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Resonance and Parametric Analysis of Planar Broad-Wall Longitudinal Slot Array Antennas

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Bonfring International Journal of Research in Communication Engineering Volume 2, Issue 4, 2012
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  Bonfring International Journal of Research in Communication Engineering, Vol. 2, No. 3, December 2012 13 Resonance and Parametric Analysis of Planar Broad-Wall Longitudinal Slot Array Antennas Rintu Kumar Gayen and Sushrut Das  Abstract--- This paper presents, method of moments based analysis of broad-wall longitudinal slots array antenna using  Multiple Cavity Modeling Technique (MCMT). Theoretical data for reflection coefficient, transmission coefficient, resonance length and resonance conductance have been obtained for two slots in two different waveguide. The theoretical data have been compared with Ansoft HFSS’s simulated data to validate the proposed method. The excellent agreement obtained between the results demonstrates that the  proposed method is able to accurately and efficiently solve complex reflection coefficients and transmission coefficients  for different slot parameters.  Keywords--- Waveguides, Slot Array Antennas, MCMT,  Moment Method, Reflection Coefficients, Transmission Coefficients, Normalized Admittance and Resonance I.   I  NTRODUCTION  TUDIES on waveguide broad-wall longitudinal slot antennas date back before World War II. Till then a number of workers have carried out considerable investigations on the admittance properties of the structure and a detail review of it will be a literature of its own. A brief survey of these literatures has been provided in [1]. However most of these analyses were carried out for a single slot element. Few literatures are also available on slot array [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. In this paper we have analyzed a two element planar  broad-wall longitudinal slots array antenna based on Multiple Cavity Modeling Technique (MCMT) [15]. The theoretical data have been compared with the Ansoft HFSS’s simulated data to validate the proposed method. The excellent agreement  between the results demonstrates that the proposed method is able to accurately and efficiently solve complex reflection coefficients and transmission coefficients for different slot  parameters. II.   P ROBLEM F ORMULATION  The 3D view of a planar two element broad-wall longitudinal slot antenna is shown in figure 1 whereas the top view of the waveguide slot antenna is shown in figure 2. The  Rintu Kumar Gayen, Senior Research Fellow, Department of Electronics  Engineering, Indian School of Mines – Dhanbad, Dhanbad, Jharkhand- 826004, India. E-mail:rintukrgayen@gmail.com Sushrut Das, Assistant Professor, Department of Electronics  Engineering, Indian School of Mines – Dhanbad, Dhanbad, Jharkhand- 826004, India. E-mail:sushrut_das@yahoo.com DOI: 10.9756/BIJRCE.3124   corresponding cavity modeling and details of magnetic current at the apertures is shown