Mathematical Analysis of Parameters of Slot Cut in a Waveguide

International Journal of Pure and Applied PhysicsISSN 0973-1776 Volume 6, Number 3 (2010), pp. 276–281© Research India Publicationshttp://www.ripublication.com/ijpap.htmMathematical Analysis of Parameters of Slot Cut in aWaveguideChakresh Kumar*, Devendra Chack**, Sandeep Kumar*and Manish Sharma**Dept. of Electronics and Communication, M.R.I.U., Faridabad, India**Dept. of Electronics and Communication, College of Science and Engg., IndiaE-mail:chakreshk@gmail.comAbstractThis paper proposes a novel analyze a longitudinal slot cut in the broad wall aswell as the narrow wall of rectangular waveguide. We have derived the fourfields (the incident field, the internally scattered field, the cavity scatteredfield, the field radiated externally from the slots) in the three different regions(Cavity, Waveguide and Half-Space).IntroductionA slot cut on the walls of a waveguide can act as an antenna. It can also act as animpedance whose value change with the dimensions of the slot .An incident waveundergoes reflection and transmission at the slot. These characteristics are veryimportant to analyze in order to fabricate a slotted waveguide antenna or a slottedwaveguide array of antenna. When studying the fields associated with a slot, we canclassify them broadly into 4 categories .The four types of field existent are1. The incident field2. The field scattered inside the waveguide3. The field inside the cavity (i.e. the slot region)4. The radiated field outside the waveguide through the slot.Antenna AnalysisSlot Parameters AnalysisThe field inside the cavity has been found using the cavity green's function which hasbeen derived by solving the Helmholtz equation for the electric vector potential forunit magnetic current source. The scattered magnetic fields in the waveguide regiondue to the presence of the transverse magnetic current densities are solved by rigorous

Mathematical Analysis of Parameters of Slot Cut in a Waveguide 277mode matching technique. The scattered field inside the waveguide due to slots in thebroad / narrow wall of the waveguide has been derived using the Green’s function forthe electric potential for such a slot. From the field existing at the aperture fed by awaveguide, the radiated field can be evaluated using plane wave spectrum approach.The radiated field is obtained by expanding the spherical waves in terms of planewave spectrum in the vector potential formulation. By applying the continuitycondition of the tangential magnetic field at the slot, and expanding the unknownmagnetic current densities in terms of piecewise linear (triangular) / entire domainsinusoidal basis function by using the Method of Moments, the problem is reduced tosolving the simultaneous linear equations.Derivation of Scattered Fields in Cavity, Half-Space and Waveguide due toApertures and SlotsThe expressions for the scattered field inside a cavity due to magnetic current sourceat the aperture has been derived from cavity Green’s function of the electric vectorpotential. The field radiated into half space by the magnetic current source at theaperture is obtained using the spectral domain approach. The expressions for scatteredfields inside the waveguide due to rectangular slots cut on the broad / narrow walls ofthe rectangular waveguide are derived using the vector potential approach.Derivation of the Field Radiated into Free Space by a Rectangular ApertureFrom the field existing at the aperture fed by a waveguide, the radiated field can beevaluated using the plane wave spectrum approach [8, 98, 100]. The radiated field isobtained by expanding the spherical waves in terms of plane wave spectrum in thevector potential formulation.The components of the magnetic field at the aperture plane are given in terms ofFourier Transform of the aperture electric field as∞∞2 21 kkx yεx+ ( k −k) εx y jkxky ( x + y )follows: Hx =− e dkdkx y2k πη∫∫k−∞−∞z∞ ∞2 21 kkx yεy+ ( k −k) εy x jkx ( x + ky y )Hy = e dkxdky2πηk∫∫k−∞ −∞z∞ ∞1 kkx zεy− kky zεxjkx ( x + ky y )Hz= e dkxdky2πkη∫∫k−∞ −∞zFor any aperture electric field distribution, therefore, the magnetic field at anypoint in space can be found out using the approach outlined so far. In the rest of thisthesis, the required component of magnetic field has been determined from theseexpressions by using suitable change of axes.

278 Chakresh Kumar et alDerivation of the Scattered Field inside a Waveguide due to a Longitudinal Sloton the Broad Wall of the WaveguideFigure 2: Longitudinal slot on the broad wall of a rectangular waveguideFigure (2) shows a longitudinal slot of dimension 2L× 2Wcut on the broad wallof a rectangular waveguide of cross section 2a× 2b. The slot is offset from the broadwall centre line by x w .If the slot is narrow, the electric field E sexisting in theaperture can be approximated asE = uE ˆs x xAnd the equivalent magnetic currentMMsis given by:uˆ xEx × uˆ ˆy= uzE xfor ⎧⎪fields radiated into free space= ⎨× − = −suˆ xEx ( uˆ y)uˆzE xfor fields scattered within waveguide⎪⎩The electromagnetic field radiated by the slot into the waveguide is therefore hasto satisfy the scalar Helmholtz equation, with source . In other words,∇ F + kF=−M2 2z z zThe green’s function is defined to be the solution of the differential equation withthe excitation function Mzbeing a unit impulse. Substitution of a delta function forMz, therefore gives,∞ ∞εεm nmπG( x,y,z/ x,y,z ′ ′ ′⎧ ⎫) = ∑∑ cos ⎨ ( x+ a)⎬×m0m0 = = 8γmnab ⎩2a⎭⎧nπ ⎫ −γmnzz − ′ ⎧mπ ⎫ ⎧nπ⎫cos⎨ ( y+ b) ⎬e cos⎨ ( x′ + a) ⎬cos⎨ ( y′+ b)⎬⎩2b ⎭ ⎩2a ⎭ ⎩2b⎭G ( xyz , , / x′ , y′ , z′ ) is the Green’s function for the electric vector potentialfunction for the internal scattered field due to a longitudinal slot on the broad wall of arectangular waveguide.Mz

