Moment Method Analysis of Standard and Reduced Height Broad-Wall Longitudinal Slot Doublets in Rectangular Waveguides

Transcription

1 IJECT Vo l. 6, Is s u e 1, Sp l- 1 Ja n - Ma r c h 015 ISSN : (Onlne) ISSN : (Prnt) Moment Method Analyss of Standard and Reduced eght Broad-Wall Longtudnal Slot Doublets n Rectangular Wavegudes 1 Rntu K. Gayen, Sushrut Das 1, Dept. of Electroncs Engg., Indan School of Mnes, Dhanbad, Jharkhand, Inda Abstract Ths paper presents a method of moments based analyss of standard and reduced heght a broad-wall longtudnal slot doublets antenna usng Multple Cavty Modelng Technque (MCMT). Theoretcal data for reflecton coeffcent, transmsson coeffcent, resonance length and resonance conductance have been obtaned. The theoretcal data have been compared wth expermental data and Ansoft FSS s smulated data to valdate the proposed method. The excellent agreement obtaned between the results valdates the analyss. Keywords Wavegudes, Slot Antennas, Moment Method, Reflecton Coeffcents, Transmsson Coeffcents, Normaled Admttance, Resonance. I. Introducton Wavegude based slot antennas are extensvely used n array antennas n ground based, ar borne and shp borne radars n varous frequency bands rangng from 1 G to 1000 G due to ther large power handlng capablty, generaton of ultra low sde lobes and excellent polaraton characterstcs. Slots mlled on the broad wall are capable of gvng vertcal and horontal polaraton, dependng on ts locaton and orentaton on the broad wall of a rectangular wavegude. Front-fed reflector or cassegran antenna systems suffer from gan reducton, due to aperture blockage due to feed structure. Slot arrays do not suffer from ths drawback. Moreover, such arrays can be fabrcated wth mnmum power loss compared to the case of reflector antennas. Due to these characterstcs, the wavegude-fed slots have found an mportant place n the array antennas as radatng elements. Studes on wavegude broad-wall longtudnal slot antennas date back before World War II. Tll then a number of workers have carred out consderable nvestgatons on the admttance propertes of the structure and a detal revew of t wll be a lterature of ts own. A bref survey of these lteratures has been provded n [1]. In addton to that, Datta, Chakraborty and Das [] analyed of a strp loaded resonant longtudnal slot n the broad wall of a rectangular wavegude. In ths paper a longtudnal slot at the centre of the broad wall of a rectangular wavegude n the presence of one or two vertcal strps was analysed. The combnaton s excted by the domnant TE01 mode, and the slot radates externally. The effect of fnte thckness of the wavegude walls was taken care of by treatng the slot as a stub wavegude. The coupled ntegral equatons were solved usng Galerkn s technque. The resultant coeffcents were used to compute the domnant mode scatterng parameters and hence the equvalent T-network. The possblty of obtanng a resonant condton for the radatng centred slot strp combnaton was establshed. Desgn data derved n ths study elmnate the need to gather ntal nput data expermentally. Matner and Amr [3] presented effcent moment method soluton for a longtudnal slot n a rectangular wavegude. In ths paper longtudnal slot n a rectangular wavegude was solved by the moment method, where the transverse varaton of the tangental electrc feld on the apertures of the slot contans the correct edge behavour. It was shown that three expanson functons are enough for an excellent convergence and accuracy of the soluton. Km and Eom [4] presented mode-matchng model for a longtudnally