390442-3-T SLOT COUPLED PATCH ANTENNAS Linda Katehi, Nihad I. Dib, J-C Cheng December 1993 390442-3-T = RL-2577

December 13,1993 Slot Coupled Patch Antennas Linda Katehi A report prepared for Texas Instruments Participating Scientists: Nihad I. Dib, Research Scientist, Ph.D. J-C Cheng, Graduate Student Brief Report During the past few months we concentrated on the characterization of patches which are electromagnetically coupled to their feeding lines. The excitation line can be of stripline, microstrip or coplanar waveguide form printed on the other side of the substrate. A schematic of this geometry is shown on Figure 1. A characteristic of this structure is that the patch may be backed by a cavity as shown in the Figure. The presence of the cavity requires the use of versatile discretization techniques as the integral equation/finite element method in frequency domain or the finite difference method in time domain. In this effort we chose to use both of the above techniques in order to gain some understanding of their capabilities and computer efficiencies in solving this type of problems. As a carriage of our experiment we chose the geometry of a microstrip patch excited by a coplanar waveguide feed through a slot aperture printed on the ground (see Figure 2). This is one of the possible geometries among the electromagnetically coupled patches. This problem has been approached with two different techniques: (a) The Finite Difference Time Domain Method, (b) the Finite Element/Integral Equation Method. The finite difference time domain method discretizes in a volume, which surrounds the antenna and the feeding line and is bounded by superabsorbing boundary conditions to simulate free-space. A description of this method is given in Appendix A. The FDTD codes used for the derivation of these data have been validated very thoroughly through numerous examples [1]-[4]. The integral equation/ 1

Brief Report finite element method applies the finite element method inside the cavity and the integral equation outside, above the patch and below the coupling slot. A detailed description of the method is given in Appendix B. Figures 4 and 5 show the FDTD results for the magnitude and phase of S1 1 as a function of the frequency for the patch with and without the cavity. The dimensions for the patch with the cavity present are given on Figure 3. The cavity has the following dimensions (see Figure 3): D =d C p W =W +2d C p P L =L +2d C p P where, DC is the depth, WC the width and Lc the length of the cavity. As shown by these results, the effect of the cavity on the performance of the patch is rather minor. This is due to the fact that the dielectric substrate on which the patch is printed is very thin, thus, minimizing the excitation of surface waves. As a result, the addition of the cavity does not disturb the fields excited under the patch. It is expected that thicker substrates will intensify cavity effects. At the present time we try to generate data for thicker substrates where the cavity effects are expected to be more noticeable. Figure 6 shows a comparison between the FDTD and the integral equation part of the hybrid FEM/LE for the case of the patch without the cavity. As we see, there is a discrepancy at the resonance of the antenna which, we believe, is due to insufficient discretization on the patch as applied by the integral equation method. In fact, only one basis has been considered on the patch as suggested by Pozar in the literature. However, the results show some discrepancies which can be eliminated if a better discretization is considered. The hybrid FEM/IE has been applied to the cavity-backed patch problem but the results have not converged adequately to the right values. At the present time, we are in the process of studying the effects of a number of parameters on convergence. Some of these parameters are the finite element meshing in the cavity and the integral equation discretization of the conductors in the outside region. Preliminary results are encouraging and indicative that we should be able to improve convergence and derive accurate solutions during the next few months. While we were trying to analyze the geometry of Figure 3 we were able to make the following interesting observations: 2 Slot Coupled Patch Antennas 2 Slot Coupled Patch Antennas

References FDTD * The finite difference time domain method was found much easier to implement and use. * The method requires a lot of memory in view of the space discretization and the time stepping. With the appropriate use of diakoptics, however, the memory requirements should be minimized as the absorbing boundaries can be brought very close to the antenna and feeding line. * The major advantage of the method is that it can generate, with one run, any desired number of frequency points. This is very important for narrow band antennas where a very dense frequency sampling is required and for wide frequency-band applications where many frequency points are required to cover the desired frequency range. Hybrid FEM * The implementation of the method is rather complex. * The hybrid FEM requires much less memory than the FDTD, but it can only provide one frequency point per run. * In view of the combination between IE and FEM, the technique is more sensitive to meshing and the relative sizes of the sub-elements created by the FEM and Integral Equation. The finite difference time domain programs are available for use and can be transferred to Texas Instruments. As it has been discussed, these programs will bet sent electronically, while at the same time a description on how to access and run the programs will be mailed to TI. Furthermore, we would like to run any other cases which are of interest to TI or for which experimental data may be available. References 1. J-G. Yook, N.I. Dib, and L.P.B. Katehi, "Time and Frequency Domain Characterization of High-Frequency Interconnections," submitted to IEEE Transactions on Microwave Theory and Techniques, special issue on Packaging and Interconnects. 2. N.I. Dib and L.P.B. Katehi, "Analysis of a Transition from a Rectangular Waveguide to a Shielded Dielectric Image Guide Using a Finite Difference Time Domain Method," IEEE Guided Wave Letters, Vol.3, Number 9, September, 1993, pp. 327-330. Slot Coupled Patch Antennas 3 Slot Coupled Patch Antennas 3

References 3. Nihad I. Dib and Linda P.B. Katehi, "Sub-mm Wave Circuit Characterization Using the Finite Difference Time Domain Method," submitted to the Special Issue of the International Journal on Numerical Modelling on The Discrete Time Domain Modeling of Electromagnetic Fields and Networks. 4. N.I. Dib, C-J. Cheng and L.P.B. Katehi, "Characterization of Cavity-Backed Patches Using Frequency and Time-Domain Techniques", submitted for presentation in the 1994 IEEE Antennas and Propagation Society. 4 Slot Coupled Patch Antennas 4 Slot Coupled Patch Antennas

Reforeino FIGURE 1. Slot Coupled Patch Antennas Radiating Patch F." Ground Plane Coupling Slot Feeding Microstrip Line Slot Coupled Patch Antennas 5 Slot Coupled Patch Antennas 5

Reference FIGURE 2. A Microstrip Patch Electromagnetically Coupled to a Coplanar Waveguide Radiating Patch Cavityi - -. p,. 6 Slot Coupled Patch Antennas

References FIGURE 3. Dimensions of the Slot-Coupled Patch. L we Lc 1 II r — LF II \ \, \\ I \ \~ I \, 11 1 I " "fL ---- -A-..i. l.I - -- -----— n1 N — ---- -.....I I\ L:(:.p: - -.,.: i H ------ I I I 1 -I I I — 1 i. -...... u \ \\ ''\, N................... \, K; Nx N\\.............................. 'bs ' iI I I wv C4-l 0- Y -09 - - " -... --- —-- `h\\'....: - -! I II.... — I: i Eb:E: ~: db da i L The transverse slot - CPW. The schematic of the original problem. Lp=7.6mm, Wp= 11.4mm, Wap=0.254mm, Lap=6.91mm, Wf =0.254mm, S=0.762mm, L,=6.858mm, Lt=2.794mm, Wt=0.762mm, XO,=O, Y0,=0, da=0.508mm, db=0.254nmm, Ea=2.2, fb=2.2. Slot Coupled Patch Antennas

01 6 6 6 01 6 0I 6 _0 w r —I0 r-* — 0 w r-I* CD CD CD C) 0 CD C) -n, (D -0 CD G) 7I N r\) 0 0 0 01i 0 CD CA) CD CD pI T I 0 I) I VN.. 'I

CD (0) (D CD Cl) K~) C0 0 N) 0) 0 0 0 0) 0 0 0 0 _0 0 (D CD 0 0 03 0 CD CD -n (D -o c CD G) N 0)CA) 0 -A 0 CD (0 CD, 0 (0 N) N) -0 0 PL) <

References FIGURE 4. Magnitude of Sll for the Geometry of Figure 2 with and without the cavity. Reflection c Defficient patch with cavity 1.20 1.20 -L --- — 0.80 - - 08 ---------- --------- 0.40 --------- ---------- 0.00 --- no cavity with,cavit 30.0 0.0 10.0 20.0 Frequency (GHz) 8 Slot Coupled Patch Antennas 8 Slot Coupled Patch Antennas

Refeno FIGURE 5. Phase of Sll for the Geometry of Figure 2 with and without the cavity. Phase of Reflecti,-1 loefficient patch with cavity i 00. 200. agrees 0. -200. no cavity with cavit 0.0 10.0 20.0 Frequency (GHz) 30.0 Slot Coupled Patch Antennas 9 Slot Coupled Patch Antennas 9

References FIGURE 6. Comparison Between FDTD and IE for the Patch without the Cavity (Yo6=2.698mm). GCPW with Offsetted Patch 1 0.8 0.6 m ", — r,.D 0.4 0.2 0 0 5 10 15 20 25 30 Frequency(GHz) 10 Slot Coupled Patch Antennas 10 Slot Coupled Patch Antennas

APPENDIX A: Finite Difference Time Domain Method APPENDIX A: Finite Difference Time Domain Method Slot Coupled Patch Antennas 11 Slot Coupled Patch Antennas 11

