Effect of conductor thickness on the mode capacitances of shielded strip transmission lines

Transcription

1 Retrospetive Theses and Dissertations Iowa State University apstones, Theses and Dissertations 1976 Effet of ondutor thikness on the mode apaitanes of shielded strip transmission lines James Leo Knighten Iowa State University Follow this and additional works at: Part of the Eletrial and Eletronis ommons, and the Oil, Gas, and Energy ommons Reommended itation Knighten, James Leo, "Effet of ondutor thikness on the mode apaitanes of shielded strip transmission lines " (1976). Retrospetive Theses and Dissertations This Dissertation is brought to you for free and open aess by the Iowa State University apstones, Theses and Dissertations at Iowa State University Digital Repository. It has been aepted for inlusion in Retrospetive Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please ontat

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3 77-2 KNIGHTEN, James Leo, 1943«EFFET OF ONDUTOR THIKNESS ON THE MODE APAITANES OF SHIELDED STRIP TRANSMISSION LINES. Iowa State University, Ph.D., 1976 Engineering, eletronis and eletrial Xerox University l^irofiims, Ann Arbor, Mihigan 48106

4 Effet of ondutor thikness on the mode apaitanes of shielded strip transmission lines by James Leo Knighten A Dissertation Submitted to the Graduate Faulty in Partial Fulfillment of The Requirements for the Degree of DOTOR OF PHILOSOPHY Major: Eletrial Engineering Approved ; Signature was redated for privay. In harge of Major Work Signature was redated for privay. Signature was redated for privay. Iowa State University Ames, Iowa 1976

5 ii TABLE OF ONTENTS Page I. INTRODUTION 1 A. Purpose 1 B. Statement of Problem 3 II. ANALYTIAL METHOD 5 A. Historial Overview 5 B. Basi Diretional oupler onepts 9. Desription of Analytial Tehnique 13 III. ANALYSIS OF THE ZERO-THIKNESS ASE 17 A. Analytial Tehnique 17 B. Numerial Analysis Tehnique 22 IV. ANALYSIS OF THE NONZERO THIKNESS ASE 27 A. Analytial Tehnique 27 B. Numerial Analysis Tehnique 29. Modified Numerial Analysis Tehnique 33 V. RESULTS 35 A. Zero-Thikness ase Single entered strip oplanar zero-thikness oupled strips Overlapped oupled strips of zerothikness 41 B. Nonzero Strip ondutor Thiknesses Single entered strip Modifiation of the hestnut method oplanar oupled strip ondutors with nonzero thikness 53

6 iii Page 4. Overlapped oupled striplines of nonzero thikness 58. Sample alulations Example of diretional oupler analysis Example of diretional oupler design 65 VI. SUMMARY AND ONLUSIONS ^ 66 VII. LITERATURE ITED 69 VIII. AKNOWLEDGMENTS 72 IX. APPENDIX A: EVALUATION OF LIMIT TERMS 73 A. Zero-Thikness ase 73 B. Nonzero Thikness ase 75 X. APPENDIX B: DERIVATION OF EXTRAPOLATION FORMULA 77 XI. APPENDIX : DERIVATION OF GREEN'S FUNTION 80 XII. APPENDIX D: OMPUTER PROGRAM LISTINGS 87

7 1 I. INTRODUTION A. Purpose The strip transmission line has beome a dominant transmission line form sine its ineption in the post World War II years. It is thought by some to be the offspring of a wartime development alled the "flat strip" oaxial transmission line [2]. It was understood, early on, that these "flat strip" oaxial lines ould be employed for purposes other than simple energy transfer from point to point. They ould be used to realize many types of mirowave passive omponents, suh as filters, diretional ouplers, et. [2]. It was also realized by manufaturers of mirowave iruitry and devies that the planar geometry of the strip transmission line was as suitable to eonomial fabriation as was the onventional low frequeny printed iruit. ertainly, the development of semiondutor devies that would operate at higher and higher frequenies gave onsiderable impetus to the widespread use of striplines. The ommon oaxial transmission line did not adapt itself well to the proess of integration of iruit omponents and onneting transmission lines at high frequenies. The printed iruit-type tehnology of the strip transmission line is also quite amenable to the fledgling mirowave integrated iruit tehnology. These appliations and developments have all been spurred