slotted wavegude array. In ths paper a mode-matchng model for a longtudnally slotted wavegude array was presented. Fourer transforms and resdue calculus were appled to obtan fast convergent seres solutons. Ths model accurately estmates the radaton patterns of a longtudnally slotted wavegude array wthout a heavy computatonal burden. Km and Eom [5] also presented radaton from longtudnal slots on the narrow wall of a rectangular wavegude. In ths paper radaton from longtudnal slots on the narrow wall of a rectangular wavegude was studed. The mode-matchng technque was used to obtan a system of smultaneous equatons for modal coeffcents. Computatons were performed and compared wth measured data at G. Montsc and Maarella [6] presented effect of the longtudnal component of the aperture electrc feld on the analyss of wavegude longtudnal slots. In ths paper Method of moment analyss of longtudnal radatng slots n the broadwall of a rectangular wavegude requred a careful choce of the bass functons. Ths s partcularly true when we need an accurate evaluaton of both longtudnal and transverse components of the Electrc feld on a slot wth fnte wall thckness. It was shown that entre-doman, two-dmensonal, snusodal bass functons are a successful choce for an accurate and effectve modelng of a radatng slot cut n a thck wavegude wall. In ths communcaton there focused focus on slender slots, whch ensure hgh polaraton purty. If a very accurate array desgn s requred, the effect of the longtudnal component of the Electrc feld, often neglected n the analyss of a narrow longtudnal slot, was shown to have a sgnfcant nfluence on the slot radatng and crcutal propertes. Immedately after fndng the nterestng propertes of a broad-wall longtudnal slot as radatng elements, studes were carred on fndng the propertes of slot doublets. Soon t was found that slots mlled on the opposte broad walls of a rectangular wavegude can provde two radaton nulls and otherwse Omn-drectonal radaton pattern. Intal reports on the resonance characterstcs of these structures [7, 8] were based on expermental results. A theoretcal descrpton remaned unavalable. Later Sangster and Wang [9, 10] presented theoretcal results on the resonance propertes of Omn-drectonal slot doublets n rectangular wavegudes. In these papers moment method was employed to analye a longtudnal slot doublets n both ar-flled and delectrcflled rectangular wavegudes n whch the slot radators were located n the opposte broad faces of the wavegude. An analyss procedure for arrays of wavegude slot doublets based on the full T-network equvalent crcut representaton of radators was presented by Coetree and Joubert [11]. Later Mondal and Chakraborty [1-13] presented a new confguraton of a wavegude broad wall longtudnal slot doublet wth postve and negatve 1 Internatonal Journal of Electroncs & Communcaton Technology

2 ISSN : (Onlne) ISSN : (Prnt) offsets wth respect to the wavegude center lne. They studed and presented the resonance and radaton propertes of the structure. In ths paper a moment method analyss of standard and reduced heght a broad-wall longtudnal slot doublets antenna n rectangular wavegudes are presented based on Multple Cavty Modelng Technque (MCMT). The methodology nvolves n replacng all the apertures and dscontnutes of the rectangular wavegude based structures, wth equvalent magnetc current denstes so that the gven structure can be analyed usng only Magnetc Feld Integral Equaton (MFIE). To make the MFIE applcable to the generaled wavegude structure problems, the gven structures are modeled usng