1 FDTD tecliicliie '.. 11'..).ell'. I t l l a' d oi ( ',' ','") I t't('l i. S 'pa a d ii,' 1' ' I t in' a:;l 1 ' 11n '1( lt) Si uI late ( ie pi())C,(1',;liuI f aII i ii i ('x cit liIll il a I 1 )flr) Iill lt'I. ]( 't". i (' Ito.1('1t ()d IIw iS bee)II suc( c'sfullly a)pp:ited to ('i<'a(t ri(' Ii I-Io(stri ) aTl (c )ol II 'g it' ( \\ i li:(s a (l dliscoil inll iit is [3]-[7'|. optical ilI,. at('(1 ('ir( uits ] ('II. TII(-\V\' ' '11 and slII 11a v / \( ' (i'elect( ric line transitions [1..9] Only a briel; s:mimna:ry of t 1lF 1I)'I ) met ho( is d(scri)t d:(( I ( In or((er to c}laracterize any planar struct lrc. prop)agaltion: of a specific tinel-dl I)epend(l It firlc 1 il l tlhroiiugh tli} structure is simulated using the 1FI1)lD tec}lique. The time (deI(cp)ndncC of thl (xcitatilon can l)e chosen arbitrarily: however, a Gaussian Iulse is often usI(d because it is solot lyI! varl.'igll in time and its Fourier transform is also a Gaussian function centered at zero freIquency. Following thle time and space discretizations of the electric and magnetic field components. tlie( F1D)TI) equivalents of Maxwell's equations are then u1Sed to uptdate the spatial distributions of t lhiese com)ponents at alternating half time steps [10]. Tlhe space steps, A:, Ay and A-, are carehflly chosen such that integral numbers of them can approxinate the various dimensions of the struct ur.e. As a rule of thumb and in order to reduce the truncation and grid dispersion errors, the maximnu:m: step size is chosen to be less than 1/20 of the smallest wavelength existing in the computational domain (i.e., at the highest frequency represented in the pulse). Then, the Courant stability criterion is used to select the time step to insure numerical stability. In order to excite the patch antenna, the vertical electric field component at the front plane (z=0) is excited and the magnetic wall source condition of [4] is used to compute the fields elsewhere in the plane z=0. For the electric field components lying on a dielectric-dielectric interface, the average between the two permittivities is used in the FDTD equations [3]. The super-absorbing first-order Mur boundary condition [11, 12] is utilized to terminate the FDTD lattice at the front (z=0) and back (z-=N Az, where NZ is the number of cells in the z-direction) planes in order to simulate infinite lines. This absorbing boundary condition requires a choice for the incident velocity of the waves, or equivalently e,ff. It has been found that an appropriate choice of Er,eff minimizes the effect of the absorbing boundary walls. On the other hand, the first-order Mur boundary condition is used on the top and side walls to simulate an open structure. In general, the frequency dependent scattering parameters, Sij, can be obtained as follows [3, 4]: SiJaP)= T-'( ) (1) Vj(w) ZZo where Vi and Vj are the voltages at ports i and j, respectively, and Zoj and Zoi are the characteristic (or wave) impedances of the lines connected to these ports. To obtain S11(w), the incident and reflected fields must be known. Since the FDTD simulation calculates the total field (i.e., the sum of the incident and reflected waveforms), the incident field is obtained from that of an infinite extent line (i.e, from the source to far absorbing wall). Then, this incident field is subtracted from the total waveform to yield the reflected field. 1

References 1 i A.: ii]. EN. N Dil) aITd L. Ea 'ti;i. "( i':I-.", i.',' a,: i,' ':,H i'I '.':' i i <(, ) '. " '' i it \\aveT: (Ie." accepted for p) (.calloll ll tI l /1.'II:- /.,..l/ /. [2] 1K. S. 'Yee (, Numerical soluti ' of iilial 'e i,111(d;varv' \ '(. Ie p-I1),.': iIvoIl (ving o.:x.{'l!' tis ' in is(,ropic media." IEE1E f rf..:l1. '1 C(2- '37..)J\ l i(t;. [3] X. Zhang and K. Mlei, "l'im e-clollliai! iiiiite (tiificreI(-c api)roa('l to t11 (I li'il&l< jon of tli(c frequency-dependent characteristics of rilicrostritp disconltiltiiles." I-'E1. 7..1l. tt. 1775-1787. Dec. 1988. [4] D. Sheen, S. Ali, M. Abouzahra aiid J. Kong, '"Application of the Tlhree-I)iInlcisional IinitcDifference Time-Domain Method to the Analysis of Planar M:\icrostrip Circuits. "IEEE 7Tn,,s. MTT, pp. 849-857, July 1990. [5] L. Wu and H. Chang, "Analysis of dispersion and series gap (liscontinuity inI shiclded suspeinded striplines with substrate mounting grooves, " IEEE Trans. AITT. pp. 279-281. Fel. 1992. [6] T. Shibata and H. Kimura, "Computer-Aided Engineering for Microwave and Millilmeter-W\ave Circuits Using the FD-TD Technique of Field Simulations," It. J. of Microwave and.i:llimeterWave Computer-Aided Engineering, Vol. 3, No. 3, pp. 238-250, 1993. [7] S. Visan, 0. Picon and V. Hanna, "3D Characterization of Air Bridges and Via Holes in Conductor-Backed Coplanar Waveguides for MMIC Applications. " 1993 IEEE MTT-S Intl. Microwave Symp. Dig., pp. 709-712. [8] S. Chu, W. Huang and S. Chaudhuri, "Simulation and Analysis of Waveguide Based Optical Integrated Circuits, " Computer Physics Communications, vol. 68, pp. 451-484, 1991. [9] N. Dib and L. Katehi, "Analysis of the Transition from Rectangular Waveguide to Shielded Dielectric Image Guide Using the Finite-Difference Time-Domain Method, " IEEE Microwave and Guided Wave Letters, pp. 327-329, Sep. 1993. [10] K. Kunz and R. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, Florida: CRC press, 1993. [11] G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the timedomain electromagnetic-field equations," IEEE Trans. EMC, pp. 377-382, Nov. 1981. [12] K. Mei and J. Fang, "Superabsorbtion-A method to improve absorbing boundary conditions," IEEE Trans. AP, pp. 1001-1010, Sep. 1992. 2

APPENDIX B: Finite Elemen/ntntegral Equation Method APPENDIX B: Finite Element/Integral Equation Method 12

