Chpter 4 Ptch Antenns A ptch ntenn is low-profile ntenn consisting of metl lyer over dielectric sustrte nd ground plne. Typiclly, ptch ntenn is fed y microstrip trnsmission line, ut other feed lines such s coxil cn e used. The dvntges of ptch ntenns re tht they rdite with modertely high gin in direction perpendiculr to the sustrte nd cn e fricted t low t on PCB. The sic operting principle of ptch ntenn is tht the spce etween the ptch nd ground plne cts like section of prllel plte wveguide. Neglecting rdition loss, the edge of the ptch is n open circuit, so tht energy reflects nd remins elow the ptch. The ptch ntenn is therefore resonnt cvity with reltively high qulity fctor. One disdvntge of high-q system is nrrow ndwidth, so ptch ntenns hve limited ndwidth, mening tht the input impednce of the ntenn only remins ner the desired vlue for smll rnge round the designed center frequency. 4.1 Resonnt Cvity Anlysis We will tret the ptch using perturtion pproch y nlyzing the fields under the ptch s if there were no rdition. This is similr to the sinusoidl current model pproch for dipole ntenn. The region under the ptch cn e modeled pproximtely s resonnt cvity with PEC oundry conditions on the top nd ottom nd n open circuit oundry condition on the sides. We cn think of the region under the ptch ntenn s section of prllel plte wveguide with open circuit lods t the sides. If we orient the coordinte system so tht the top nd ottom pltes re prllel to the x-y plne, the dominnt mode in prllel plte wveguide hs n electric field tht is in the z direction. The z component of the electric field stisfies the differentil eqution ( 2 + k 2 E z = jωµj z (4.1 We cn find the homogeneous solutions (i.e., with J z = using the method of seprtion of vriles. Seprtion of vriles is sed on expnding E z s product of functions of ech of the coordintes, so tht Inserting this into the Helmholtz eqution leds to E z (x, y, z = f(xg(yh(z (4.2 = f gh + fg h + fgh + k 2 fgh = f f + g g + h h + k2 (4.3 54
ECEn 665: Antenns nd Propgtion for Wireless Communictions 55 Ech term in this sum must e constnt, or else we could chnge one of the independent vriles x, y, or z nd the equlity would no longer hold. It follows tht f + k 2 xf = g + k 2 yg = h + k 2 zh = (4.4 (4.4 (4.4c where k 2 x, k 2 y, nd k 2 z re the constnt vlues of ech of the terms in (4.3. From (4.3, the constnts must stisfy k 2 x + k 2 y + k 2 z = k 2 (4.5 This constrint on the seprtion constnts is referred to s dispersion reltion. The generl solution to ech of the ordinry differentil equtions in (4.4 is liner comintion of sine nd ine functions or complex exponentils, with f(x = A (k x x + B sin(k x x (4.6 for the x dependence nd similr forms for the y nd z functions. Since the dielectric lyer is typiclly thin, we cn ignore the smll z dependence of the fields, nd set k z =. 4.1.1 Boundry Conditions At the ptch edges, due to the open circuit oundry condition we hve so tht E z x = t x =, x = (4.7 k x A sin(k x x + k x B (k x x =, x =, x = (4.8 It follows tht B = nd k x = mπ, where m is n integer. Similrly, we hve tht g(y = C (nπy/, where n is n integer. The solution for ech pir m, n is E z,mn = A mn (mπx/ (nπy/ (4.9 In order to otin the complete solution to the originl differentil eqution, we must sum over the indices m nd n, so tht E z = A mn (mπx/ (nπy/ (4.1 m= n= Ech term in the sum is mode of the structure. The mplitude A mn of the m, nth mode is determined y the driving source J z. 4.1.2 Modl Solution Sustituting the generl solution (4.1 into the inhomogeneous Helmholtz eqution (J z leds to ( 2 + k 2 E z = 2 2 A mn [ + k 2] E z,mn m n = jωµj z (x, y Wrnick & Jensen Jnury 29, 215
ECEn 665: Antenns nd Propgtion for Wireless Communictions 56 Defining kmn 2 = (mπ/ 2 + (nπ/ 2 leds to (k 2 kmna 2 mn m,n = jωµj z (x, y (4.11 Becuse the left-hnd side of this expression is Fourier series, the unknown coefficients A mn cn e found from the Fourier series of the driving source J z (x, y. To find the unknown modl coefficients, we multiply y nother Fourier function nd integrte to otin dx ( rπx Using the identity we hve dy ( sπy = jωµ (k 2 kmna 2 mn m,n dx dx ( rπx dy ( sπy J z (x, y = jωµj rs (4.12 ( nπx = 2 δ mn (4.13 A mn = jωµ 4 k 2 kmn 2 J mn (4.14 At this point, it remins to choose specific model J for the ntenn feed nd compute the Fourier coefficients J rs, from which we cn otin the electric field in the cvity etween the ptch nd the ground plne. A ptch ntenn is typiclly operted t resonnce, so tht k = k mn for some m nd n, in order to mke E z s lrge s possile t the