rectangular cavtes. As t s necessary to use a number of such cavtes n order to study these complcated wavegude structures, the present method s named as Multple Cavty Modelng Technque (MCMT). The nterfacng apertures between dfferent regons (wavegude cavty, cavty half space) are then replaced by equvalent magnetc current denstes. By applyng the contnuty condton of the tangental magnetc feld at the nterfacng apertures, and expandng the unknown magnetc current denstes n terms of entre doman snusodal bass functon by usng the Method of Moments, the problem s reduced to solvng the smultaneous lnear equatons. II. Problem Formulaton The 3D vew of a broad-wall longtudnal slot doublet s shown n fg. 1. The correspondng cavty modelng and detals of magnetc current at the apertures are shown n fg. respectvely. Fg. 1: Three Dmensonal Vew of Wavegude Slot Doublets Proposed Antenna Structure Fg. : Detals of dfferent regons and magnetc currents at the apertures of a wavegude slot doublet. The electrc feld at the slot may be assumed to be X-drected and can be expressed n terms of a sum of weghted snusodal IJECT Vo l. 6, Is s u e 1, Sp l-1 Ja n - Ma r c h 015 bass functons, e defned over the entre length of the slot as p, follows: Where e s defned as: p, pπ sn ( + L ) On aperture "" e ( p, x,y, ) = L 0 Elsewhere Where L s the slot length, W s the slot wdth and b s the gude heght. Wth the gven electrc feld dstrbuton the magnetc currents can be obtaned usng equvalence prncple. At the regon of slot, the tangental components of the magnetc feld should be dentcal. Ths wll gve M + M M M = wvg 1 cav1 1 cav1 wvg 3 nc cav1 1 cav1 uhs M + M + M = 0 M M M + M = wvg 1 wvg 3 cav 3 cav 4 nc cav 3 cav 4 lhs 4 M + M + M = 0 (4) The magnetc feld scattered nsde the cavty regon due to the source s determned by usng cavty Green s functon of the electrc vector potental. The cavty Green s functon has been derved by solvng the elmholt equaton for the electrc vector potental for unt magnetc current source. The scattered feld nsde the wavegude due to slots n the broad wall of the wavegude also has been derved usng the Green s functon for the electrc potental for such a slot. From the feld exstng at the apertures fed by the wavegude, the radated feld can be derved usng plane wave spectrum approach. The radated feld s obtaned by expandng the sphercal waves n terms of plane wave spectrum n the vector potental formulaton. The feld components are gven by M lhs uhs WL k k ( M) = E p, 1/ π kη p= 1 k k k ( x ) ( ) ( ) j( k x+ k ) j sn k L for p even cos k L for p odd x snc( k W ) e dk dk pπ Lk 1 pπ x x jε ε W ( M ) = E ab M wvg m n p, p= 1 m= 0 n= 0kηγmn ( π) cos n mπ cos ( xw + a) 1+ S p a mπ mπ nπ sn c W cos ( x + a) cos ( y + b) a a b pπ pπ k sn ( + L) + L L ( γmn ) ( γ ) snh for p even cosh mn for p odd (1) () (3) Internatonal Journal of Electroncs & Communcaton Technology 13

3 IJECT Vo l. 6, Is s u e 1, Sp l- 1 Ja n - Ma r c h 015 ISSN : (Onlne) ISSN : (Prnt) nc πx = j sn e a m0 jβ M cav jωε mπ ( M) = E p, k k p= 1 m= p= 1 L 1 mπ nπ + + Γ sn Γ t L W { sn m0 } ( L) cos ( x W ) { Γm0 ( )} { Γ m0 ( + )} { Γm0 ( )} { Γ m0 ( + )} cos y t cos y t y > y cos y t cos y t y < y Where a s the gude wdth, t s the slot / wavegude wall thckness, x w s the sot offset from the center of the wavegude and S( p) = pπ ( Lγ mn ) Rests of the symbols have ther usual meanng. The method of moments s appled wth Galerkn s specalaton [14] to obtan M dfferent equatons from the boundary condtons to enable l determnaton of the E p,. The weghtng functon wq, ( x, y, ) are defned as follows: qπ sn ( + L ) On aperture "l" l wq, ( x,y,) = L 0 Elsewhere for all q (q = 1,, 3,.., M). After takng moment of each of the terms n boundary condtons (1) (4), wth w l q,, we obtan a set of smultaneous equatons whch upon solvng gves the unknown bass coeffcents. The (p, q)th elements of the moment matrces can be derved as follows: wvg l m n p, ( ) q, m= 0 n= 0 ηγ mn + M,w ( γmn ) ( γ ) jε ε W 1 = k ab 1 S p ( k mn ) γ mn ( + ) S p S q +γ pπ γmnl k δ pql+ e L 1 S q snh L for p, q both even cosh mnl for p, q both odd 0 otherwse mπ 1 f =l cos ( xw + a) a cos( n π) f l where δ pq s the Kronecker delta functon. cav jωε mπ p, ( M ),wq, = k k m= p= q= 1 L LW cos( Γ ) m0t f y = y Γ { Γ } m0 sn m0t 1 f y y M,w π/π ( 1 sn cos ) θ= 0 φ= 0 θ φ 16W L = λ η lhs / uhs l p, q, vs. reg. sn Lk sn cos for p, q both even cos Lk sn cos for p, q both odd 0 otherwse pπ qπ Lk sn θcos Lk sn θcos 1 1 pπ qπ snc Wk snθsnφ snθdθdφ 14 Internatonal Journal of Electroncs & Communcaton Technology lhs uhs l 16W L p,,wq, = j nvs. reg. λ η π ( 1 cosh cos ) θ= 0 φ= 0 θ φ sn Lk cosh cos for p, q both even cos Lk cosh cos for p, q both odd 0 otherwse pπ qπ Lk cosh θcos 1 pπ Lk cosh θcos 1 qπ θ φ θ θ φ snc Wk cosh sn cosh d d ( β ) nc l πwq πxw πw,wq, = j sn snc L a a cos βl for q odd j sn L for q even qπ β L The expresson for the reflecton coeffcents are obtaned as: πx Γ = = ηβ π W w πw sn snc nc 3 4a bk a a = 0 S p ( β ) cos βl for p odd M M 1 3 Ep, Ep, p= 1 p= 1 1+S p jsn L for p even at the = 0 plane The admttance for the shunt network models are obtaned from the Γ usng the followng relatons [15] Y = Γ 1+Γ III. Results On the bass of the formulaton, MATLAB codes have been wrtten to compute the reflecton coeffcents, transmsson coeffcents, resonance length and resonance conductance of the structures. The results are presented below: The magntude of the S parameters of the structures for slots length = 16 mm, wdth = 1 mm, thckness = 1.7 mm, offset = 8.43 mm, mm have been obtaned and compared wth FSS and measured data n fgure 3 over a frequency range 8. G to 1.4 G. Fg. 3: Comparson of theoretcal data for the magntude of reflecton and transmsson coeffcent wth FSS and measured data for the proposed structures.

4 ISSN : (Onlne) ISSN : (Prnt) IJECT Vo l. 6, Is s u e 1, Sp l-1 Ja n - Ma r c h 015 The magntude of the S parameters of the antenna structures for slots length = 16 mm, wdth = 1 mm, offset = 1.54 mm, mm/ 3.94 mm, mm and mm, mm, thckness = 1.7 mm mlled on a wavegude wth a =.86 mm, b = mm/ 5.08 mm and.54 mm are plotted wth frequency n fg. 4 and fg. 5 respectvely. Fg. 5: Varaton of complex transmsson coeffcents (S1) wth frequency for the proposed slot antennas developed on standard and reduced heght wavegudes over 8 to 1 G. The magntude of the S parameters of the antenna structures for slots length = 16 mm, slot dstance along the propagaton drecton = 3.4 mm and.7 mm, wdth = 1 mm, offset = 4 mm, - 4 mm and 6 mm, - 6mm, thckness = 1.7 mm mlled on a wavegude wth a =.86 mm, b = mm/ 5.08 mm and.54 mm are plotted wth frequency n fgure 6 and fgure 7 respectvely. Fg. 4: Varaton of complex reflecton coeffcents (S11) wth frequency for the proposed slot array antennas developed on standard and reduced heght wavegudes over 8 to 1 G. Fg. 6: Varaton of complex reflecton coeffcents (S11) wth frequency for the proposed slot antennas developed on standard and reduced heght wavegudes over 8 to 1 G. Internatonal Journal of Electroncs & Communcaton Technology 15