Analysis of a Slot Coupled Coplanar Waveguide Fed Patch Antenna Jui-Ching Cheng November 29, 1993 1 Introduction Fig. 1 shows a patch antenna that is fed by an open-ended coplanar waveguide(CPW) through a slot on the ground plane between the patch and the coplanar waveguide. A transverse slot on the CPW is used to increase the coupling efficiency. Full wave analysis and moment method are used in the analysis of this structure. Similar structures have been fully analyzed by the same method[1][3][4]. This analysis may be further extended to combine finite element method for more complex structures. As shown in Fig. 2, a patch is put on top of a cavity. If the shape of the cavity is complicated or the media inside the cavity is not simple, i.e., inhomogeneous or anisotropic, it is difficult to apply moment method in the cavity. Thus, the combination of finite element method and moment method is necessary to analyze this structure. Section 2 will give a rigorous analysis of the problem. Section 3 includes the finite element method in the analysis of the cavity-backed patch antenna. Section 4 shows the preliminary numerical results. Details of finite element method formulation and Green's functions are given in appendix. 2 Theoretical Analysis The schematic of the antenna and feed line is shown in Fig. 3. The ground plane and dielectric substrates extend to infinity in the x and y directions. 1

M.rco.qt r 1 Pat ci / i:::::':::::.::::::::::::::::::: / Aperture on ground plane / /: / Transverse slot on CPW CPW feed line Figure 1: The structure of a slot coupled and CPW fed patch antenna. By using equivalence principle, the original problem may be changed to an equivalent one which is shown in Fig 4. Jp denotes the induced current on the patch. The slot and the coplanar waveguide are closed by conductor such that the original structure is separated to 3 regions. In order to keep the same field distribution as the original one, magnetic surface currents Map must be added on the surface of the slot and also Mi,, the equivalent incident traveling magnetic current mode on the CPW, and Mf the induced equivalent magnetic current on the CPW due to the field scattered by this structure. Furthermore, Map satisfy Map = x Eap (1) Let region a denote the lower dielectric slab, region b denote the upper dielectric slab and region c denote the free space below the CPW. The total 2

aec io: b 7 ConaucLor Region d CPW line(not shown) Figure 2: The structure of a slot coupled and cavity backed patch antenna. electric and magnetic fields can be represented by superposition of fields due to the various currents as following equations: E't = Ea (Minc) + Ea(Mf) + Ea(Ma) (2) Htot = Ha(Minc) + Ha(Mf) + Ha(Map) (3) Et~t = Eb(Jp)- Eb(Map) (4) Ht = Hb(Jp) - Hb(Map) (5) Ett =-E(Minc) - Ec(Mf7) (6) HCt = -Hc(Minc) - HMf). (7) Each field in equations (2)-(7) can be represented by dyadic Green's function for each structure such as Eb (Map) = JJ GEM (x y, yzo, Yo, o,) Map do dyo, (8) slot where G'jM is the electric field at (x, y, z) due to an infinitesimal magnetic current at (xo, Yo, zo) radiating in the presence of a grounded dielectric slab. This and other Green's functions needed for the analysis are obtained by 3

using spectral domain methods so th1iat C;EA. ('. y, z o y o ^ o ) = JJ QEM (k 1 k; z'.,o) * e(iJr(- )(,Jk (yy-) IA dl'. -OC (9) Three coupled integral equations are obtained for the three unknown currents AMf, Jp, Map by enforcing the boundary conditions: 1) Hta, is continuous on the CPW, 2) Htan is continuous through the slot, and 3) Et'L7 = 0 on the patch. That is, Eb(Jp) - Eb(Map) = 0 on the patch (10) Ha(Minc) + Ha(Mf) = -Hc(Mnc)- Hc(Mf) on the CPW (11) Ha(Minc) + Ha(Mf) + Ha(Map) = Hb(Jp) - Hb(Map) on the slot. (12) Galerkin moment method is used to obtain the integral equations linking the unknown currents in region a and b. The unknown currents are formulated by choosing expansion functions as follows: Nb Jp(x, y)= E JbJ (x, y) (13) n=l Nap Map (x, y) = M nPM (x, y) (14) n=l N1 N2 Mf (x, y) = E MlM (x, y) + E M2M22 (x, y) + RMf (x, y), (15) n=l n=l where Mref is the reflected traveling current mode on the CPW, M1 is the high order modes near the end of the CPW, Ml2 is the high order modes on the transverse slot on the CPW. Define an inner product (FG)S J F Gds. (16) S The three boundary conditions lead to [Zb][Ib] + [Tb][Ma] 0 (17) [Cb][Ib] + ([yb] + [ya])[Map] + [yapl][M1] + [Yap,2][M2] 4