in figure 3. Figure 1: Three Dimensional View of a Two Element Planer Waveguide Slots Antenna Figure 2: Top Views of the Proposed Waveguide Slot Antenna Figure 3: Details of Different Regions and Magnetic Currents at the Apertures of a Two Element Planar Waveguide Slot Antenna The electric field at the slot may be assumed to be X-directed and can be expressed in terms of a sum of weighted sinusoidal basis functions, , ie pz  defined over the entire length of the slot as follows:  M iiˆ  E(x,y,z)uEei xp,zp,z p1 ′ ′ ′ = ∑=  r  Where , ie pz  is defined as: S ISSN 2277 - 5080 | © 2012 Bonfring  Bonfring International Journal of Research in Communication Engineering, Vol. 2, No. 3, December 2012 14 ( ) ( )  ( ) iex,y,z p,z p' sinzzLyb On aperture i  wss2Ls0 Elsewhere ′ ′ ′ =⎧ ⎧ ⎫π′− + δ −⎪ ⎨ ⎬⎨ ⎩ ⎭⎪⎩  Where 2Ls is the slot length, 2Ws is the slot width, 2Z WS  is the distance between the slots and 2b is the guide height. With the given electric field distribution the magnetic currents can  be obtained using equivalence principle. At the region of slot, the tangential components of the magnetic field should be identical. This results in the fallowing boundary condition. ( ) ( ) ( ) 111112 wvgcavcavinc HMHMHMH  zzzzzzz + − =  (1) ( ) ( ) ( )( ) 1112230 cavcavext  HMHMHM  zzzzzzext  HM  zz − + +=+ ( ) ( ) ( ) 2323 extextcav HMHMHM  zzzzzz + + − ) 40  M  z  = ( )  (2) ( ) 240 cav HM  zz  = ( ) ( ) ( 22324 wvgcavcav HMHMH  zzzzz − + +  (3) (4) The Green’s function, derived by solving the Helmholtz equation for the electric vector potential for unit magnetic current source is used for obtaining the magnetic field spread inside the cavity region. The scattered field inside the waveguide due to slots in the broad wall of the waveguide also has been derived using the Green’s function for the electric  potential for such a slot [1]. From the field existing at the apertures fed by the waveguide, the radiated field can be derived using plane wave spectrum approach. The radiated field is obtained by expanding the spherical waves in terms of  plane wave spectrum in the vector potential formulation [1]. The field components are given by: ( ) ( )( )( ) ( )  ( )( ) ( )( )  ( ) ( )  ( )( )  M cosn jW wvgiimns HME  zz p,z22 p1m0n02kab1SpmnmmcosxasincW ss2a2a pp22ksinzLk smn2L2Lsssinhz if p even LmnmnsSpecoshz if p odd mn2 ∞ ∞ πε ε= ×∑ ∑ ∑= = = ηγ +π π⎧ ⎫ ⎛ ⎞+ ×⎨ ⎬ ⎜ ⎟⎩ ⎭ ⎝ ⎠⎧ ⎫⎛ ⎞ ⎧ ⎫π π⎪ ⎪− + + + γ⎨ ⎬ ⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠ ⎩ ⎭⎩ ⎭⎧ ⎫− γ−γ ⎪ ⎪⎨ ⎬γ⎪ ⎪⎩ ⎭× ( ) ( ) mncosxacosyb2a2b ⎡ ⎤⎢ ⎥⎢ ⎥×⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦π π⎧ ⎫ ⎧ ⎫+ +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭    x jzinc Hjsine z2a π⎛ ⎞ − β= − ⎜ ⎟⎝ ⎠   ( ) ( ) ( )  ( ) { } ( ) { }  ( ) { } ( ) { }  ( ) { } 2 M  jmcavii2 HMEk  zz p,z22L p1mp1k s1mnsinzLcosxW ss2L2Wsin2ssmnmn'' cosytcosytyymnmn for mp and n0'' cosytcosytyymnmn0 ⎧ ⎫∞ ⎛ ⎞ωε π⎪ ⎪= − ×∑ ∑ ⎨ ⎬⎜ ⎟= = = ⎝ ⎠⎪ ⎪⎩ ⎭−⎧ ⎫ ⎧ ⎫π π+ + ×⎨ ⎬ ⎨ ⎬Γ Γ⎩ ⎭ ⎩ ⎭⎛ ⎞Γ − Γ + >⎜ ⎟= =⎜ ⎟⎜ ⎟Γ − Γ + >⎜ ⎟⎝ ⎠  otherwise ⎧⎪⎪⎨⎪⎪⎩  Where 2a is the guide width, 2t is the slot / waveguide wall thickness, s  