rect_patch_cavity.doc Page 1 of 12 Microstrip Antennas- Rectangular Patch Chapter 14 in Antenna Theory, Analysis and Design (4th Edition) by Balanis

Transcription

1 ect_patch_cavit.doc Page 1 of 1 Micostip Antennas- Rectangula Patch Chapte 14 in Antenna Theo, Analsis and Design (4th dition) b Balanis Cavit model Micostip antennas esemble dielectic-loaded cavities that ae bounded b conductos on the top & bottom (i.e., tangential electic fields ae eo) and magnetic walls (i.e., tangential magnetic fields ae eo) on the sides, simulate open cicuits. A pue cavit model does not take into account that pat of the field is adiated. Radiation loss is woked into the model b intoducing an effective loss tangent δ eff = 1/Q whee Q is the antenna qualit facto. Since h <<, the electic field is neal nomal to the patch (neglect finging) inside the cavit. This leads us to conside onl the tansvese magnetic (TM ) field configuations o modes. h L W Rectangula micostip patch geomet.

2 ect_patch_cavit.doc Page of 1 TM field configuations o modes The wave equation that will be solved, fo the dielectic cavit is A k A whee A is the -component of the vecto magnetic potential and k is the wave numbe. The geneal solution fo A is 3 3 A A1 cos( k) B1 sin( k) A cos( k ) B sin( k ) A cos( k ) B sin( k ) whee k, k, and k ae the wave numbes in the indicated diections. The applicable bounda conditions ae Top and Bottom of cavit (conductos) ( =, L, W ) = ( = h, L, W ) = Sides of cavit (magnetic walls) ( h, L, = ) = ( h, L, = W ) = ( h, =, W ) = ( h, = L, W ) = whee the pimed coodinates epesent the inside of the cavit. Appling these bounda conditions leads to A A cos( k ')cos( k ')cos( k ') mnp whee A mnp is the poduct of the amplitude coefficients and the wave numbes ae

3 ect_patch_cavit.doc Page 3 of 1 and k k k m m,1,, h n m n p n,1,, L (can't all be eo) p p,1,, W k k k k. The subscipt efes to the esonant fequenc. Theefoe, the esonant fequenc is f mnp 1 m n p h L W. Afte solving fo A, the electic and magnetic fields can be found fom the vecto magnetic potential using and 1 A j k A 1 A j 1 A j

4 ect_patch_cavit.doc Page 4 of 1 (tansvese magnetic) 1 A 1 A which ields k k mnp j A cos( k ')cos( k ')cos( k ') kk j Amnp sin( k')sin( k ')cos( k ') kk j Amnp sin( k')cos( k ')sin( k ') and k Amnp cos( k ')cos( k ')sin( k ') k Amnp cos( k ')sin( k ')cos( k ') The electic field configuations fo the lowest few cavit modes ae shown in Figue The dominant mode (i.e., the mode with the lowest esonant fequenc) depends on the dimensions of the cavit (patch). Since the cavit height h << L and h << W, the length L and width W of the patch will contol the dominant mode.

5 ect_patch_cavit.doc Page 5 of 1 Figue Field configuations (modes) fo ectangula micostip patch. [Fom Balanis, Antenna Theo, Analsis and Design (Fouth dition)] If L > W > h, the dominant mode (i.e., the desied mode) is the TM 1 whee the esonant fequenc is c. L L 1 f 1 Futhe, if L > W > L/ > h, the net highest mode (afte TM 1) is the TM 1 1 c f 1. W W