+~ ~~0R(Y~T~] ~In. -( -'Y + 7i,I!, O]['] iI;(,K!"LlSj~ '~] ~-[YPI'~"[AraP] - -~I i([1~' 1c - j{~~lS] [Y2-41] [AJ] + [)1'22~] [AJ2] + I~n R [y~2c] + j [Y.2S]) t[y2,ap][fl[Aap] - - j[y2c y s )I (IS) (20) The last two equations come from the Ioundarv condition 2. and vector elements are defined as follows: The matrix ya, = (_M~p,Ha(IM$ar))ap Ymn (M=,Ha(MRap))ap = a (MaPI Ha (Mn)a ya = ( bMaP, Ha(MC))ap yap~= (_MaP, H(MS)),p = (M4m (Ha + Hc)(1Mtf Ymn Afan H fll))a (,2m1 (Ha + H )(2M yP = ( IMH(Ma))ap Ymn a (Mn Ymn = (-M7a, Ha+MHc)())) ) =- (Mm, (Ha + H,)(M7) = (-Mt, (Ha + Hc)(M())j ymn = (-m2H(MaP))1nl y = (-mi (Ha + Hc)(Mn))f =n (Mt (Ha + H)(M) Ymn lam= (M,-+,Hb(tn))a Ymn (-Mmii (Ha + Hc)(Msf 2,1m H,(1 Ymn km i (Ra + c) (Rnl)) 212 MM2,MN M2ap (- _MM2_tl -,aM 2, s 2 M2 ~~~,M t b E MnaP)) p a iV,,,, x N,, N,, x N N,,p x N, +px V Nj+l N2 N x N2, Na +lxl Nl+l xl N2 ~ N, N2 ~ N2 N2 + Nap N2 xN N2 xN Nb x Nb Nap X Nap Nap X Nb Nb X 1 N1 x 1 N2 x 1 Np xl matrix matrix matrix matrix vector vector matrix matrix matrix vector vector matrix matrix matrix vector vector matrix matrix matrix column column column column (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) vector vector vector vector 5

(-4-1 ).Note that on the ('P\V extra higher order current mode is used as a test fiiunctionI to avoid using traveling current mode such that comlputationI coInlllexitv is reduced. From reciprocity, the following relations exist. yap, = yla (45) 21 12 (46) Recast some of the quantities above into new forms: [Mtot] = M2 (47) yl,1 v1,2 (48) [y12]= y2,1 y2,2 (48) [yI2,c]= ["C (49) y,c] yl,s [yl2Xs] = [2,] (50) yl,ap (5) [yl2,ap] y2,ap (51) [ytot] = [yl21yl2,c + jyl2,s] (52) [yap,12] [yap,llyap,2] (53) [yap] = [ya + yb] (54) [yap,tot] = [yap,121yap,c + jyap,s] (55) [Vinc] -Vinc([y12,c] - j[y12,s]) (56) [Minc] = -Vin([ya"p,] - j[yaps]). (57) Substituting the above equations to (17)-(20) and rearranging yield [yl2,tot[Mtot] + [yl2,ap][Map] [Vinc] (58) [Zb][Ib] + [Tb][Map] = 0 (59) [Cb][Ib] + [ya][Map] + [yaPtt][Mtot] =[Mnc]. (60) 6

Solving (5S)-(60) simultaneously. w\( goet [/i] - - [Z] [T - ] [r[ ] ( i 1 [:,llto t] = [is 12,tot]-1 l([Iinc] _ [}"12.'a][ ICIP]) t(i'2) [A p] ([ap] - [C]a[Zb]1 [Tb] - [y.1aptot][- '2.tot]-1 [.-12,ap] )-I ([,1i7lC] - [yapot] [,l12,tot] -1 [',,IlC] ) ((3) Such the unknown quantities [AlPP] [Altot] and [P6] are solved in termns of matrix equations (61)-(63). Assume the CPW is very narrow such that only y-directed magnetic currents exist, i.e. Minc, Mref and M1 only have y component. Also assume the slots on ground plane and CPW are small such that only x directed magnetic currents exist. the current modes on patch are the same as that in [1] and are not shown here. The incident and reflected currents are represented by traveling current mode which is the fundamental CPW mode. The traveling current mode is further separated to sine and cosine part as follows Minc(x,y) = Y(MC(x,y)-jM(x, y)) (64) Mref(x, y) = yR(MC(x, y) + jMs(x, y)), (65) where fx f(x) cos ka(y- L), for - mA, e- a < y - L < -A Mv (x, y) - 66) { 0, elsewhere 66) Ms (x y) { fx(x)sin kea(y-L), for -mAe < y-L <0 (67) MS1x'y) ~ o 0elsewhere (67) [ 1 for < < + W -1 f (x) = for - > > -W (68) j 0, elsewhere. kea is the propagation constant of the fundamental mode of CPW. It is derived by the method of [5]. \e is the equivalent wavelength. m is a parameter to be chosen. The Fourier transforms of Mc and Ms are f -L) jkymAe (69) M- = f~kcka Le k2 - (69) e 7

and aIS I kd 2 /2 C J 77A., _ 1 wfhs = s te f-rjkyLr _t where f, is the fourier transformn of fi and (70) - 2j cos k( s + 1f) - cos. f 4 J z - =k, (-1) Most of the basis functions are represented by piecewise sinusoidal(PW\S) modes(see Fig 5) as follows. For M1 not on the boundary of the slot, -,f k h -Iy-yn k ) for I-i <hi, — ~ sinkh ' - i= { xf(x) sik(h-ly-yh ) for ly - Y. < h 0, elsewhere and k is chosen to be (kea + k0)/2, h is half of the width of the The fourier transform of M,/ is (72) PWS mode. - 1k k) - ( 2ke-jyt* " cos kyh - cos kh ' sin kh k2 - k2 (73) For M, on the boundary of the slot, Af (x) sin k(hi -ly-y.\) x sin khl,1 - f(x), sink(h2-Y-yn.) O, for n - hl < y < Yn for yn < Y < Yn + h2 elsewhere (74) and Mj (k k ) fx(kx)ejkyyn ekyhl - kcoskhl -jkysinkhi L' sink hl(k2 - k)2) ke-j'h2 - k cos kh2 + jky sin kh2 sin kh2(k2- - ) J For M2 not on the boundary of the CPW, (75) ^ sink(h-lx-xn|I) M2 XJ y y) sin kh n \0, for Ix ~ xnI < h elsewhere (76) 8