operting frequency. If the ptch were truly lossless nd the ove derivtion exct, k = k mn would men tht the mplitude A mn of the m, nth mode is infinite. In relity, the pth ntenn rdites, nd the ove derivtion is only pproximte, so A mn is lrge ut finite for k = k mn. If we ssume tht the resonnt mode is lrge enough tht the contriutions of other modes cn e neglected, then the series for E z reduces to single term, E z (x, y A mn (4.15 where m nd n re the mode numers of the resonnt mode. In designing ptch ntenn, we choose the ptch dimensions nd such tht k mn = k for some m nd n t the desired operting frequency. To keep the ptch s smll s possile, the mode numers should e low, lthough lrger ptch cn e used to otin multiple resonnces ner the operting frequency nd increse the ntenn ndwidth. 4.1.3 Feed t Edge Let us choose the feed model J = ẑδ(y, c x d, t z (4.16 Wrnick & Jensen Jnury 29, 215
ECEn 665: Antenns nd Propgtion for Wireless Communictions 57 This represents microstrip feed line. Computing the Fourier coefficients of J z nd using them to find E z leds to { 4(d c 2(d c E z (x, y = jωµ k 2 + (k 2 kn 2 + 4.1.4 Fr Fields + m=1 m=1 n=1 n=1 4 sin[mπ(d c/(2] [mπ(d + c/(2] mπ(k 2 k 2 m 8 sin[mπ(d c/(2] [mπ(d + c/(2] mπ(k 2 k 2 mn } (4.17 In order to find the rdited fields from the ntenn, we will crete n equivlent source from the fields t the edges of the region under the ptch using ( ˆx E z ẑ = ŷ (nπy/ x = (ˆx E M s = ˆn E = z ẑ = ŷ( 1 m (nπy/ x = (4.18 ( ŷ E z ẑ = ˆx (mπx/ y = (ŷ E z ẑ = ˆx( 1 n (mπx/ y = No equivlent electric current is required since ˆn E t the wlls of the cvity. In order to tke into ccount the ground plne elow the ptch, we use imge theory to replce the ground plne with imge sources for the current M s. The width of the skirt of current round the region under the ptch increses to 2t, nd the equivlent source then lies in free spce. The electric vector potentil is F = ϵ e jkr 4πr F x = ϵ e jkr 4πr 2t e jkˆr r M(r dr [ ( e j k x x mπx dx ( 1 n e j k y = ϵ e jkr 4πr 2tg 1(θ, ϕ F y = ϵ e jkr 4πr 2tg 2(θ, ϕ ( ] e j k x x mπx dx in terms of which the electric fr field is E = jωηˆr F. If the ptch ntenn is designed to operte t the lowest order mode, then m = nd n = 1. At θ =, ( πy g 2 dy = (4.19 so tht the edges t x = nd x = do not rdite significntly. For the other edges, g 1 = [1 ( 1 1 e j k y ] where k x = k sin θ ϕ nd k y = k sin θ sin ϕ. (e j k xx dx = 2e j k y /2 ( k y /2 ej k x 1 j k x = 2e j k x/2 e j k y/2 ( k y /2 sin( k x /2 k x /2 Wrnick & Jensen Jnury 29, 215
ECEn 665: Antenns nd Propgtion for Wireless Communictions 58 4.1.5 Input Impednce Ner the feed point, ll the modes in the series solution for E z must e tken into ccount in order to otin n ccurte vlue for the input impednce. The input impednce is Z in = V in I in (4.2 We cn find the voltge t the input terminl from the electric field (4.17. Since the z dependence of the field is pproximtely constnt, V in E z (x, yt (4.21 where the point (x, y is locted t the feed. The resulting input impednce is [6] Z in = m=1 n=1 jωα mn δ m δ n ω 2 mn (1 jδ eff ω 2 where δ m = { 1 m = 2 m > (4.22 nd ω mn = c k mn ϵr (4.23 α mn = t ( 2 mπx ϵ ϵ r ( 2 nπy ( sinc 2 mπw 2 (4.24 In this expression, w is the width of the feed line. The lst fctor ccounts for the dependence of the input impednce on the feed line width. To get more ccurte results, the ptch length nd width cn e replced with djusted lengths tht tke into ccount fringing fields t the edges of the ptch. Since the series solution for the fields under the ptch ws derived y pproximting the rdition s zero, the resonnt mode leds to n infinite term, ut rdition loss mens tht effectively the wvenumer hs smll imginry prt tht ccounts for rdition loss. In the expression for input impednce, δ eff is n effective loss tngent for the ptch ntenn. This cn e estimted using δ = 1/Q, where Q is the qulity fctor of the ptch ntenn. The input impednce cn lso e estimted using empiricl formuls or computed using numericl lgorithm such s the method of moments. Since the series solution for the fields under the ptch ws derived y pproximting the rdition s zero, the resonnt mode leds to n infinite term, ut rdition loss mens tht effectively the wvenumer hs smll imginry prt tht ccounts for rdition loss. In the expression for input impednce, δ is n effective loss tngent. This cn e estimted using δ = 1/Q, where Q is the qulity fctor of the ptch ntenn. The input impednce cn lso e estimted using empiricl formuls or computed using numericl lgorithm such s the method of moments. Wrnick & Jensen Jnury 29, 215