Sensor and Simulation Notes. Note June A Family of Canonical Examples for High Frequency Propagation on unit Cell of Wave-Launcher Array

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1 !. Sensr and Simulatin Ntes Nte June 1989 WL-1051?-sSN-318 A Family f Cannical Examples fr High Frequency Prpagatin n unit Cell f Wave-Launcher Array D. V. Giri Pr-Tech, 125 University Avenue. Berkeley, CA Abstract The available analytic slutins fr a peridic array f symmetric in-line wave launchers have been used in characterizing the high-frequency prpagatin alng the launchers. These analytic slutins are based n a high-frequency asympttic frmulatin f the multicnductr transmissin-line equatins. In particular, a family f curves that permit a desired highfrequency vltage transfer rati frm the input at the apex t the aperture plane are cmputed and pltted. symmetric transmissin lines CLEAEEDFORPU3LIC RELEASE W?Q y7-35a y-w$j

2 s Preface J The authr wishes t thank Dr. Carl Bauinf the Weapns Labratry, Mr. I Ian Smith and Mr. Vic Carbni f Pulse Sciences Inc., and Dr. R. Pixtn and Mr. M. Dinall f the BDM Crpratin fr valuable discussins, encuragement and supprt. Sectin Cntents Page 1. Intrductin 2. High-Frequency Unit Cell f a 3 Vltage Transfer Characteristics Acrss the 4 Peridic Array f Wave Launchers 3. Family f Cannical Examples f Wave Launchers 6 A. Special Case f n = 1 6 B. General Case f Varying n 10 c. Realizing a Particular Impedance Prfile V(C) Summary 17 References.,....:.;,,,,. 4,.,.,.,.,. r,,,,.; : ;,.,, _..?,.

3 1. Intrductin Recent papers [1, 2 and 3] have cnsidered the high-frequency (r equivalently early-time) prpagatin f a step pulse surce at the apex, alng the unit cell f a symmetric in-line, peridic array f wave launchers. Starting frm multicnductr transmissin line equatins specialized t a tw-cnductr (plus reference) mdel fr the unit cell, [3] derives an expressin fr the vltage transfer rati frm the surce t the aperture plane in the high frequency limit. It is nted that, the transmissin line equatins are strictly valid when the crss sectinal dimensins f the unit cell are small and the launcher length is large cmpared with the shrtest significant wavelength f interest. This results in the requirement that the launcher length be large cmpared t its crss sectinal dimensins. Under the stated assumptins, the asympttic transmissin-line thery is useful in estimating the early-time prpagatin acrss the launchers. Of curse, at late times (r equivalently lw frequency prpagatin), the vltage transfer rati frm the apex t aperture plane tends t unity. The present interest hwever is in the high-frequency prpagatin which is a strng functin f the characteristic impedance matrix (2x2) elements. It was bserved in earlier papers [1 and 2], that three f the fur elements f this matrix are explicitly knwn in terms f the crss-sectinal dimensins fr an incremental length f the launcher and the single unknwn element can be numerically evaluated [2] fr a given set f launcher crss sectinal gemetry. Cnversely, a given imped?nce prfile (within practical realizability cnstraints) can be realized by evaluating the gemetry that prduces it. Denting the nrmalized characteristic impedance matrix (2x2) by (Fn,m(c)), we culd be.a linear functin f the nrmalized length bserve hat 1,2 = Fz,l crdinate L, which varies frm O t 1 as ne mves frm the apex t aperture planes. has been shwn t be a cnstant [1 and 2]. Under these cnditins, [3] has 2,2 analytically cnsidered tw special cases f cnstant and quadratically varying in deriving the vltage transfer ratis in the limit f high ituatins r 1,1 frequencies. Under the same cnditins fr the ff-diagnal and the(2,2) element, Fl,l(c) is allwed t vary linearly in G, and the vltage transfer rati is derived analytically in this paper. In additin, a family f impedance prfiles fr V(C)=F1,l are treated numerically. The usefulness f the results reprted in this paper lies in the ability t pre-seiect a launcher gemetry that prduces a desired highfrequency prpagatin characteristics. 3