wbere = " k is chiosen To be (ks, + ko)/2 and for I7 —<y< I V# elsewhere. AI,(, 2~ f,(ky)2 cos k~x,, *kcs -csL sin kh k2 k where f, is the fourier transforma of fl, and sinkY 2 Fo 2~on the boundary of the CPW, (78) ( 79) Mn, (kx Iky) fv(~ [2k cos(k~hi - k~x,) - 2k cos, kh, cos k,,x, - 2kx sin kh1 sin k~xx [ ky sin khl(k2 - k2 ) 2k cos(kxh2 + k~xx) - 2k cos kh2cos k~xx + 2kx sin kh2 (80)xn + ~~sin kh2(k2- k2 ) sink8x ) For Map Ma~ ~fsin k(h - Ix - x, J) Mn= A() sin kh k is chosen as that of [1], and f~k)2kejkxnco kxh - Cos kh Mn ~~sin kh k (81) (82) Since there are poles in the Green's functions. Pole extraction method [3] is used. The integrand always takes the form [2r [00 f (flI 0) /d3 ~Joko ksin k~d (83) w here k, = kT ~-# 3Rewrite I as = 2ir j0o [k- Grf,b22 /f3 0 d3 + j21 f Jo Gr0,i)2k2 f (84)) where GTr is the residue of 1 /(k, sin k, d) at ~3=k which is equal to - IJ/2kd. The third term of I can be evaluated analytically, which is equal to f02j f (k, q$)G r(-j-rk)dq5. 9

3 Combination of Finite Element Method and Moment method The schematic of the ca-vity-ba.cked patch antenna and feed line is shownI inl Fig. 6 with impressed and induced currents indicated. Following the samie notation and procedure as the previous section, the boundary conditions call be expressed as Ha (Minc) + Ha(Mlf) + Ha(Map) = -Hd(Ml,,) - Hd(Alf) on the CPW (85) Ha(Mlinc) + Ha(Mf) + Ha(Ml]ap) = Hc(Mlp) - Hc(Mlap) on the slot (86) -Hb(Mp) H ) - H(Mp)- (Map) on cavity opening, (87) where subscripts a,b,c and d denote the regions shown in Fig 2, and Mi,,,, Mf, Map and Mp are the equivalent sources shown in Fig 7. The fields in the cavity are obtained by using finite element method, which is shown in appendix. Greens function technique is used to obtain the fields in other regions. Since the structures of Fig 1 and Fig 2 only differ in the cavity and patch parts, only small modification is necessary to the formulations in previous section. Thus, the following matrix equation is obtain. [yb][M] = - [y][Mp] + [Yp,ap[Map] (88) [YaP][MaP] + [ya][MaP] - [YaPP][Mp] + [yapl][M1] + [YaP2][M2] +VincR([Yap,C] + j[yap,s]) = -Vc([YapC] - j[yap,s]) (89) [Y1'][iM1] + [Y1,2][M2] + VincR([Y11c] + j[Y1S]) +[ylIaP][MaP] = -Vi([Ylc] - j[Y1']) (90) [y2,1][M1] + [y2,2][M2] + Vinc R([y2,C] + j[y2,]) +[y2,ap][Map] _V([y2.c] - j[Y2,s]). (91) The definition of the matrices and vectors which are different from previous section are shown as follows: Ymn (-Mm, Hb(MP))p Np x Np matrix (92) Ymn(-Mm, THc(MnP))p Np x Np matrix (93) Yan - (-MaP, Hc(MnaP))ap Nap x Nap matrix (94) 10

~1717' = ( —,l. ' r; \ a> ~ mtrix 9 I, Al. =H- ( H.a ) ), \', x \', atrix. ( E(luations (93)-(96) are calculated by finite element mIetlhod wIichl is shownI in ap)pendix. Recast some of the quantities above into new forms: [Iyb to [yb +yP] P p X p matrix (97) [ot] = [ya + yap] Nap x NIap matrix. (98) Substituting the above equations to (88)-(91) and rearranging yield [Ytb ][Mp] = [YPS][MS] (99) [y'l2tot][Mt~t] + [Yl2'aP][Map] = [Vin] (100) [yot][M P]- [yapp][Ma] + [yaptot[tot] [Mi] (101) Solving (99)-(101) simultaneously, we get [Mp] = [Yt11t]-[YP ][MS] (102) [Mtt] [yl2,tot]-l([inc] - [yl2,aP][Map]) (103) [Map] = ([yt]j - [yaP][Yotb]-1 [ya] - [yap^tot][yl2.tot]-1[y12,ap])-1 ([M nc] - [yaptot] [y12,tot]-1[vinc]). (104) Such the unknown quantities are solved. 4 Numerical Results Since the CPW line is very narrow, only y-directed magnetic currents are used to model the currents on it. Also assume the aperture on the ground plane and the transverse slot on the CPW are narrow enough that only xdirected magnetic currents are necessary to model the currents on them. The currents on the patch are assumed to be only y-directed too. The incident and reflected currents are represented by traveling wave modes which are the fundamental CPW modes. It is further separated to sine and cosine parts as in [1]. Since infinite traveling wave modes produce extra poles in spectral domain, it is truncated at sufficient large distance from the end of the CPW. 11