8 2 by the burgeoning usage of the spetrum thus inreasing the need for better design tehniques at higher and higher frequenies. The purpose of this work is to analyze the behavior of a oupled pair (in general, offset) of strip transmission lines. Speifially, the effets of real, nonzero thiknesses of the strip ondutors are to be examined. Past analyses of the overlapped oupled stripline struture have onsidered the use of idealized strip ondutors of zero thikness. These studies have not adequately onsidered the effets of nonzero ondutor thiknesses. The basi stripline struture is apable of supporting a transverse eletromagneti wave (TEM) that propagates longitudinally along the strip transmission line. The full wave analytial solution for the existing eletromagneti waves redues to a solution of Poisson's equation in a rosssetional plane and is equivalent to an eletrostati solution in that region. The struture is nondispersive in nature. The general tehnique of analysis is an eletrostati approah involving numerial integration on a digital omputer. This integration tehnique is applied to a Green's funtion potential integral. The integral is of the Fredholm type with the produt of a Green's funtion and the harge distribution on the strip ondutor surfaes as an integrand.

9 3 It is possible to assume a potential on the perfetly onduting strips and then to numerially invert, or unfold, the integral to solve for the kernel, the surfae harge distribution that exists on the strip ondutors. From this knowledge, the total harge existing on the ondutors may be numerially approximated and the apaitane or the impedane of the struture may be obtained. The transmission line apaitane is a ommon parameter, along with impedane, in desribing the eletrial behavior of a transmission line. These results allow analysis and, indiretly by an iterative method, design of the stripline diretional oupler. B. Statement of Problem A ross-setion view of the overlapped oupled stripline is shown in Figure 1. The struture is bounded on the top and on the bottom by parallel perfetly onduting ground planes of infinite breadth and of vertial spaing B. The two parallel strip ondutors are idential. The pair is positioned with 180 rotational symmetry about the enter axis of the struture. The ondutors are eah of width W, thikness T, and lateral overlap W^. The vertial gap between the strips is designated by S. The struture is filled with a homogeneous nonmagneti dieletri material of relative permittivity e^.

10 4 y i I x=b ^l'^l ± 4 W -y=o B x=a i -4 ^ Figure 1. ross-setional view of oupled overlapped strip transmission lines

11 5 II. ANALYTIAL METHOD A. Historial Overview The origins of strip transmission lines are unlear as are the origins of their use in realizing diretional ouplers. It is ertain that the early 1950's saw an interest develop by mirowave engineers in ethed sheets of both single and double ondutor lines. A rigorous solution for the harateristi impedane of the single ondutor stripline of zero thikness was obtainable by a onformai mapping tehnique and was well known. The effets of strip thiknesses greater than zero on the line impedane were attaked in the mid-1950's by S. B. ohn [8] using an approximate formula for line apaitane that onsisted of a parallel plate omponent plus a term that aounted for the fringing eletri fields near the edges of the strip ondutor. The fringing term was based on the fringing apaitane of a semi-infinite slab and hene was muh more aurate for wide strip ondutors than for narrow ones. A separate tehnique was applied to narrow strips. Other tehniques for desribing the behavior of the thik single ondutor stripline, inluding a series mathing tehnique by Begovih [3] and a variational tehnique by ollin [12] were published in the late 1950's and early 1960's. The development of greater digital omputer apability and availability in the

