siβ ŷ siα ẑ ad ˆ β si β ˆ + csβ ŷ. B substitutig the PWE f the vect spheical haics, Eq. ad Eq. 5, it the SWE, Eq. 1, ad b itechagig the de f itegati ad suati, sice the duble itegal is uifl cveget [6], the PWE f the electic field i the spectal α,β-dai, valid f eve >, ca be fud as π j jŝ E Ê ŝ e α dα dβ π si 8 πc whee the spectu cple aplitude Ê ŝ 1 jq 1 + j π Q η + 1 Y α, β ]. [ ŝ Y E ˆ s ˆ 6 7 is give b α, β + 8 Eq. 8 ca thus be calculated f the wledge f the 3 SWE cefficiets ad. Q 1 Q j csα The spectu i the spectal α,β-dai, Ê ŝ e, ca w be taslated it the, -dai t btai j T, e b usig the elati [7] j j csα T, e Ê ŝ e π 1 whee α,β the ight had side ust be epessed as fuctis f the spectal vaiables ad accdig t the elati ŝ /. Havig btaied the plae wave spectu f Eq. f the SWE f Eq. 1 a give plae, we ca calculate the field a ew plae ew beig f eaple ew - L, with L > 0, as depicted i Fig., aivig at 9 C + E,, ew 1 T, e j L e j + d d 10 Fig. 1. Dai f the vaiable α with ctu C +.

3 The btaied spatial esluti f the field is give b δ π /, δ π / ad thus ca be achieved b a a selectig a ad a apppiatel i the SWE-t- PWE tasfati. We ca theefe suaie the equied steps f this atea diagstics techique as fllws: 1. Evaluate the Q cefficiets thugh a spheical ea-field easueet f the SWE f the adiated field f the atea ude test AUT.. Calculate the plae wave spectu i the, dai a give plae, >, accdig t the SWE-t-PWE tasfati, i.e. thugh Eqs Bac-tasf the spectu t a ew plae clse t the atea <.. Cpute the field the desied plae as the ivese Fuie tasf f the spectu thugh Eq.. The ifiite seies i i Eq. 8 ca be tucated t a fiite ube N, chse sufficietl lage t esue the desied accuac f the spectu. This value ca be diffeet f the taditial N +10. Give the equied N, the saplig desit the easueet sphee is give b 3. Test cases θ ϕ N + 1 Nw that the steps f the ew diagstics techique ae clea, we ca illustate the pcedue b csideig a siple atea cfiguati. A set f fu -ieted Hetia diples the - plae, equall displaced at the distace f the igi, ae csideed, see Fig. 3. plae L ew plae Fig.. Bac- tasfati f PWE f a plae utside the iiu sphee t a ew plae iside the iiu sphee. 3. IMPLEMENTATION 3.1 Pactical csideatis The ew atea diagstics techique has bee peseted, but w se pactical csideatis have t be ade. The plae f the cputati f the specta as well as the fields has t be selected sufficietl clse t the AUT t ealie a efficiet diagstics. O that plae, the etesi f the spectal dai is chse i de t esue the desied spatial esluti f the apetue field, ad t iiie the tucati e. Fig. 3. Fu -ieted Hetia diples the - plae, the iiu sphee with adius ad the cstat plae. The chice f this cfiguati is due t a facts. Fist f all, b lcatig the diples the - plae, it is pssible t ivestigate the effect f vig the plae iside the iiu sphee. Secd, f this atea the Q cefficiets ca be calculated aalticall avidig the use f eal easueets data, the use f which wuld have bee peatue at this pit. Thid, as it will be shw late, the adiated field ctais abitail high-de des i ad. Futh, the aaltical epessi f the plae wave spectu i the, -dai the plae, see Eq. 3, ca be used as efeece, sice the diple cuets ae w. I this wa the tucati i the spectu seies f Eq. 8 ca be aaled. Fifth, the aaltical epessi f the adiated field ca easil be calculated ad used as a efeece f the esult f the IFT f the specta. Diffeet value f have bee studied, hee we peset the esults f λ, beig the diple psiti as well as the iiu sphee adius. Als, we will cside L 1λ, ad thus the plaes + L3λ ad ew - L 1λ.