4 I 2. High-Frequency Vltage Transfer Characteristics Acrss the Unit Cell f a Peridic Array f Nave Launchers 8aum [3] has derived an expressin fr the high-frequency vltage transfer 1 rati T(<) in general, and T(1) in particular gives the hi-frequency vltage transfe~ frm the surce at the apex t the aperture plane f the unit cell f a peridic array f wave launchers. J T(1) = [1 f(l) =Cs(g(l) +T/4)/fi (1) where nrmalized launcher length crdinate (O SL ~ 1) v,(l) = V2(1) z vltage at z = 1 (aperture plane) in the limit f high frequencies F,,, k) = v(c) = g(l) = step functin input at the apex initial value f F,, (nrmalized characteristic 1,1 impedance matrix element) v(~) *l,l(~)/[z(b/a)] ; U:V(C) ~1 (1,1) element f the characteristic impedance matrix plate separatin in the aperture plane in the E-field directin plate width in the aperture plane in the H-field directin,1 (2) V(C) with O :C :1 v(o) and V(l)=l (5) Frm the abve set f equatins, it is evident that T(l) Is in principle, \ readily cmputable fr a given impedance prfile v(c). Cnsistent with 4

5 .! 0 previus wrk [1 and 3], ne can use an empirical frm fr V(C) such as v(~) =a+ (l-ci)cn ; (n~o) (6) and vary the initial value CYand the expnent n, Cnsequently, we can think f the vltage transfer rati T t be a functin f three variables and dente it by T(c, u, n). Our interest is in the vltage r fields in the aperture plane (C =1) s that, we seek T(l, ~, n). Observe that if a = 1, v becmes 1 fr any n, crrespnding t a cnstant impedance prfile acrss the launcher. Anther example is when n = 2, the impedance prfile starts frm Q and quadratically increases t 1 in the aperture plane. These tw examples f cnstant and quadratic variatin were cnsidered in [3] and the results can be summarized as fllws T(l, 1, n) = (1/~) T(1> a, O) = (l/v%) (asympttic hybrid case[3];limit fn-o+) (7) 1 T(l, a,2)= cs~ & 4& () recalling that the three arguments f the vltage transfer rati at high frequencies are indicated in T(c, a, n). The result fr n = O in (7) cmes frm bserving that g(1) = Owhen n = O and T(l, a, O) is simply [cs(it/4)/fi]. In the next sectin, we cnsider yet anther special case f linear variatin in impedance prfile given by V(C) = a + (1-ci)~r n=l. 5

6 1 3. Fam~ly f Cannical Examples f Wave Launchers A. Special Case f n = 1 When the expnent n equals 1 in (6], we have v(i) =Cl+(l-a)~=a+~~ with B= (1-u) (8) Fr such a linear impedance prfile, we have the terms in (4) given by with [(1-V)2 + 4#1= (1-ci-iz)z+ 4# = #. 2B2G + (B2+4)i? = (fa.4)(@(c+q) P = -13/(f3-i2) and q = -9/(S+12) = p* (9) (10) (V.C2) = a + ~~ - C* = (~+u)(l-~) (11) (12) Cmbining we have frm (4), (13) = B [hl(d - h2(dl 2(B2+4) where (14) h2(d = what we require is [7& 1 (15) (16).. (17) j

7 ! 0 Cnsider g, fr the present, (18) (9(1) V=[2J+4) 91] (19) Using a tabulated integral ( in [4]), we have +$$-WI (c+p);:+a),l-aj [(l:i-q ]L (,+q,4%mt Denting the tw integrals in the abve by 1P and Iq respectively, we have 20) (62+4) [(l+a) - pb] 1P - ~ [(l+a)- qf3] Iq 1 = i46 1 [ 1[ (21) Nte that since p is the cmplex cnjugate f q (see (10)), the tw terms in (21) are als cnjugates f each ther resulting in 91 = (6;;4),m [[(l+a) - Pq j] (22) 7