5 IJECT Vo l. 6, Is s u e 1, Sp l- 1 Ja n - Ma r c h 015 ISSN : (Onlne) ISSN : (Prnt) Fg. 8: Varaton of complex reflecton coeffcents (S11) wth frequency for the antenna system wth dfferent slot length and gude heght. Fg. 7: Varaton of complex transmsson coeffcents (S1) wth frequency for the proposed slot antennas developed on standard and reduced heght wavegudes over 8 to 1 G. The magntude of the S parameters of the antenna structures for slots length = mm and mm, wdth = 1 mm, offset = mm, mm, thckness = 1.7 mm mlled on a wavegude wth a =.86 mm, b = mm/ 5.08 mm and.54 mm are plotted wth frequency n fgure 8 and fgure 9 respectvely. Fg. 9: Varaton of complex transmsson coeffcents (S1) wth frequency for the antenna system wth dfferent slot length and gude heght. The varaton of normaled resonant conductance and resonant slot length for the proposed slot array wth slot wdth = 1 mm, thckness = 1.7 mm, offset = mm, mm/ 3.94 mm, mm and 1.54 mm, mm mlled as a wavegude wth a =.86 mm, b = mm/ 5.08 mm and.54 mm are plotted wth frequency n fgure 10 and fgure 11 respectvely. 16 Internatonal Journal of Electroncs & Communcaton Technology

6 ISSN : (Onlne) ISSN : (Prnt) Fg. 10: Varaton of normaled conductance for the proposed slot array antennas developed on standard and reduced heght wavegudes over 8 to 1 G. Fg. 11: Varaton of resonant length for the proposed slot array antennas developed on standard and reduced heght wavegudes over 8 to 1 G. IV. Conclusons Ths paper presents, method of moments based analyss of reduced heght as well as of standard heght broad-wall longtudnal slots antenna usng Multple Cavty Modelng Technque (MCMT). Theoretcal data for reflecton coeffcent and transmsson coeffcent have been obtaned for the antenna system and also have been compared wth FSS and measured data n fgure 3 to valdate the analyss. The excellent agreement obtaned between them valdates analyss. The varatons of the complex S-parameter wth frequency for dfferent slot offset and gude heght have been shown n fgure 4 5. Fgure 4 shows that as the slot offset ncreases (keepng gude heght constant) the magntude of reflecton coeffcent and resonance frequency ncreases. In addton f the gude heght decreases (keepng slot offset constant) the magntude of reflecton coeffcent and resonance frequency ncreases. Fgure 5 shows as the slot offset ncreases (keepng gude heght constant) the S1 also decreases. Smlarly f the gude heght decreases (keepng slot offset constant) the S1 also decreases. Fgure 6 7 shows that the antenna network parameters depend on slot offset and gude heght. Fgure 6 shows that as the slot offset ncreases (keepng gude heght constant) the magntude of reflecton coeffcent and resonance frequency ncreases. owever f the gude heght decreases (keepng slot offset constant) the magntude of reflecton coeffcent and resonance frequency IJECT Vo l. 6, Is s u e 1, Sp l-1 Ja n - Ma r c h 015 ncreases. Fgure 7 shows as the slot offset ncreases (keepng gude heght constant) the S 1 also decreases. Smlarly f the gude heght decreases (keepng slot offset constant) the S 1 also decreases and resonance frequency ncreases. Fgure 8-9 shows that the antenna network parameters depends both on the slot length and gude heght. Fgure 8 shows that f the slot length s ncreased, the magntude of reflecton coeffcent ncreases and the resonance frequency decreases (for same gude heght). Reflecton coeffcent and the resonance frequency also ncrease wth decrease n gude heght for the same slot length. Fgure 9 shows that as slot length s ncreased S 1 decreases for the same gude heght. S 1 also decreases wth the ncrease n gude heght f slot length s kept constant. As evdent from fgure 