Truncation errors in cylindrical near to far field transform. A plane wave synthesis approach

TRUNCATION ERRORS IN CYLINDRICAL NEAR TO FAR FIELD TRANSFORM. A PLANE WAVE SYNTHESIS APPROACH J. Romeu, L. Jo e. ABSTACI Cylind ical Nea o a ield an enna ans o ma ion is exp essed as a plane wa e syn hesis p oblem whe e a plane wa e is p oduced by a cylind ical cu en dis ibu ion ha encloses he An enna Unde Tes (AUT). The cu en dis ibu ion ha p oduces he plane wa e is di ec ly de i ed om he nea o a ield ans o ma ion algo i hm. IN l DUClIN In nea o a ield an enna measu emen s he An enna Unde Tes (AUT) nea ields a e measu ed on an enclosing su ace close o he AUT. The AUT a ields a e ound a e p ocessing hese nea ields. Plana , cylind ical and sphe ical scanning su aces a e used in p ac ice. In o de o pe o m he nea o a ield ans o ma ion, he nea ields mus be known on he en i e su ace ha encloses he AUT. In he plana and cylind ical case, his condi ion implies ha he su ace mus sp ead o -in ini y. In p ac ice an e o is associa ed o he limi ed ex en o he measu emen su ace. This e o is usually known as unca ion e o . The e ec o his e o on he an enna a ield pa e n is among o he ac o s hea ily dependen on he AUT cha ac e is ics. The o mula ion o he nea o a ield ans o ma ion as a plane wa e syn hesis p ocess allows o exp ess he unca ion e o as a de e io a ion o he syn hesized wa e. In his way he pe o mance o he nea ield ange can be exp essed independen ly o he AUT. On he o he hand, plane wa e quali y is a common igu e o me i in compac o a ield anges. EXPRESSION OF THE CYLINDRICAL NEAR TO FAR FIELD TRANSFORMATON AS A PLANLWAVE SYNIEEIS All an enna pa e n measu emen schemes ind he AUT esponse o an inciden plane wa e. In nea ield measu emen s he AUT esponse o a plane wa e is somehow ob ained, al hough i does no appea explici ly in he ans o ma ion algo i hm. In cylind ical an enna nea ield measu emen s, a p obe scans he AUT nea ields on a cylind ical su ace ( ig. 1). Conside a ecip ocal case whe e he p obe is emi ing and he AUT ecei ing. Le us de ine c ( , z ( , 4 ')as he open ol age measu ed on he AUT access po when he p obe is placed a p o , z ' and ed wi h an uni a y ol age a i s inpu . The supe index (p) can be 1 o 2 and e lec s he ac ha wo p obes a e equi ed o pe o m he nea o a ield ans o ma ion. Conside now ha a cylind ical a ay o p obes is cons uc ed and each p obe is ed a i s inpu by a ol age gi en by w ( z X, '). The AUT esponse o he cylind ical a ay can be ound as: Uni e si a Poli ecnica de Ca alunya. Ap a . 30002, 08080 Ba celona, SPAIN 659 (p)= X ( ~,) (P)(z`, ¢`)d`dz (1) o X~~~~~~o Conside now ha w ( z k, ' ; , zO , o ) exp esses he weigh ha has o be gi en o he each elemen o he cylind ical a ay placed a poin po' ' , z', o p oduce a plane wa e p opaga ing in di ec ion k z0 = k cos O o and a gi en pola iza ion e, whe e ek = O (2) ko=ksinOosin O*+ksenOocos o5+kcosOoZ In his case he ol age up) induced a he AUT e minals is he AUT esponse o a plane wa e p opaga ing in he k o di ec ion and ha gi en pola iza ion e. Fo a ecip ocal an enna his ol age is he same as he ampli ude o he adia ed ield in he same di ec ion and pola iza ion. The gene al p oblem o inding he weigh s w ( z * , - " ; k zc , ¢ o )does no ha e a solu ion o a ini e leng h cylinde . An i e a ing app oach o ind weigh unc ions ha p oduce good app oxima ions o plane wa es in a ce ain olume wi h ini e sou ces is ound in [11. In ou case we a e in e es ed in inding he weigh s ha a e inhe en ly implici in he ans o ma ion algo i hm. The cylind ical nea o a ield ans o ma ion is well desc ibed elsewhe e [2,3], and can be w i en in he ollowing way: [F0 (kzo, 00) J . 0zkF) (2 'z1 d z L ~~, (k20,~~0) j -~~ J0J o (4K z')J wi h - =6 3senO0 ejk20z E Jn(+0-*') -an2) an)' 1 (4) 16) 3sen 0A (n, k,O) j bjb2) jb(l ) and A (n, k ) a (l) k ):b (2) k a (2) (-k ()b (5))(-k =) In hese exp essions E.(k0, . ¢0) and E,(kzo . 0 a e he AUT a ields in di ec ion k,o . 0, and pola iza ionOand @ espec i ely. The ol ages Q (c) ., )and 4c (2 z Z )a e he measu ed ol ages by each o he p obes equi ed o pe o m he ans o ma ion in posi ion , z'. And he coe icien s a() ), b ('), a (2), b (2) a e ela ed o he cylind ical wa e expansion o he p obe adia ed ields. No ice ha (w z ;k , . 