8 I Using the abve in (19), we have, (23) ) It nw remains t evaluate lp, which is given by (24) Once again, using a tabulated integral [5] as nted belw, we have J dx = 2 (x-a)tix-b) (c-x) ~b-a)(c-a) arctan(,~) (25) p=& [arctan~-)] 1 0 = [](2 p-a)(p+l) -u arctan a p+] 6)1 Using (26) in (23), we have (27) By substituting fr p frm (10) in the factr in frnt f the arctan( ) in the abve and after sme algebraic manipulatins, we find (28) Substituting fr p frm (10) int the abve, we find 8

9 t..re[a..t.. (-~)] [9(1) +;] [( = Re arctan. (1-a) - )1 i (l+ci) 2~ =i arctan 2JIY 2 (cl-l) () (29) In writing (29) use is made f the identity Re arctan( [ x+iy)1 = (1/2) arctan 2X l-x2-y2 Cntinuing the simplificatin, () (30) [1 g(l)+; = (1/2) arccs[$~ * 1/2 () = (1/2) arccs (a-l) (a+l)2 Use has been made f the identity 1/2 1 arctan (x) = arccs 1+X2 () Taking csine f bth sides f (31) we have = (1/2) (31) ~ +? () (32) ( u ( )][ a-1 cs g(1) + (m/4) = cs + arccs ~ = = [a/(ci+l)]1/2 _+l (a-l) 1/ Tw (33) T(1, a, 1) = Cs(g(l) + (T/4))/G= (a + 1 )- /2 (34) which indeed is a cmpact T(c, a, n),we have T(I, 1, n)=l/@ T(I, ci,o) = l/(~) expressin. Summarizing the analytical results fr T(l,a, 1) = l/(vgi) T(l,cxj 2)=~cs ~ & () 46 {35) 9

10 Fr the cases f n = 1 and n = 2, g(1) was evaluated numerically by emplying a Simpsn s rule fr integratin and the last tw results in (34) were evaluated numerically as well. The analytical and numerical results fr the fur cases f (34) are reprted in Table 1. The analytical results f (34) are als pltted in figure 1. It is nted that n = O and a = 0.5 leads t a high-frequency vltage transfer rati f 1. Baum [3] had earlier cnsidered a hybrid launcher which is made up f tw sectins. The first sectin has a transfrmer actin resulting in a vltage f VO ~ and the secnd sectin with a cnstant impedance has a transfer rati f 1//2, resulting in a net rati f 1 frm surce t aperture plane. The relative lengths f the tw sectins is an pen issue. In the cntext f such a hybrid launcher, it is bserved thatthe n = O case cnsidered here is the asympttic limit f the hybrid case. Fr example, starting with an a =0.5, and n= 0.1 say, the impedance v (i.e., nrmalized Zl,l) will start frm 0.5 and rapidly rise and asympttically g t 1 as c tends t 1. Such a launcher array has the desired high-frequency prpagatin characteristics. ) B. General Case f Varying n Me are nw in a psitin t cmpute T(l, CL,n) fr a parametric set f values. The high-frequency vltage transfer rati T is cmputed numerically, tabulated (see Table 2) and pltted as a functin f~ ranging frm O t 1, fr different values f n in the range f O t 2, in figure 2. In this figure T= 1 line leads ne t pick ut pairs f values f u and n fr which the highfrequency vltage transfer rati is nminally 1. Such values fci and n fr which T = 1, are shwn pltted in figure 3 and it is bserved that such pairs f ciand n frm a smth curve, while resulting in a T f nminally 1. It is als evident that u = 0.5 and n = O is a desirable set f values frm purely high-frequency prpagatin characteristics. c. Realizing a Particular Impedance Prfile v(g) In [2], nrmalized impedance v(c), therein called Fl,l was evaluated fr a given launcher gemetry. Cnversely ne can determine the gemetry f the launchers t attain a prescribed impedance prfile using the nmgraphy in [21, within certain practical cnstraints. Me have thus established that certain ptimal launcher gemetries exist and they are cmputable t accmplish desired results at high frequencies, prpagating alng the launchers. In cncluding this sectin, it is bserved that prpagatin f intermediate times r frequencies are nt addressed yet. Cnsequently, actual cmputatins f launcher gemetries and their eventual ptimizatin shuld be perfrmed after

