Contribution to the cavity model for analysis of microstrip patch antennas

Transcription

1 JOURNAL OF OPTOELECTRONICS AND ADVANCED MATERIALS Vol. 8, No. 1, Febauy 006, p Contibution to the cavity model fo analysis of micostip patch antennas D. D. SANDU *, O. G. AVADANEI, A. IOACHIM a, D. IONESI b Faculty of Physics, Univesity Al. I. Cuza of Ia i, Romania a National Inst. of Mateial Physics, P. O. Box MG 7, Buchaest,Romania B FERA Bac u, Romania In this communication we popose a futhe impovement of the cavity model fo micostip antennas, given by Richads et al. [1]. The main attention is paid to the calculation of the adiated fields E θ, E ϕ and to the vaiation of the input impedance as a function of the fequency and the feed point. In ou model we have suppose that the magnetic walls ae not pefect and the patch acts as a cavity with the length coesponding to its esonance fequency. The validity of the poposed model was poved by compaison of theoetical computations and expeimental esults. Received Septembe 13, 005; accepted Januay 6, 006 Keywods: Micowaves, Micostip, Patch antennas, Impedance, Radiation patten 1. Intoduction Duing the design pocedue of patch antennas thee ae two impotant featues: the calculation of the adiation patten fo a single patch and fo an aay; detemination of the input impedance that can assue a good matching of the feed line with the patch. In this aspect wee poposed many models. In this communication we deal mainly with the impoved cavity model given by W. F. Richads et al. [1], which is elated to the classical cavity model developed by Y. T. Lo et al. []. Besides these models othe appoaches ae known in the liteatue; among these we can mention: the vecto potential appoach [3], the dyadic Geen s function model [4], the wie gid model [5], and the tansmission line model [6,7]. In Fig. 1 we pesent a ectangula patch of width a and length b ove a gound plane with a dielectic substate of thickness h and the elative pemittivity ε. Inside this cavity the z-diected electic field satisfy the equation + kp Ez Je ˆz whee is the laplacian and J e is the excitation cuent. The solution of the homogeneous wave equation fo TM p mode is given by [8]: 1 Ez E0 kmx kn y k pz whee E 0 is an amplitude coefficient depending on the excitation condition; the eigenvalues satisfy the equation. Geneal elations fo the classical cavity model k p p km + kn + k p 3 The Classical Cavity Model [] is based on the following consideations: a The close poximity between the micostip patch and the gound plane suggest that E has only z component and H has only xy-components in the egion bounded by the micostip and gound plane. b The fields ae independent of the z-coodinate fo all fequencies of inteest. c The electic cuent in the micostip must have no component nomal to the edge at any point on the edge, implying a negligible tangent component of the H along the edge. Thus the egion between the patch and gound plane may be teated as a cavity bounded by electic walls above and below, and magnetic walls along the edges. The fields inside the antenna ae assumed to be the fields inside of this cavity. whee fo a non adiating cavity k m m / a, k n n / b, k p p / h ; 4 is the pemittivity of the substate and its pemeability. Tacking into account the small thickness of the substate, p must be zeo fo usually fequencies and in this case E x, E y, ae zeo and the magnetic fields components ae: H x kn E kmx sin kn y k pz 0 k p 5 H y km E sin kmx kn y k pz 0 k p