Mathematical Analysis of Parameters of Slot Cut in a Waveguide 279“The same Green’s function holds good for a longitudinal slot in the narrow wallof a rectangular waveguide too”.Derivation of the Cavity Scattered FieldWith the cavity Green’s function of electric vector potential “F” being determined, thenext step is to find the scattered magnetic field inside the cavity region due to themagnetic currents.Derivation of the Externally Radiated FieldThe magnetic field at the plane of the aperture (i.e. X-Y plane) is evaluated using theplane wave spectrum approach.∞ ∞2 2( k −k)ext 1x jkx ( x + ky y )H =− ε ( k , k , k)e dk dk2πηk∫∫kx y x y x y−∞ −∞ zSince E at the aperture is assumed to only the Y-component, there is no εxtermin above equation . The Fourier Transform of the electric field at the aperture is givenby equation12π∞∞−∞ −∞∫∫( x ′ ky y ′) ′ ′= ε ( )− jkx+s s x y zE e dxdy k ,k ,kCarrying out the necessary integration, εyandε2WM2,y( k,k,k) = ∑ E sinckW ( )⎪⎧extHxare obtained as:( x )( )j sin k L for p evencos k L for p odd⎨⎬x⎪⎪⎩⎭y x y p y 2⎫π⎧⎪⎪ ⎞⎛p1 =π 2Lkxp 1 − ⎨2L⎜pπ⎟ ⎬⎪⎫⎝⎠⎪⎪⎩⎭M ∞ ∞ 2 2ext WL 2,y k −kxHx =−2 ∑Ep sinc1 ( kWy ) ×π kηp1 = 2 2 2 2−∞ −∞ k k k∫∫( −x−y)( x ) ⎫⎪⎬( x ) ⎪ jkxky { x + y }⎧⎪j sin k L for p even⎨⎪⎩cos k L for p odd ⎭2 e dkdkxpπ⎧⎪⎛2Lk1x⎞ ⎫⎪⎨ −2⎜ ⎬pπ⎟⎪⎩⎝ ⎠ ⎪⎭Derivation of the Externally Radiated FieldThe plane wave spectrum approach, is used to determine the magnetic field at the slot,due to the electric field existing at the slot aperture. For the Z-component of they

Mathematical Analysis of Parameters of Slot Cut in a Waveguide 281ConclusionThe necessary equations have been derived for basically 4 fields associated with thewaveguide and the slot. Namely, the incident field, the internally scattered field due tothe magnetic current , the cavity scattered field and the field radiated externally fromthe slots. After this extensive theoretical understanding the task of finding theequivalent circuit parameters has been handled.References[1] Liang. J. F., Chang. H. C. and Zaki. K. A., “Design And Tolerance AnalysisOf Thick Iris Waveguide Bandpass Filters,” IEEE Transactions on Magnetics,Vol. 29, No. 2, pp. 1605-1608, March 1993.[2] Oliner, A.A., “The Impedance Properties Of Narrow Radiating Slots In TheBroad Face Of Rectangular Waveguide. Pt. 1-Theory. Pt. 2- ComparisionWith Measurement”, IRE Transactions on Antennas and Propagation, Vol.AP-5(1), pp. 4-20, 1957.[3] Das, B.N. and Sanyal, G.S., “Network Parameters Of A Waveguide BroadWall Slot Radiator”, Proc. IEE, Vol.117, No.1, pp.41-44, 1970.[4] Prasad, S.M. and Das, B.N., “Studies On Waveguide Fed Slot Antenna”, Proc.IEE, Part H, Vol. 120, No. 5, pp. 539-540, 1973.[5] Das, B.N. and Sinha, M., “Admittance Characteristics Of Longitudinal SlotsIn The Broad Wall Of Rectangular Waveguide”, JIETE (India), Vol. 21, pp.32-37, January 1975.[6] Bailey. M. C., “Design of Dielectric-Covered Resonant Slots In A RectangularWaveguide,” IEEE Transactions on Antennas and Propagation, Vol. 15, No. 5,pp. 594-598, September 1967.[7] Yee, H.Y, “Impedance of Narrow Longitudinal Shunt Slot In A SlottedWaveguide Array”, IEEE Trans. on Antennas and Propagation, Vol. AP-22,pp. 589-592, July 1974.[8] Bailey, M.C., “The Impedance Properties of Dielectric Covered NarrowRadiating Slots on the Broad Face Of A Rectangular Waveguide”, IEEETrans. on Antennas and Propagation, Vol. 18, pp. 590-603, 1970.

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