9 o ) a e he weigh s ha ha e o be gi en o p obes l and 2, in posi ion * z - o p oduce a plane wa e p opaga ing in di ec ion k zO' O. I we conside ha p obe 1 is a e ical in ini esimal dipole and p obe 2 a ho izon al in ini esimal dipole, he cu en dis ibu ion o p oduce a e ical plane wa e is: JkzoZ V I -ii (#oZ) 1 e(p - po) (6) 2 y 2 ' senOoH () (k~00 660 and o p oduce a ho izon al plane wa e is: 1 jk j ~~~~~~kzon (7) J (Z4 )=--- kezozzin+ ej(0* 2 2 n k2 poH (2) (klppO) 1 ~ b6(p-po) Hn 2)kpopPo) P wi hk o+k2o -k2. Each one o hese cu en dis ibu ions p oduce he ollowing ields inside o he cylinde E= 411 e 6(8) 00=cosO0coslo0 + cosO0sin Oh-sin0 A and E = ) We Eo ;+ (9) 411 = -sinO* + cosOO5 Howe e when he cylinde leng h is ini e hese ideal plane wa es de e io a e. The esul ing syn hesized wa e esul s om conside ing he cu en dis ibu ions gi en by equa ions 6 and 7, bu wi h a ini e leng h along he z axis FRRR CRIJEA In o de o quan i y he syn hesized plane wa e e o , i is con enien o de ine he ollowing e o unc ion [4]: _- I E e ( ) 12+2I He ( )12 (10) = E-1( ) 12 +Ti2 1 Hdpi ) 12 whe e he de ia ion om he ideal plane wa e is de ined as e =-E- (11) E e =E - E pi He HHPI This e o unc ion is a scala unc ion ha conside s he e o in he elec ical and magne ic ield in magni ude and phase. I is also in e es ing o de ine he oo mean squa e alue o he plane wa e e o unc ion as: 661 1 ~~~~~~~~~~~~~~~~(12) T ms S[ IT( ) I2dS]( REE;QLTS By using exp essions (1) (2) i is possible o compu e he syn hesized plane wa e inside he measu emen cylinde . Fo his pu pose he cu en dis ibu ions ha e been disc e ized, and he ield inside o he cylinde has been compu ed as he supe posi ion o he adia ed ields o a cylind ical a ay o in ini esimal dipoles. Figu e 2 shows he elec ical ield componen s along he z axis when a e ical plane wa e is syn hesized. Figu e 3 shows he plane wa e e o unc ion along he z axis, o e ical plane wa e, o di e en inciden angles. Finally, igu e 4 shows he ms alue o he plane wa e e o compu ed along he z axis o di e en cylinde leng hs and in unc ion o he inciden angle. The as e isk shows he alidi y angle (0 ,) o he di e en cylinde s de ined as: anO =3 Z m - (13) 2(pO-a) No ice ha o he di e en cylinde leng hs, he ms alue o he plane wa e e o unc ion is app oxima ely he same when he plane wa e inciden angle is he alidi y angle. On he o he hand, in igu e 4 he op ho izon al line is he ms alue o he plane wa e e o conside ing only a quad a ic phase e o along he z axis. The maximum e o is n / 8 as in he usual a ield c i e ia. CQBCL<UQIQNS The cylind ical nea o a ield ans o ma ion has been exp essed as a plane wa e syn hesis p ocess. In his way he unca ion e o due o he ini e leng h o he measu emen cylinde can be s udied as a de e io a ion o he syn hesized plane wa e. By using an e o c i e ia i is possible o compa e a cylind ical nea ield an enna measu emen acili y wi h a compac o a a ield ange on plane wa e quali y basis. REEERENCL [1] Benne . J.C, Schoessow. E.P., ' An enna nea ield/ a ield ans o ma ion using plane wa e syn hesis echnique', IEE P oc., ol. 125, No. 3, Ma ch 1978. [2] Leach. W.M., Pa is. D.T., 'P obe Compensa ed Nea -Field Measu emen s on a Cylinde .' IEEE T ans. An ennas P opaga ., ol. AP-21, July 1973. [3] Bo gio i. G.V. 'In eg al Equa ion Fo mula ion o P obe Co ec ed Fa -Field Recons uc ion om Measu emen s on a Cylinde ', ol. AP-26, July 1978. [4] Hansen. J.E., 'Sphe ical Nea Field An enna Measu emen s', Pe e Pe eg inus L d., London 1988, Cap.7. 662 This wo k has been suppo ed by he Spanish Commi ee o Scien i ic and Technical Resea ch (CICYT) unde g an 91-1034. Figu e 1. Syn hesis geome y. PLANE WAVE SYNTHESIS plane wa e e o , e leal pola Iza on Nn *O -. n ," a -- _ e To" A --- Wn0 .,. ea 60 a _-.. an 40 A --- -4 - 2 4 .a/ Figu e 3. Plane wa e e o unc ion. Figu e 2. Syn hesized elec ical ield. PLANE WAVE SYNTHESIS RIMS alue o he plane wa e e o , e ical pola iza ion L-W1A L-UA L- IA L..UA L-MA -.-. -_ Figu e 4. Rms alue o he plane wa e e o . 663 PLANE WAVE SYNTHESIS Elec ical ield, e cal pola iza lon VMS -- a/A sIdwmeM mA ACKNQMaJMGM