x  is the slot offset from the center of the waveguide and ( )  ( ) Spp2Lsmn = π γ  Rests of the symbols have their usual meaning. The method of moments is applied with Galerkin’s specialization [16] to obtain 2M different equations from the  boundary conditions to enable determination of the . The weighting functions are defined as follows: , i pz  E  ( ,,, lwxyzqz M WLextiiss HME  zz p,z2 p1k 22 = − ×∑=π η∞ ∞ kk  xSinckW  xs1/2222kkk  z x jsinkL if p is even zscoskL if p odd  jkxkz zs z xwswse22Lk  p zs12p''  jkzkx z xedkdk  z x ⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠ −×∫ ∫−∞−∞− −⎧⎪⎨− + ⎪⎩×⎧ ⎫⎛ ⎞π ⎪ ⎪−⎨ ⎬⎜ ⎟π⎝ ⎠⎪ ⎪⎩ ⎭+   ) ( ) ( )  ( ) lwx,y,zq,zqsinzzLyb On aperture l  w002L00 Elsewhere =⎧ ⎧ ⎫π⎪ ⎪⎪ − + δ −⎪ ⎨ ⎬⎨ ⎪ ⎪⎩ ⎭⎪⎪⎩  for all q (q = 1, 2, 3, …….., M). After taking moment of each of the terms in boundary conditions (1) – (4), with, we obtain a set of simultaneous equations which upon solving gives the unknown basis coefficients. Here we have assumed that weighting function is defines over a slot having slot length 2L 0 , slot width 2W 0 , and 2Z W0  is the distance between lq,z w ISSN 2277 - 5080 | © 2012 Bonfring  Bonfring International Journal of Research in Communication Engineering, Vol. 2, No. 3, December 2012 15 the slots. The (p, q)th elements of the moment matrices can be derived as follows: ( ) ( )( ) 2,,sinsinc22cos for sin for 222 WqxW incixsss Hwj zq Laas Lqodd s jLqeven jzswseq Ls π π π  β  β  β π  β  〈 〉 =−− ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭ ×   ( ){ }( ) { }  ( ) { } ( ) { }  ( ) { } 222,2121sin2coscosfor mpq 0and n0coscos00 otherw  LW mcavxss Hwjk  zqmp Lk st mnmn ytyt mnmn ytyt mnmn ωε  π  ∞〈 〉 = − −∑= =−×Γ ΓΓ − Γ +×= ==Γ − Γ + ⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭⎛ ⎞⎛ ⎜ ⎟⎜⎜ ⎟⎝ ⎝ ⎠ ise ⎧⎪⎪⎨⎪⎪⎩   ( )( )  ( ) ( )  ( ) ( ) ( ) ( ) 163200,2/22221sincossinsinsin00sinsinsincossinsinsin0sinsinsincos00sinsincos if , sinsincos0 WWLLssext  HMwvisible zzqregionckW s jkxkzwswsckWe jkxkzwwekLs pqkL λ η π π θ φ θ φ θ φ θ φ θ θ φ θ φ θ φ θ φ θ φ  = − ×− ×∫ ∫= =+ ( )( )  ( ) ( )  ( ) ( ) ( ) ( ) 163200,2/22221coshcossincoshsin00coshsincoshcossincoshsin0coshsincoshcos00sincoshcossincoshcos0 WWLLssext  HMwinvisible zzqregionckW s jkxkzwswsckWe jkxkzwwekLskL λ η π π θ φ θ φ θ φ θ φ θ θ φ θ φ θ φ θ φ θ φ  = − ×− ×∫ ∫= =+ φ  ×− +×× ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠ ( ) ( )  both cossincos if , both cossincos0sin22sincos1222sincos012 evenkLs pqodd kLdd  Lk  ps p Lk qq θ φ θ φ θ θ φ θ φ π π θ φ π π  ×− ×− ⎧⎪⎪⎪⎨⎛ ⎞⎪⎜ ⎟⎪⎜ ⎟⎪⎝ ⎠⎩⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭   φ  ×− +×× ⎛ ⎞⎜⎜⎝ ⎠ ( ) ( )  for , both coscoshcos for , both coscoshcos0cosh22coshcos1222coshcos012  pqevenkLs pqodd kLdd  Lk  ps p Lk qq θ φ θ φ θ θ φ θ φ π π θ φ π π  ×− ×− ⎧⎪ ⎟⎟⎪⎪⎨⎛ ⎞⎪⎜ ⎟⎪⎜ ⎟⎪⎝ ⎠⎩⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭ ( ) ( )  ( )( )( ) 2216,2/222221sincossinsinsin002sinsincos for p,q both even2cossincos for p,q both odd  0 otherwise WLextiiss HMwvisible zqregioncWk s Lk s Lk s λ η π π θ φ θ φ θ φ θ φ θ φ  = − ×− ×∫ ∫= = ⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪⎩ ⎭ sin22sincos p12222sincos1 dd  Lk qs p Lk sq θ θ φ θ φ π π π θ φ π  − ×− ⎪⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭   ISSN 2277 - 5080 | © 2012 Bonfring  Bonfring International Journal of Research in Communication Engineering, Vol. 2, No. 3, December 2012 16 ( ) ( ) ( )( )( ) 2216,cosh2/22221coshcos002sinccoshsin2sincoshcos for p,q both even2coscoshcos for p,q both odd  0 otherwise WLextiiss HMwjinvisible zqregionWk s Lk s Lk s θ λ η π π θ φ θ φ θ φ θ φ θ φ  = − ×− ×∫ ∫= =× 22coshcos p12222coshcos1 dd  Lk qs p Lk sq θ φ θ φ π π π θ φ π  − ×− ⎧⎪⎪⎨⎪⎪⎩⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭⎫⎪⎪⎬⎪⎪⎭   ( )  ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2,20011122cossin1222111222222121sinh for , both cosh for , bot W wvg xmns Hwj zpmnkabmnmm xacW ssaaSp pkL pqs LsSpSqk  LmnmneSqmn Lpqevenmns Lpqmns ε ε ηγ π π π δ γ γ γ γ γ  ∞ ∞= − ×∑ ∑= =+ ×+− ++ −×+ ⎧ ⎫ ⎧⎪ ⎪ ⎪⎨ ⎬ ⎨⎪ ⎪ ⎪⎩ ⎭ ⎩⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭ h 0 odd  ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢ ⎥⎧ ⎫⎢ ⎥⎪ ⎪⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥⎢ ⎥⎩ ⎭⎣ ⎦⎫⎪⎬⎪⎭⎤⎥⎥⎥⎥⎥⎥⎥⎥  The expression for the reflection coefficients Γ   and transmission coefficients T at the z = 0 plane are obtained as: ( )( ) ( )( ) 2WxW  H sss zsinsincinc322a2a H 4abk  z z0cosL for p odd  M sps1 E  p,z2 jsinL for p even p11sps −π π π⎛ ⎞ ⎛ ⎞Γ = = ×⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠β=⎧ ⎫β⎪ ⎪∑ ⎨ ⎬β= + ⎪ ⎪⎩ ⎭   ( )( ) ( )( ) Where 2L 0  and 2W 0  are the length and width of the slot on which weighting function is defined. The admittance for the shunt network models are obtained from the Y using the following relations [11]. ( ) Y21 = − Γ +Γ . III.    N UMERICAL R  ESULTS  On the basis of the formulation, MATLAB codes have  been written to compute the reflection coefficients and transmission coefficients of the structures. The magnitude of the S – parameters of the structure for slots length = 16 mm, width = 1 mm, thickness = 1.27 mm, Xs = - 8.43 mm, X 0  = - 8.43 mm,  | Z WS  –Z W0 | = 0 mm and | X WS  – X W0 | = 24.13 mm milled as a waveguide with 2a = 22.86 mm, 2b = 10.16 mm have been obtained and compared with HFSS data over a frequency range 8 GHz to 12 GHz. The theoretical and HFSS data results have been plotted in Figure 4. Figure 4: Comparison of Theoretical Data with HFSS Data for Two Broad wall Longitudinal Slots Antenna System The variation of the complex S – parameters for slots with length = 16 mm, width = 1mm, thickness = 1.27 mm, width = 1 mm, X S  = 4 mm, X 0  = - 4 mm, | X WS  –X W0 | = 24.13 mm, 32.13 mm, 48.26 mm, 56.26mm, and | Z WS  – Z W0 | = 0 mm milled as a waveguide with 2a = 22.86 mm, 2b = 10.16 mm are plotted with frequency in figure 5 and figure 6 respectively. 2incWxW  HH sss zz1sinsincinc322a2a H 4abk  zcosL for p odd  M sps1 E  p,z2 jsinL for p even p11sps ⎛ ⎞ ⎛ ⎞Τ = = + ×⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠β⎧ ⎫β⎪ ⎪∑ ⎨ ⎬β= + ⎪ ⎪⎩ ⎭+π π π+   ISSN 2277 - 5080 | © 2012 Bonfring
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