6 ect_patch_cavit.doc Page 6 of 1 oweve, if W > L > h, the dominant mode is the TM 1 whose esonant fequenc has alead been given. Fotunatel, if a centeed micostip feed is used, the TM 1 mode can be ecited, even if it is not the dominant mode (see Fig 14.13a). If L > L/ >W > h, the dominant mode would be the TM 1 f c. L L If W > W/ > L > h, the net highest mode (afte TM 1) is the TM c. W W 1 f Note: These calculations ignoe the effects of finging and assume that the dielectic substate is onl unde the patch. Radiation (TM 1 mode) Assuming the active o dominant mode is the TM 1, the fields in the cavit ae cos ' and sin ' L L whee n = 1, m = p =, = -ja 1, and = (π/µl) A 1. See Figue 14.13a fo pictues of the electic field distibution. Radiation occus fom the two end slots (located at = and = L). The ectangula slots have dimensions of W h, and ae sepaated b about / at esonance. The side slots (located at = and = W ) ae non-adiating because the adiation fom the fields along the sides cancel each othe in the fa-field (note that along half the side slots the electic field points up and on the

7 ect_patch_cavit.doc Page 7 of 1 othe half it points down). The fa-field adiated electic fields adiated b each slot (see Chapte 1) ae j jk k hw e X Z sin( ) sin( ) sin X Z whee and ae the standad spheical coodinate angles, and kh X sincos. kw Z cos If k h << 1, then educes to jk kw he sin cos j sin cos Note, the voltage acoss the slot is V =h. Modeling the adiating slots as a two-element aa (see Chapte 6) of ectangula apetue antennas leads to jk tot khw e sin( X) sin( Z) k Leff j sin cos sin sin X Z Again, if k h << 1, this educes to jk kw tot he sin cos k Leff j cos sinsin sin cos Note, the voltage acoss the slot is V = h.

8 ect_patch_cavit.doc Page 8 of 1 The adiated electical field in the two pincipal planes is kh sin cos sin cos jk tot kwh e k Leff j cos kh in the -plane (- plane above the gound, = 9), and kh kw jk sin sin sin cos k Wh e sin kh kw sin cos tot j in the -plane (- plane above the gound, = o 18). Figue shows eamples of tpical -plane and -plane adiation pattens. Note that the epeimental, theoetical, and MoM esults agee well in the -plane. oweve, thee ae some diffeences at low angles (nea the dielectic substate) between the methods in the -plane. This pimail because the cavit theo assumed the dielectic substate was tuncated at the edges of the cavit, which does not happen in ealit.

9 ect_patch_cavit.doc Page 9 of 1 Figue Pedicted and measued - and -plane pattens of ectangula micostip patch (L=.96cm, W=1.186cm, =.316cm, ε =., f =1G). [Fom Balanis, Antenna Theo, Analsis & Design (Second dition)]

10 ect_patch_cavit.doc Page 1 of 1 Diectivit Knowing the fields allows the diectivit of the ectangula patch to be calculated. In paticula, we ae inteested in the maimum diectivit D U 4U ma ma ma D. U Pad Fo the tpical case that k h << 1, the maimum adiation intensit and the powe adiated b a single ectangula slot ae and P U ma V W V kw sin cos 3 ad sin d. cos The maimum diectivit of a single ectangula slot is then whee I 1 is I 1 D W 1 D I1 ma kw sin cos 3 sin d cos sin( kw ) cos( kw ) kw Si ( kw ) kw The maimum diectivit of a single slot is shown in Figue

11 ect_patch_cavit.doc Page 11 of 1 The maimum diectivit of a ectangula patch ( adiating slots) is D tot tot ma W W D I 15G ad whee G ad is the adiation conductance and I kw sin cos 3 kl eff sin cos sin sin cos d d. A slightl simple epession fo the maimum diectivit is tot tot Dma D D 1 G1 / G. 1 The diectivities fo two slots (i.e., ectangula patch) ae shown in Figues and 14. as functions of slot width W and substate height h. The PBWs in the -plane and -planes ae (ve) appoimatel and 7.3 sin (14-58) 1 4 3Leff h 1 kw 1 sin (14-59).

12 ect_patch_cavit.doc Page 1 of 1 Figue 14. Computed diectivit of one and two slots as a function of the slot width. [Fom Balanis, Antenna Theo, Analsis and Design (Fouth dition)] Figue 14.3 Diectivit vaiations as a function of substate height fo a squae micostip patch antenna (coutes of D. M. Poa). [Fom Balanis, Antenna Theo, Analsis and Design (Fouth dition)]