10, the resonance conductance ncrease wth decrease n gude heght, provded slot offset s kept constant. If slot offset s ncreased, keepng gude heght constant, resonance conductance ncreases. Fnally fg. 11 reveals that the resonance slot length depends on gude heght and slot offset. If the slot offset ncreases (keepng gude heght constant) the resonance slot length ncreases. Smlarly f the gude heght decreases (keepng slot offset constant) the resonance slot length ncreases. Fnally, ths paper demonstrates that MCMT can also effectvely ncorparete the nternal mutual couplng, essental for the analyss of antenna arrays. Therefore we can conclude that MCMT can be appled for the analyss of antenna arrays. V. Acknowledgment The measurements were carred out n the Department of Electroncs and Electrcal Communcaton Engneerng, Indan Insttute of Technology Kharagpur and the authors wsh to express ther grattude to Prof. A. Bhattacharya of Indan Insttute of Technology, Kharagpur, for allowng us to use the lab faclty. References [1] S. Gupta,"Electromagnetc feld estmaton n aperture and slot Antennas wth ther equvalent network representaton", Phd Dssertaton, Department of Electroncs & Electrcal Communcaton Engneerng, I.I.T. Kharagpur, Inda, [] A. Datta, A. Chakraborty, B.N. Das, Analyss of a strp loaded resonant longtudnal slot n the broad wall of a rectangular wavegude, IEE Proceedngs-h, Vol. 140, No., Aprl [3]. Matner, N. Amr, Effcent moment method soluton for a longtudnal slot n a rectangular wavegude, IEE Proc.- Mcrow. Antennas Propag., Vol. 153, No. 4, August 006. [4] Yang. Km, yo J. Eom,"Mode-Matchng Model for a Longtudnally Slotted Wavegude Array, IEEE Antenna and wreless propagaton letters, Vol. 6, 007. [5] Yang. Km, yo J. Eom, Radaton from Longtudnal Slots on the Narrow Wall of a Rectangular Wavegude, IEEE Antenna and wreless propagaton letters, Vol. 7, 008. [6] Gorgo Montsc, Guseppe Maarella, Effect of the Longtudnal Component of the Aperture Electrc Feld on the Analyss of Wavegude Longtudnal Slots, IEEE Transacton on Antenna and Propagaton, Vol. 59, No. 11, November 011. [7] I. P. Kamnow, R. J. Stegen., Wavegude slot array desgn, NTIS Report, AD63600, [8] T. Takeshma, X-band omndrectonal double-slot array antenna, Electroncs Engneerng., pp , October Internatonal Journal of Electroncs & Communcaton Technology 17

7 IJECT Vo l. 6, Is s u e 1, Sp l- 1 Ja n - Ma r c h 015 ISSN : (Onlne) ISSN : (Prnt) [9] A. J. Sangstar,. Wang, Resonance propertes of omndrectonal slot doublet n rectangular wavegude, Electroncs Letters., Vol. 9, No. 1, pp , January [10] A. J. Sangstar,. Wang, Moment method analyss of a horontally polared omndrectonal slot array antenna, Proc. of IEE Mcrowave Antenna Propagaton., Vol. 14, No. 1, pp. 1 6, February [11] J. C. Coetree, J. Joubert, Analyss procedure for arrays of wavegude slot doublets based on the full T-network equvalent crcut representaton of radators, Proceedngs of IEE Mcrowave Antenna Propagaton., Vol. 147, No. 3, pp , February 000. [1] P. Mondal, A. Chakrabarty, Equvalent crcut representaton of wavegude slot doublet, Proceednngs of APSYM 006, pp , December, 006. [13] P. Mondal, A. Chakrabarty, Slotted wavegude antenna wth two radaton nulls, IEEE transactons on Antennas and propagaton, Vol. 56, No. 9, pp , Sepetember 008. [14] R. F. arrngton, Feld Computaton By Moment Methods, Roger E. Kreger Publshng Company, USA, pp [15] B. N. Das, G. S. Sanyal., Network Parameters of A Wavegude Broad Wall Slot Radator, Proc. IEE, Vol. 117, No. 1, pp , Internatonal Journal of Electroncs & Communcaton Technology