4 F the calculati f the Q cefficiets we efe t the esults epted i [3; p.339] Q 1 jp 8π + 1 η dp csθ dθ δ δ, 1 j, 1 θ π / 11 ae the efeece spectal cpets f Eqs ad the e cputed thugh Eqs with a tucati value f the -seies equal t N +0. All quatities ae i liea scale ad the selected plae is 3λ. It is see that all cpets, als thse affected b sigulait i, ae ecstucted, shwig l a sall diffeece i the aplitude f the de f 10-7, see Fig. 5. If we decease N, the diffeece iceases: f N +10 it beces f the de f Q P 8π + 1 η + 1 j P 1 d + d δ δ ], 1, 1 { j } θ π / csθ δ + δ, 1, 1 P csθ siθ + 1 I Eqs. 11-1, P detes the diple et, the spheical Bessel fucti, while, defied as j δ, µ ± 1 is µ, is 1 d µ, 0 δ,µ 13 0 thewise B use f Eqs , we ca theefe slve the fist ad secd steps f the pcedue. T cpae the btaied esults with se efeece values, we use Eq. 3 T P µ ε cs + cs P µ T cs + cs ε 1 15 Fig.. Aplitude f the spectu cpets the plae 3λ: the left the efeece value f Eqs , the ight the spectu cputed thugh the pcedue f N +0. P T µ cs + cs. 16 ε It is ted that a sigulait f is peset i the - ad -cpet. It ca be pved that such a sigulait will alwas be peset at least i e f the spectu cpets ad that the ecessa, but t sufficiet, cditi t avid this sigulait is that the atea fa-field patte has a ull f θ π / Specta cputati We chse a, dai equal t [-3 : 3] [-3 : 3] ad we saple it with pits. I Fig. Fig. 5. Aplitude diffeece f T -spectu 3λ, the left f N +0, the ight f N +10. F the sae spectu we cpute als the phase: i Fig. 6 the esult f the cpet calculated with N +0 is shw. As we see, the phase is t ecstucted utside the visible egi, eve if we icease the ube f saplig pits. This happes t the the cpets as well. Althugh the ve lw aplitude f the spectu i the ivisible egi, see

5 Fig., eas that a accuate ecstucti f the phase is t f high iptace, we will biefl shw hw the phase ecstucti is iflueced b the tucati ube N. Fig. 6. Phase f the spectu cpet 3λ: the left the efeece value f Eq. 16, the ight the spectu cputed thugh the pcedue f N +0. We theefe icease N util N +50 ad we cpute agai the phase, see Fig. 7: w the phase is ecstucted iside the egi ± ad the cespdig diffeece i aplitude is equal t Fig. 8. Aplitude f the field cpet 3λ: tp the aaltical value, the left the IFT f the spectu cputed thugh the pcedue f N +10, the ight the IFT f the efeece spectu f Eq. 16. Fig. 7. Phase ad aplitude f the spectu cpet 3λ f N Fields cputati Nw that the specta calculated thugh the ew pcedue have bee ivestigated, we ca ivese Fuie tasf the t cpute the field a - plae utside ad iside the iiu sphee. All figues ae i liea scale. a Outside the iiu sphee 3λ The spectu cputed thugh Eqs is ivese Fuie tasfed the plae 3λ with the tucati value N +10. The, -dai is deceased t the value f [-:], btaiig theefe a spatial esluti equal t λ /. Just the cpet will be aaled t avid difficulties i the IFT f fuctis with sigulaities. The esults ae shw i Figs The IFT esults ae i gd ageeet with the aaltical efeece field epessi shwig the sae level f diffeece. Fig. 9. Aplitude diffeece f the IFTs field cpet, i espect f the aaltical field 3λ. B lig at Fig. 9 we awa see a diffeece i the IFT esults f se % with espect t the aaltical field, which is t quite satisfact. Sice bth IFTs shw the sae diffeece value, the iaccuac is t due t a t sall ube N f -des i the spectu epesetati. The pble is als t due t a tucati i the spectal dai, sice, see Fig., the spectu utside the visible egi ges sthl t e ad is t tucated t a fiite value. The iaccuac culd theefe be due t a isufficiet ube f pits i the spectal dai. T aale this fact, we ivestigate theefe the esults f ad pits, see the esults i Fig. 10. The diffeece cputed with pits is alst fu ties highe tha the e with pits, while the e with is half f the diffeece give b pits.

6 Fig. 1. Aplitude diffeece f the IFTs field cpet just visible egi, i espect f the aaltical field 3λ. Fig. 10. Aplitude diffeece f the IFTs field cpet, i espect f the aaltical field 3λ, tp with 100 saplig pits, belw with 300 saplig pits. O the left the IFT f the spectu cputed thugh the pcedue f N +10, the ight the IFT f the efeece spectu f Eq. 16. We chse theefe as the ube f saplig pits i the, -dai ad we w ivestigate the iptace f the ivisible egi f the spectu i the field ecstucti. I Figs ae the ivese Fuie tasfs f l the visible egi f the specta. Agai the tw IFTs l the sae, but thei diffeeces with espect t the aaltical value ae alst twice the e i Fig. 10. O this plae the wledge f the ivisible spectu, besides pvidig a highe spatial esluti, ca ipve the accuac f the field cputati. b Iside the iiu sphee 1λ Nw the spectu is bac-ppagated t a -plae iside the iiu sphee ad the ivese Fuie tasfed. The tucati value N +10 is used ad the esults appea i Fig. 13. Iside the iiu sphee the csideed ube f des is isufficiet f a ecstucti f the field. The IFT f the spectu calculated thugh the Q cefficiets is ttall diffeet f bth the efeece aaltical field ad the IFT f the efeece spectu, while the IFT f the efeece spectu ecstucts the field with a eve bette accuac tha utside the iiu sphee, see Figs Fig. 13. Aplitude f the field cpet 1λ: tp the aaltical value, the left the IFT f the spectu cputed thugh the pcedue f N +10, the ight the IFT f the efeece spectu f Eq. 16. Fig. 11. Aplitude f the field cpet 3λ: the left the IFT f the spectu cputed thugh the pcedue f N +10 just visible egi, the ight the IFT f the efeece spectu f Eq. 16 just visible egi. Fig. 1. Aplitude diffeece f the IFT f the efeece spectu f Eq. 16 i espect f the aaltical field 1λ. T disciiate the suce f e shw i Fig. 13 we ivese Fuie tasf just the visible pat f the spectu btaied with N +10. As we ca see f Fig. 15, the esult is w cpletel diffeet eaig that the pble lies i the ecstucti f the ivisible pat f the plae wave spectu. We cpute theefe the spectu at 1λ, see Figs The spectu cputed with N +10 is ttall