The cosine part of the traveling waV Modeo is end(led at a Iquarter wavlien'lt h fromi the eind of the CP\\ to avoid aI not zero currenIIt at tlie eind of CPW\\. The prop)agation constant of the fundamental miode of CP\NV is calculateId by) tlhe method of [5]. Althlough there is a small imaginary part in the propagation constant, it is not taken into account because it is much smaller than the real part. Typically, the ratio is smaller than 1/100. The dimension of the structure is the same as [6] as shown in Fig. 3. All the basis functions used are piecewise sinusoidal(PXWNS) modes. Fig. 8 shows the variation of the amplitude of S5ln v.s. the number of basis functions used on CPW. Good convergence is reached with 7 basis functions. Fig. 9 shows the amplitude of Sll v.s. frequency. After the correctness of the analysis of section 2 is confirmed by comparison of the numerical results with other methods, the finite element method part will be included in the programs to analyze the cavity-backed patch antenna. 12

Appendix A Finite Element Formulations In the cavity, tie magnetic field satisfies JWlc V x (- V x H) = jo'cH. (10') Let (i denotes the test functions. Integrating the test functions with equation (105) in the whole volume of the cavity leads to JJ[v x V x Hc)] i dv - (jw cHc i) dv = 0 (106) VI Jjcc V_ Integration by parts the first term in (106), we obtain: JJ( --- x Hc) V x q dv -JJ(jw Hc H ) dv Vc Jwc VI = JJ(i x E~).ds = - JJM O ds, (107) S S where S is the surface of the cavity. Because the cavity is surrounded by conductor except the slot and the opening around the patch, MC is Mp or -Map. Let Hc represented by basis functions 4i, N, HC E ii. (108) i=l Then the following equations is obtained Hc(Map) [4][S][MaP] (109) H,(Mp) [4][P] [MP]. (110) Where [FD] = [41D243"-.] is a row vector, [P] and [S] are No x Np and AD x Ns matrices respectively. Column j of [S] and [P] are the finite element solutions 13

of equation (107) due to ilJ ali.\' rIes)ectively. Now xc caln expes [VS]. [ '"]. ["YsP] and [) PS] as follows: [Is] = [TS 1[i;] [YP] = [~P][P] [y' p] [=s][p] [Yps] = [4TP][S]. (111) (112) (11:3) (114) (115) (116) whrere n)L = (-M1n ~1)S Pz = M(-AP, 4n)p NA, x N ' ANp x AN matrix matrix. B Green's Functions The spectral domain kernels that are used for the Green's functions of the analysis are presented below. The following definitions are used in the expressions: k = W2 o0E (1 kla = (ak2 - f2)1/2 Im{kla} < 0, Re{kla} > 0 (1 klb = (ek2 - 2)1/2 Im{klb} < 0, Re{klb} > 0 (1 k2= (k2 - 2)1/2, Im{k2} < O, Re{k2} > 0 (1 f2 = k2 + k2 (1 Te = klb cos(klbdb) + jk2 sin(klbdb) (1 Tb= E k2 cos(klbdb) + jklbsin(klbdb) (1 Zo (o/6o)1/2 (1 The required kernel functions are listed below. For GJyy(x, y, dblxo,yo, db):.Zo QEJyy(k ky) = -j3 (ec2 - kc)k2 cos(klbdb) + j(kO - k)klb sin(klbdb) TbTbm x sin(klbdb) (1 17) 18) 19) 120) 121) 22) 23).24).25) 14

ForI (GHJ2-(.(x!. y xO. yo. db ): 1 -_(:lb k2 cos(/'1 bdi, ) + J(A( ( - I) - f, )J SiIl (1 ( QIIJXY(/ r k,%) = - 4, 2 12(i) For GHAlx(x. y. rxo, Yo, 0): QHMx r kxy) -j 1 " 2 4w72kZk klbTT * [ik 1b(4 - 1) + (cbk - k-) 47rX l2(o + 1T s(T ) l) xl{k bk2 (kC. + 1) sin(klbdb) cos(kbdb ) +j( k2 sin2(klbdb) - k cos (k'lbdb))}] l 21bb (127) For GEMy( y, dbxo,y O,0) QEMyx(kx ky)(kx, ky) = QHJ (x, ky) For GHM(x,y,0 IXOxo, y,0): (128) Q HMxxkx k) = J ~ (ako - k2) COs(klada ) kla sin(klada) (129) For GaMYY( x, y, 0xo, yo,0): Q HMyy (x ky) (cak2 - k2) For GaHMX(x, y, dlxo, yo, 0): cos(klada) kia sin(kiada) 1 kla sin(klada) (130) Qa Mx(kxj k) = * (ako - k) (131) For GHMXY(x Iy,| 0lxo, yo, 0): QMxyv(kx, ky) - a o~ (k2 ky) COS(klada) kla sin(kiadda) (132) For GaMyx(x,y, 0xoyo, d): QMyx(kx, ky) - -J-weo 47r2k2 * (kkky). d kla sin(kiada) (133) 15

