Development of closed form design formulae for aperture coupled microstrip antenna

Transcription

1 Journal of Scientific & Industrial Research Vol. 64, July 2005, pp Development of closed form design formulae for aperture coupled microstrip antenna Samik Chakraborty, Bhaskar Gupta* and D R Poddar Department of ETCE, Jadavpur University, Kolkata Received 06 September 2004; revised 11 April 2005; accepted 27 May 2005 Closed form equations are presented for designing aperture coupled microstrip antennas to ensure impedance matching with the feed network with a low return loss over a wide frequency band. The features like compactness, integrability with printed circuits and shielding of the radiating patch from the radiation emanating from the feed structure etc. make these antennas attractive for present day scientific and industrial applications in fields like mobile computing and communication. Keywords: Microstrip, Antenna, Impedance matching IPC Code: H 01 P 3/08; H 01 Q Introduction An aperture coupled microstrip antenna 1 (Fig. 1) consists of a rectangular patch of dimensions a x b printed on a substrate of thickness h and dielectric constant Є ra. A microstrip line feeds the microstrip patch through an aperture or slot in the common ground plane. The aperture is of dimensions l a X l w and centered at (X o, Y o ). The width of the microstrip line is W and it is printed on a substrate described by thickness t and dielectric constant Є rf. The characteristic impedance of the microstrip line is denoted by Z om and that of the slot line corresponding to the coupling slot by Z os. Coupling of the slot to the dominant mode of the patch and the microstrip line occurs because the slot interrupts the longitudinal current flow in them 2. The coupling slot is nearly centered with respect to the patch where the magnetic field of the patch is maximum. This is done to enhance coupling between the magnetic field of the patch and the equivalent magnetic current near the slot. The coupling amplitude can be determined 3 from the following expression: The biggest problem encountered by engineers, while working with aperture coupled microstrip antennas, is that no closed form design formula is available. The paper presents closed form design expressions for optimal design of the feed when the patch is designed according to the relations available readily. Theory for Structural Analysis In a simplified equivalent circuit of an aperturecoupled microstrip antenna (Figs 2 & 3), the patch is characterized by admittance, Y patch, and the aperture by an admittance, Y ap. The patch admittance is determined at the centre of the slot. Coupling ( M r. Hdv r π x 0 ) sin v la... (1) where x 0 is the offset of the slot from the patch edge. *Author for correspondence gupta_bh@yahoo.com Fig. 1 Expanded view of an aperture-coupled microstrip antenna