2 # 340 D. D. Sandu, O. G. Avadanei, A. Ioachim, D. Ionesi Fig. 1. The epesentation of adiating cuents. The cavity model assumes that the field stuctue in the patch antenna is essentially the same as that in the cavity. With those fields we calculate the cuents geneated by the field E z, on the side walls: J E zˆ xˆ E n y / b y fo x 0 and x a 6 m y z 0 ˆ Jmx Ezˆz ˆy ± E0 m x / a xˆ fo y 0 and b whee xˆ, yˆ, ˆz ae unity vectos. In this model only the electic fields geneates adiating cuents because the magnetic fields ae zeo at the lateal walls. Fo the oscillation mode TM 100 the cuents J my on the walls, x0 and xa, ae adiating only; the othe two J mx cuents on the walls y0 and yb ae nonadiating. The input conductance is give by the following elation [] P P G + d 7 V whee P is the adiated powe, P d is the powe lost in dielectic, and V is the voltage at the feeding point. 3. Futhe impovements of the cavity model Fo the eal case of a patch excited by a coaxial feed line the field is a supeposition of all TM modes and theefoe the z-diected electic field will be E z x, y A e x, y 8 m n whee A ae the mode amplitude coefficient and e ae the z-diected othonomalized electic field mode vectos. Fo the elementay case of a nonadiating cavity with pefect open-cicuits walls, we have [9] with e x, y kmx kn y 9 abh 1, m 0 m 0 m 0 and o and n 0 n 0 n 0 If the excitation cuent J 0 is a z-diected cuent pobe I 0 of small ectangula coss-section d x, d y at the point x 0, y 0 and zeo elsewhee the mode amplitude coefficient is h k A I0 0 0 G k x k y 10 ab m n k k whee G m d / a sin n d y / b sin x m d x / a n d y / b k is the wave numbe and k is the k p fo p0. The facto G accounts fo the width of the feed, dx, dy which, fo a coaxial line feeding is five times geate than the physical dimension of the excitation cable [1]. Substituting 10 into 9 we obtain Ez I 0 x, y x0 Z0 k G m 0 n 0 k k whee Z 0 /, k and 11 kmx kn y 1 ab Tacking into account that the voltage at the feed point is V in he z x 0, y 0 Richads et al. [1] poposed fo the input impedance the elation V x0 Z in in Z0 kh G 13 I0 m 0 n 0 k k But accoding 13 with k eal the impedance would be puely imaginay, and that is in contadiction with expeiment. To solve this poblem Richads et al. lump all losses into a single effective dielectic loss with effective tangent loss δ eff. In this case the wavenumbe k would be eplaced by an effective wavenumbe keff 1 eff k0. In an ideal cavity δ1/q, whee Q is the quality! facto, theefoe eff P l / Wc 14 whee P l P +P d +P m ae the total losses, P is the powe lost by adiation, P d is the powe lost in dielectic, P m the powe lost in metallic walls and W c is the time-aveaged electic enegy stoed by cavity. The total adiated powe is [5]: $%$ '' / P Re E H E H sin" d d" Besides these impovements we have tied to futhe develop the cavity model. One of the most impotant dawback of the cavity model was the fact that fist it