7 diffeet f the efeece spectu, while b iceasig N t the value N +50, the spectu is ecstucted with a accuac f the de f Fig. 15. Aplitude f the field cpet 1λ. O the left the IFT f the spectu cputed f N +10, just visible egi, the ight its diffeece with the aaltical value. Fig. 18. Aplitude f the field cpet 1λ. O the left the aplitude f the IFT f the spectu cputed f N +50, the ight its diffeece with the aaltical value. T cplete u ivestigatis, we shw i Fig. 19 a eaple f apetue field ecstucti ve clse t the atea, the plae 0.λ. The spectu is cputed with the tucati value N +50 givig a diffeece i aplitude f alst 10%. This high value is due t a accuac f the de f l 10-3 i the specta ecstucti. The accuac f the field ca f cuse be deceased b iceasig the value f N. Fig. 16. Aplitude f the spectu cpet the plae 1λ: the left the efeece value f Eq. 16, the ight the spectu cputed thugh the pcedue f N +10. Fig. 19. Aplitude f the field cpet 0.λ: tp the aaltical value, the left the IFT f the spectu cputed thugh the pcedue f N +50, the ight the diffeece with the aaltical value. Fig. 17. Aplitude f the spectu cpet the plae 1λ: the left the spectu cputed thugh the pcedue f N +50, the ight the diffeece with the efeece value f Eq. 16. We theefe use N +50 ad we calculate agai the IFT, see Fig. 18. Nw the diffeece is alst equal t the e give b the IFT f the efeece spectu i Fig. 1. This shws that a accuate spectu ad apetue field ca be btaied iside the iiu sphee if the cect tucati ube i the -seies f Eq. 8 is csideed.. CONCLUSIONS A ew diagstics techique f spheical ea-field atea easueets, that ca pvide a high spatial esluti f the ecstucted apetue field, has bee peseted. The plae wave spectu is ecstucted i the visible as well as i pat f the ivisible egi f the spectal dai. The achieved accuac, a -plae utside ad iside the atea iiu sphee, is high beig i ecellet ageeet with the aaltical efeece values. It has bee shw that the -seies ivlved i the specta epessis ca be tucated t a fiite ube N. Regadig the apetue field, the spatial esluti btaied hee eaches the value f λ /. It has bee pved that a isufficiet tucati value N i the spectu ca have diffeet iflueces the field cputatis. I paticula, es ae ecveed whe a spectu with a lw value f N is ivese Fuie tasfed a plae utside the iiu sphee. But whe the field is cputed iside the sphee, l a uch highe value f N ca ecstuct the field with gd accuac. The iptace f the

8 wledge f the ivisible egi f the spectu i the field ecstucti has als bee destated. Ma ivestigatis eai t be caied ut i the futue. I paticula, f a give atea f a cetai sie, a give plae, a equied accuac ad esluti, it is eeded t establish a elati t deteie the ptiu N f the seies f Eq. 8. Als, the IFT f specta with a sigulait will be ptiied ad e ealistic atea dels will be ivestigated. REFERENCES 1. Kapla L., Haflig J. D., Bgitti G. V., The Bacwad Tasf f the Nea-Field f Recstucti f Apetue Field, IEEE Tas. Ateas ad Ppagati Sc. Sp. Dig., , J E. B., Gule M. G., High Resluti Spheical Micwave Hlgaph, IEEE Tas. Ateas ad Ppagati, vl. 3, 6-7, Hepage f the DTU-ESA Facilit: Hase J. E., Spheical Nea-Field Atea Measueets, Pete Peegius Ltd. Ld Hase T. B., Yaghjia A. D., Plae Wave The f Tie-Dai Fields, Nea-Field Scaig Applicatis, IEEE PRESS, Devae A. J., Wlf E., Multiple Epasi ad Plae Wave Repesetatis f the Electagetic Field, Jual f Math. ad Phsics, Vl. 15, 3-, Febua Cappelli C., Atea Diagstics i Spheical Nea- Field Atea Measueets b Plae Wave Epasi, M. Sc. Thesis, Østed.DTU, Electagetic Sstes, Techical Uivesit f Dea, Apil 00.