For CJGAi (r. yJ.O o..o, 0): H l - LJ~n 1 QHAMyrT(k, k) = 4ir2k (L':) - (1:34) QHAISS. = a;2 (lo ri) ( li)) For GHA~(.T, y,|Oox. yo. 0): QHMx(kk) = (- 4k-) - (135) For G'MY(x, y, 01xo yo, Y0): Q.MYY(k,Y) = (k2 - k2) (136) Because of reciprocity and symmetry, the following equations also hold. GMAr(x, Y, dlxo, yo, d) = GM (a(x, y,0 Ixo, Yo, 0) (137) GHMyy(,, dlxo, YO, d) = GHMyy(, y, OXOY, O ) (138) GHMX(, y, dlxo, yo, 0) = GHMX(X, y, OIX Yo, d) (139) GHM (x, Y, dlxo, YO, 0) = GHMys(x, y, xo, Yo,d) (140) GHMXy(x, Y, xo, o, 0) = GMYx(x, Y, 0Ixo, YO, O) (141) GHMxy(X, Y, 0xo, yo, 0) )= GH MYx(X,, 0, Yo, 0) (142) References [1] P.L. Sullivan and D.H. Schaubert, "Analysis of an Aperture Coupled Microstrip Antenna," IEEE Trans. Antennas Propagat., vol. AP-34, pp. 977-984, Aug. 1986. [2] D. M. Pozar, "Input impedance and mutual coupling of rectangular microstrip antennas," IEEE Trans. Antennas Propagat., vol. AP-30, pp. 1191-1196, Nov. 1982. [3] N.K. Das and D.M. Pozar, "Multiport Scattering Analysis of General Multilayered Printed Antennas Fed by Multiple Feed Ports: Part I Theory," IEEE Trans. Antennas Progagat., vol. AP-40, pp. 469-481, May 1992. 16

[-1] N.I. )as aIl( D.M. Pozar. "MiIlltiport ScatteriIn Anialysis of (;General Multilayered Printed Antennas Fed by NMultitple Feed Ports: Part II -- Applicatiols," IEEE Trans. Anten2nas Progagat.. vol. AP-40. pp. 482 -490. iMay 1992. [5] R.\. Jackson and D.M. Pozar, "Full-wave analysis of microstrip openend and gap discontinuities," IEEE Trans. Alicrouwave Theory Tech., vol. MTT-33, pp. 1036-1042, Oct. 1985. [6] R.N. Simons and R.Q. Lee, "Coplanar waveguide aperture coupled patcl antennas with ground plane/substrate of finite extent." 17

- 'F " -I eb Sa The transverse slot L CPW Wt Figure 3: The schematic of the original problem. Figure 3: The schematic of the original problem. db da 18

Reg i on Region II Region III jp 00 ~aa M Mnc + Mf -.-M -M - M Inc f Y Free space Figure 4: The equivalent problem after closing the slot by conductor. Suitable magnetic currents are added to ensure the equivalence. Yn-1 Y Boundary of transvers slot //xyn hi h2 - -a W2 h PWS modes on CPW h h hi h n L2 Wf S Wf L2 PWS modes on the transvers slot of CPW Figure 5: The PWS modes on the CPW. x 19

Region b Region c Region a C b Ep E ~ a Minc + M f I — II Y Region d Figure 6: The schematic of the original problem... 1z Region b Region c Region a Region d -Mp -Mp Ec -Map -0 - E a Map Minc + Mf -Minc - Mf y Figure 7: The equivalent problem after closing the opening above the cavity and slot by conductor. Suitable magnetic currents are added to ensure the equivalence. 20

1 I I I I If=3.Nw Nf=3,Nw=2,Ns=4 --- I I I I I I Nf=6,Nw —3,Ns=t, OF 0.8 Nf=3,Nw=l,Ns=4 Nf=3,Nw=O,Ns-4 Nf=6.Nw= Ns=8 Nf=9,Nw=3.Ns=12. 0 -. 0.6 H 0.4 - 0.2 k 0 I I I I I I I I I I 6 8 10 12 14 16 18 Number of basis functions 20 22 24 1 2 3 4....Nf-1 Nf 1 2 3.....Nw 1 2 3 4..... Ns-1 Ns CPW Figure 8: The amplitude of ISl| v.s. number of basis functions on the CPW. Nf, N, and Ns are as shown in figure. 5 PWS modes are used on the patch, 3 on the transverse slot, 5 on the aperture. The PWS modes on the CPW extends from the end to 3/4 wavelength. The traveling wave mode extends 3 wavelengths. 21

1 I I I 0.8 0.6 - C, 0.4 - 0.2 0 I I I I, 10 11 12 13 Frequency(GHz) 14 15 Figure 9: The amplitude of IS,,I v.s. frequency. Nf, Ns and 4 respectively. Other parameters are the same as Fig. 8. Ns are 3, 0 and 22