3 ? D H 10 :98 NML ' /. > 5 76 C B KJ I > ' B Contibution to the cavity model fo analysis of micostip patch antennas 341 suppose the cavity pefect, and calculate the fields inside it, and then with this fields we calculate the adiation cuents at the sides walls. In ou model we have suppose that the fields extend outside the cavity, that the magnetic walls ae not pefect and the magnetic field amplitude along the x axis has the vaiation as in Fig.. The esonant fequency of a patch with the length a is the same that of a cavity with the length a+dl, whee [8] dl 0. 41h eff b / h b / h eff 16 and eff is the dielectic pemittivity fo a micostip, consideed as a tansmission line. Fig.. The epesentation of the magnetic field H y along x axis. Fo this eason we may assume that the fields ae extending outside the patch on a distance dl, and the patch act as a cavity with the length a+dl. In this case, if we assume mode TM 100 fo which in the fame of this assumption, the fields ae: x a dl E z E0 / + H y ae0 sin / + x a dl 17 and the length of the patch would be taken between dl and a+dl. Now it is obvious that the magnetic field is zeo at the coodinate 0 and a+dl but it would not be zeo at the ends of the patch: dl and a+dl. This means that besides the magnetic cuents geneated by E z at the bodes thee would be also electic cuents J geneated by H y. In this appoximation we calculate the cuents on the side walls: x Jm x Ezzˆ yˆ ± E0 ˆx fo y 0 and y b 18a a 18b dl Jm y Ezzˆ xˆ E0 yˆ fo x dl and x a + dl a + dl πdl Jz xˆ H yŷ H0 sin ẑ fo x dl and x a + dl a + dl 18c and afte the otation of the adiating apetues in a position paallel to the gound plane the elation 18c become: πdl x H sin ẑ fo x dl and x a + dl a + dl J 0 18d Now we suppose that the cavity adiate as fou magnetic cuents, and two electic cuents situated above the gound plane. The J mx cuents could be each decomposed as two equal magnetic cuents oiented in opposite diections, and so thei total adiated field would be zeo. The adiation patten of a patch situated ove a lage gound plane may be calculated by modeling the adiato as two paallel unifom magnetic line souces of length b and width h sepaated by distance a, and two unifom electic souces of length h and width b, sepaated by the same distance a. Fo an apetue containing both electic and magnetic cuents, the adiated fields ae [10]: E φ E θ e we obtain: e k E θ - exp k 0 E π * 343 k - φ J me 4π ωµ θ ˆ Je 4π k k θ ˆ J me 4π ωµ φ ˆ Je 4π k h$%$ b / kô kô k 0ysinθ sinφ z e 0 b / kô kô ds ds ds ds "!!#φ k asin θ dydh φ 1 h b /, + 0 k0ysin sin k0asin exp k0 H y e -dydh 0 b / h b - ;<; A k0ysin sin k0asin E exp k0 Ez e dydh sin 0 b / E F h b / G 0 k0ysin sin k0asin exp k0 H y O4O e dydh sin 0 b / In the equations 1 and the two ines tems unde the integation epesent the aay facto fo the two magnetic cuents, k 0 is the wave numbe in fee space and we also have take account of the gound plane. Afte

4 # " 34 D. D. Sandu, O. G. Avadanei, A. Ioachim, D. Ionesi solving the integals we obtain the adiating fields. Fom these elations we can detemine the adiated fields until a constant E 0 that depend of the excitation. In ou case the powe dissipated in dielectic P d is [11] a + dl b h Pd E E dv 3 dl 0 0 and losses in the electic conducting walls P m is defined by the elation a + dl b Pm H H ds 4 dl 0 whee δ is the dielectic loss tangent and σ is the electical conductivity. The time-aveaged electic enegy stoed by cavity is [1] a + dl b h 4 E Wc / E dv 5 dl 0 0 Intoducing, 3, 4 and 5 in 1 we compute δ eff and then intoducing k eff in 13 we obtain 0hc x0 Z in G m 0 n 0 1 eff m 0 n 0 C + L whee: ck /, c is the speed of light, 1/R ωδ eff /α, L αn ω, C 1/α hc and 0 x0 G ; R epesent the eal pat of the impedance, L is the inductance and C is the capacity at the feed point. Because all micostip antennas ae naow-band and they wok on one of the cavity mode the R ω can be simply appoximated by R ω whee ω is the esonant pulsation of the TM mode. This means that the summation in 6 would disappea and would emain only the m and n coesponding to the oscillation mode of the cavity. The input impedance will be Z X whee: hc R 0 $%! x0 G 8 nm eff hc X 0 x, y G +* ' / Fo the TM 100 mode excitation we can easily neglect the othe modes because A 010 and A 110 ae zeo and at this mode fequency the othe modes contibution to the impedance value can be neglected. 4. Expeimental esults Theoetical computations and measuements wee made fo a ectangula patch of dimension a.9 cm, b1.93 cm, h0.1 cm with a eal pemittivity ε.8 and a loss tangent appoximately The measuements wee made with a vectoial netwok analyze. The patch was fed at x 0 1. cm and y cm with a 50 Ω coaxial line. The measued impedance has the fom ZR+JX. Putting the computed impedance in a simila fom we obtain: X R 30 X + X X 31 X + In Table 1 we pesent the measued and the computed eal pat of the input impedance fo the cavity model [1] and ou model aound the esonant fequency fo the TM 100 mode. Fequency MHz Computed impedance [1]Ω Table 1. Computed impedance [ou model] Ω Measued impedance Ω It can be seen that aound the esonant fequency, that we found 985 MHz, both models have an excellent concodance with the expeiment. It can be also seen that if we chose a fequency fa fom the esonance the concodance is no longe as good, but emembe that in elation 8 we supposed that ωω nm, and is nomal that the model woks well only aound the esonance. Anyway this peculiaity has little pactical impotance because the patches ae vey naow band antennas and they adiate only at esonant fequencies. Fo the same patch, using elations 1 and, we have pesented in Fig. 3 the computed and the measued adiation pattens fo ϕ 0 and ϕ 90 planes.