11 b., (, a = 1 n ; v= 1 T(l,l,n) (numerical ) Asympttic hybrid n*o+; v=cx+(l-q)g a T(l,a,O) = il/(v 2E ) ; see [3] infinity , 0.1 2, , a, T(l,cx,l) numerical analytical , U T(1,u,2) numerical analytical infinity Table 1. Sme Special Cases f T(<,a,n) 11

12 .. : 2.0! 2.0 T(l,u,n) \,fl \ \ \ n= \ \ \ = (1 V i=z / H _. --% ~ -1.C ( -2.0 Figure 1. Three special cases f impedance prfiles V(C) fr which the high-frequency vltage transfer ratis are analytically derived. 12

13 @ TABLE 2. High-Frequency Vltage Transfer Rati T(l,a,n) \ x n * * * + (XI 1, , ,221 1, , , , , D D , * analytical expressins in clsed frm are available fr these three cases nly

14 1 n = O*5 T(l,a,n) ( 0.8 ti O*6O 0.7 () VI(l) r 1.25 ( I / 0.2/ / ci~ 0.4 0, ,. / I Figure 2. Family f high-frequency vltage transfer ratis as a functin f u fr varying n. Nte: n = 0, 1 and 2 pltted in brken lines have been analytically derived (here and [3]).

15 .I 1.OO c n O* ,5 0.6 O* a O* \ Figure 3. Pairs f values f (n and a) that result in (Vi(l)/V) = T = 1. 15

16 . the intermediate times prpagatin characteristics are analyzed and understd. * Such an analysis valid fr all time regimes can be perfrmed by numerically slvi the Telegrapher s equatins fr vltages and currents in the time dmain. The tw cnductr (plus reference) mdel is available and ne can cnsider a variety f impedance prfiles and cmpute the aperture vltage ( VI(I) and V2(1) ) fr an ideallzed unit step input in the time dmain. This analysis will be the subject f future studies. )

17 (, 4. Summary This paper has extended the earlier wrk f Baum [3] which cnsidered the high-frequency prpagatin characteristics acrss the unit cell f a peridic array f wave launchers. A special case f linear variatin f the launcher impedance as a functin f the length crdinate alng the launcher is analyzed fr its high-frequency prpagatin characteristics. In additin, a family f curves, indicative f the high-frequency vltage transfer characteristics frm the surce t the aperture plane are cmputed numerically, tabulated and pltted. Such results are useful in designing launcher gemetries that are ptimal frm purely high-frequency cnsiderating. Additinal analysis is required t evaluate the prpagatin characteristics in the intermediate times (r equivalently mid-frequency regime). 17

18 . References [1] C. E. Baum, Cupled Transmissin-Line Mdel f Peridic Array f Wave Launchers, Sensr and Simulatin Nte 313, December [2] D. V. Giri, Impedance Matrix Characterizatin f an Incremental Length f a Peridic Array f Wave Launchers, Sensr and Simulatin Nte 316, April 19!39. [3] C. E.!3aum,Cannical Examples fr High-Frequency Prpagatin n Unit Cell f Wave-Launcher Array, Sensr and Simulatin Nte 317, April [4] I. S. Gradshteyn and 1. M. Ryzhik, Table f Integrals, Series and Prducts, Academic Press, [51 Handbk f Mathematical, Scientific and Engineering Frmulas, Tables, Functins, Graphs and Transfrms, Staff f Research and Educatin Assciatin, New Yrk 1984 (pp. 366).! 18