2 CHAKRABORTY et al: APERTURE COUPLED MICROSTRIP ANTENNA 483 Y 1 =Y o [{(G r +jb open )+jy o tan(βl 1 )}/{Y o +j(g r +jb open ) tan(βl 1 )}] (3) L 1 = X o (3a) Y 2 = Y o [{(G r +jb open )+jy o tan(βl 2 )}/(Y o +j(g r +jb open ) tan(βl 2 )}]... (4) L 2 = a - L 1 (4a) Here (Y o, β) characterize the rectangular patch antenna as a microstrip line of width b, and (G r +jb open ) is the edge admittance of the patch. The value of Y ap can be obtained from transmission line model of a slot and is given by 4 Y ap = - j2y os cot (β s l a / 2 )... (5) Fig. 2 Equivalent circuit of an aperture-coupled microstrip antenna A transformer of turns ratio n 2, used to describe the coupling of the patch to the microstrip feed line is modeled from the discontinuity ΔV in modal voltage of the feed microstrip line, that is n 2 = ΔV/V o where V o is the slot voltage. Thus n 2 =[{(J o (β s w/2) J o (β m l w /2)}/(β s 2 β m 2 )][β m 2 k 2 Є rf / {(k 2.Є rf.cos(k 1 h) k 1.sin(k 1 h)}+(β m 2.k 1 /{k 1 cos(k 1 h) + k 2 sin(k 1 h)}]... (6) Fig. 3 Transmission line model of a rectangular patch fed by a slot in the ground plane In feed configuration, the patch antenna appears in series with the feed because of slot coupling. The nonresonant slot is represented as an inductor in series with the R-L-C network representing the patch resonator. The open circuited microstrip stub of length l s can be replaced by a shunt capacitor C s such that 1/ωC s = Z o cot (β l s ), where Z o is the characteristic impedance and β is the propagation constant of the microstrip feed line. The coupling of the patch to the aperture is described by an impedance transformer of turns ratio n 1 = l a /b. If Z 1 and Z 2 are the impedances looking toward the left and right of the aperture (Fig. 3), then Z patch = Z 1 +Z 2 = 1/Y 1 + 1/Y 2... (2) where, J o (.) is the zeroeth order Bessel function; k 1 = k o ( Є rf Є res Є rem ) 1/2 ; k 2 = k o ( Є rf + Є rem - 1 ) 1/2 ; and β s = k o res and β m = k o rem. Here (w, Є rem, β m, Z om ) are microstrip line parameters and (l w, Є res, β s,y os ) are the corresponding slot line parameters. Taking into account the impedance of the microstrip open circuited stub of length l s, the input impedance of the antenna at the centre of the slot 5 is given by Z in = [(n 2 2 /(n 1 2 Y patch + Y ap )] jz om cot(β m l s ) (7) Setting n 1 2 B patch + B ap = 0 for resonance and using Eqs (3) and (4) yields the following condition for resonance: B patch = -B ap / n 1 2 4b 2 / (Z os β s l a 3 ) (8) Therefore increasing l a results in a decrease in B patch and consequently the resonant frequency decreases. Development of Closed Form Design Formulae In the analysis, series expansion of Bessel function 6 of first kind was taken as

3 484 J SCI IND RES VOL. 64 JULY 2005 J n (x) = [1 / {2 n (n+1) }] α (-1) r (x n+2r ) / r= 0 {2 2r r!(n+1)(n+2) (n+r)}... (10) l a = (2n + 1). / β s l w = 4/β m ((1 (β s w/4) 2 ) / B) 2 (15a) (15b) where r = 0,1,2,3,4., and terms involving third or higher degrees of the argument were neglected. Further for inverse cotangent function, first six terms of corresponding Maclaurin Series 7 are considered as cot -1 Z= ( / 2) Z + (Z 3 /3)- (Z 5 /5) + (Z 7 /7) - (Z 9 /9) +... (11) This, according to Eqs (2)-(7), gives (l a /b) 2 Y patch j(2cot(β s l a / 2))/A = [(1- β s 2. w 2 /16 - β m 2 l w 2 /16) 2.(X 1 +X 2 ) 2 ]/Z in where, X 1 = β m 2 k 2 Є rf /[k 2 Є rf cos(k 1 h) k 1 sin(k 1 h)], and X 2 = β s 2 k 1 /[k 1 cos(k 1 h) + k 2 sin(k 1 h)]. The expressions for the characteristic impedance and guide wavelength of a slot line on a substrate of low Є r have been obtained 8 by curve fitting of the numerical results obtained using Galerkin s method in the Fourier transform domain. These expressions are for w/λ o 1.0 and 2.22 Є r 3.8. λ s /λ o = lnє r [(0.62 Є r (w/λ o ) 0.48 ]/ (1.344+w/h) [1.91-(Є r +2)/Є r ]ln(h/λ o ) (12) average error=0.69%, max error = 2.6%(evaluated at two points, for w/λ 0 > 0.8) and Z os = (Є r -1.8) {2.96+(Є r ) 2 } [(l w /h+2.32є r -0.56){( Є r )(100h/λ o ) 2-1)}] 1/2 - (684.45h/λ o )(Є r -1.35) {(Єr-1.722) l w /λ o } 2 (13) average error = 1.9 %, max error = 5.4% (evaluated at three points, for w/λ o > 0.8). Therefore [(1- ß s.w /16 - ß m. l w /16) (X 1+X 2 )] Z in = 2 2 (l /b) Y - j cot(ß.l / 2)/Z a patch s a os (14) Equating real and imaginary parts of Z in as given by Eq. (14) to those of the characteristic impedance of the feed line, required design equations are obtained as la b Here B = ( X 1 + X 2) Y patch and l s = 1/ β m ( cot -1 (Y/Z om )) Z in (15c) where Y and Z om are the values of the real and imaginary parts of the microstrip feed line characteristic impedance and the other parameters are as defined earlier. Thus a set of closed form design equations is obtained for the aperture length (l a ), width (l w ) and corresponding stub length (l s ) of the aperture coupled microstrip patch antenna. Validation of Design Procedure In order to check the validity of the design equations developed, an aperture coupled rectangular microstrip patch antenna was designed for operation at K u band with the following parameters: for antenna substrate, thickness= mm, Є r =2.4; and, for feed line substrate, thickness= mm, Є r =2.4. The corresponding resonant patch length and width are 3.70 mm and 5.60 mm respectively. For a 50 Ohm feed line, the aperture parameters, according to the design equations developed in this paper, are obtained as l a = mm, l s = 2.79 mm and l w = 2.98 mm. Computer simulation using IE3D software package 9 was run for the designed antenna, which yielded a 9.5 db (VSWR<2:1) bandwidth of GHz and undistorted radiation pattern providing half power beam width of degrees. The simulated plots for return loss vs frequency and radiation pattern at resonance are shown in Figs 4 and 5 respectively. To extend the validation, further investigations were carried out in array environment. A 4 4 uniform array was investigated at a frequency band centered around 18.5 GHz with inter-element spacing of 12 mm. along both orthogonal directions (Fig. 6). The individual antennas had configuration as mentioned earlier, obtained by using the design procedure developed. Since the memory requirement for computer simulation of such a large structure is prohibitively large, the entire array was fabricated on a single substrate and tested experimentally. For