5 Contibution to the cavity model fo analysis of micostip patch antennas 343 Fig. 5. Impedance dependence with the feeding point fo a.3 GHz patch. In Fig. 6 we pesent the esults fo a patch with: a 11,43 cm, b 7,6 cm, h mm, and dielectic pemittivity ε.6 [1]. Fig. 6. Impedance dependence with the feeding point fo a 803 MHz patch. 5. Conclusions Fig. 3. Radiation patten fo a 3 GHz antenna: a- ϕ 0 plane; b- ϕ 90 plane. We have also compaed the esults obtained with the cavity model [1] and ou model with the expeimental esults published in othe aticles. In Fig. 4 we pesent the esults fo a patch with the following dimensions: a 7.6 cm, b cm, h mm, and elative dielectic pemittivity ε.6 [1]. Radiation of patch antennas is efficient only when the excitation is made at the esonant fequency of a mode. The field is usually dominated by that single mode in the fequency ange of inteest. Thus the adiation patten and the input impedance can be detemined and they ae in good coelation with expeiment. It can be seen fom Fig. 3 that ou model gives bette esults fo the adiation patten of a patch antenna, mainly in the ϕ0 0 plane. Fom Table 1 we can see that ou model gives bette esults in 6 cases and wost in 1 case. Fom Fig. 4, 5 and 6 we can see that ou model descibes bette the dependence of the impedance with the feeding points compaed with the impoved cavity model developed by Richads et al. Refeences Fig. 4. Impedance dependence with the feeding point fo a GHz patch. In Fig. 5 ae pesented the esults fo a patch with a 4.04 cm, b 5.94, h 1.7mm, ε.4, [13]. [1] W. F. Richads, Y. T. Lo, D. Haison, IEEE Tans. Antennas Popagation, vol. AP -9 pp , [] Y. T. Lo, D. Solomon, W. F. Richads, IEEE Tans. Antennas Popagation, vol. AP-7, pp , [3] I. J. Bahl, P. Batia, Atech House, [4] N. G. Alexopoulos, N. K. Uzunoghn, I. Rana, IEEE AP-S Int. Symp. Digest, pp. 7-77, [5] P. K. Agawal, M. C. Bailey, IEEE Tans. Antennas Popagation, vol. AP. 5, pp , 1977.

6 344 D. D. Sandu, O. G. Avadanei, A. Ioachim, D. Ionesi [6] R. E. Munson, IEEE Tans. Antennas Popagation, vol. AP-, 1974, pp [7] A. K. Bhattachayya, R. Gag, IEEE Tans. Antennas Popagation Repinted in Micostip Antenna Design edited by K. C. Gupta, A. Benalla. [8] N. Balanis Antenna theoy: Analysis and design, sec. edition, John Wiley Sons, N.Y. 1997, chapte 14. [9] K. R. Cave, J. W. Mink, IEEE Tans. Antennas Popagation, vol. AP-9, pp. -4, Jan [10] A. Ishimau, Pentice Hall Intenational Edition, chapte 9, [11] W. F. Richads, Y. T. Lo, Fellow, J. Bewe, IEEE Tans. Antennas Popagation vol. AP-9 pp [1] M. Poza Micowave engineeing, Adison-Wesly [13] L. I.Basilio, M. A. Khayat, J. T. Williams, S. A. Long, IEEE Tans. Antennas Popagation. vol. AP-49 no. 1, 001. * Coesponding autho: ddsandu@uaic.o