4 CHAKRABORTY et al: APERTURE COUPLED MICROSTRIP ANTENNA 485 Fig. 4 Return loss vs. frequency plot for single aperture coupled patch (simulated) Fig. 7 Return loss vs. frequency plot for aperture-coupled microstrip array (experimental) Fig. 5 Radiation pattern for single aperture coupled patch (simulated) Fig. 8 Radiation pattern for aperture-coupled microstrip array (experimental) Fig array of aperture coupled microstrip patches

5 486 J SCI IND RES VOL. 64 JULY 2005 return loss (Fig. 7), measured values show very good impedance matching with the feed indicated by a less than 10 db return loss over GHz. The measured radiation pattern at resonance (Fig. 8) expectedly provides half power beam width of about 25 degrees and side lobe level of 8.5 db. Conclusions A set of closed form design formulae is presented for the feed to an aperture coupled microstrip antenna. Its validity is also checked through simulation of a single patch as well as fabrication and measurement in an array environment. It is established that usage of the relations developed results in wideband performance with very low return loss, thereby indicating a good design from practical point of view. However, this work was carried out for rectangular coupling slots only. Closed form design equations for coupling slots of other shapes can be formulated in a similar fashion. References 1 Garg R, Bhartia P, Bahl I & Ittipiboon A, Microstrip Antenna Hand Book (Artech House, Boston) 2001, Das B N & Joshi K K, Impedance of a radiating slot in the ground plane of a microstrip line, IEEE Trans Ant & Propagat, AP-30 (1982) Pozar D M, Microstrip Antennas, Proc IEEE, 80 (1992) Ryder J D, Networks, Lines and Fields (Prentice hall, New Delhi) 1998, Garg R, Bhartia P, Bahl I & Ittipiboon A, Microstrip Antenna Hand Book (Artech House, Boston) 2001, Rajput B S, Mathematical Physics (Pragati Prakashan, Meerut) 1996, / inverse cotangent.htm 8 Janaswamy R & Shaubert D H, Characteristic impedance of a wide slotline on low permittivity substrates, IEEE Trans Microwave Theory & Tech, MTT 34 (1986) IE3D, release 10.1, Zeland Software Inc, Fremont, CA 94538, USA.