12 6 middle 1960's led to the implementation of finite differene methods in solving the stripline problem [17, 26]. These methods, although powerful, required muh omputer time and required a new omputer program for eah hange in rosssetional geometry. In reent years, analytial interest in the single ondutor stripline has been elipsed by interest in the less well understood mirostrip. The two ondutor oupled stripline and its use in the diretional oupler also has its origins somewhat shrouded in time. Major artiles on the diretional oupler of the late 1940's [23, 25] onern themselves only with strutures that exhibit oupling at disrete points, suh as the Bethehole oupler. In a brief, but interesting, historial sketh, however. Young [29, pp. v-vii] reports a quarter wave ontradiretional parallel wire oupler as early as By the middle 1950's interest in parallel wire diretional ouplers, and more pertinently, parallel strip ondutor diretional ouplers, had burgeoned. ohn [9] onsidered the oplanar struture (W^ < 0, S=T=0) and derived rigorous solutions for zero-thikness strips by onformally mapping with a Shwarz-hristoffel transformation. The resulting equations were funtions of omplete ellipti integrals of the first kind, so he developed a series of nomograms that ould be onveniently used for oupler design. onsidering

13 7 the possibility of ondutor thikness, he introdued a fringing apaitane term modified from his previous work on the single strip line. This analysis was only valid for wide strips, W/B ^ In 1960, in a series of two artiles published simultaneously, ohn [10, 11] looked at the broadside (W=W^, S > 0, T=0) oupler, among others. Simplifying approximations were possible in the onformai mapping proess that eliminated the ellipti integrals and yielded more manageable equations. It was required that [(W/B)/(1-S/B)] > 0.35 for permissible auray. Sine these results ould only be applied to ases where T>0 with great are, a apaitive orretion term was devised from semi-infinite slab fringing field behavior. Getsinger [16] onsidered the ase of very thik parallel bar ondutors plaed side by side. The method applied was onformai mapping, yielding a solution involving ellipti integrals. The results were presented graphially so that one ould design from the urves. A maximum error of 1.24% was assumed if [(W/B)/(1-T/B)] > Gupta [18] pursued this solution further providing urves suitable for more tightly oupled bars than did Getsinger. Shelton [27] onsidered the ase of parallel strip ondutors not in the same plane but offset from one another as shown in Figure 1. This sort of struture grew out of neessities to ross lines in stripline iruits. Shelton

14 8 onsidered two models. Loose oupling was defined as [2(W-W^)/(1+S)] > Tight oupling was restrited to W^/S _> 0.7. Applying the Shwarz-hristoffel transformation, he arrived at an approximate set of design equations that were valid for relatively wide strips only [W/(l-S)] > Shelton's equations have been applied numerially to several speifi ases of stripline ouplers and the results plotted [22, in 29, pp ]. Employing a method muh removed from the previous onformal mapping work, Kammler [20] set up a Green's funtion integral equation and used it to solve for the matrix of oeffiients of apaitane. Approximate values of the harge density existing on the strip ondutors were determined at disrete points. This harge distribution was adjusted with the aid of a digital omputer so that the potential funtion made the best fit, in the manner of the least squares, to the exat potential. Numerial omparisons were made to exat solutions available from onformai mapping for the single ondutor stripline. onsiderable auray was obtained. Strip thikness was not onsidered, but no restritions were plaed on strip width. An outgrowth of Kammler's pereptions was the work by hestnut [6], who found a different numerial tehnique for solving the Green's funtion integral problem. This tehnique will be entral to the work presented in this thesis.

15 9 This disussion of the historial development is by no means exhaustive, but is merely an attempt to trae the general trend in the development of analytial methods. The reader will find an exellent bibliography on the general subjet of the diretional oupler, urrent to 1966, by aswell and Shwartz [5]. B. Basi Diretional oupler onepts Fundamentally, the diretional oupler is a four port devie that allows inoming power to be tapped, or split, before it exits. The tapped, or oupled, portion of the inoming power is direted to a speifi exit port, the identity of whih is determined by the diretion of the inoming power. It is, in this sense, a diretional devie. This phenomenon ours when all four ports of the devie are impedane mathed. A fourth port is isolated, meaning that no power exits that port. The identities of the oupled and the isolated ports are determined, in general, by the diretion of the inoming signal. An important desriptive parameter is that of oupling, ^, measured in deibels, given by [13] = io ^ (1) where is the inident power to the oupler and is the power oupled off of the main line. Thus, P^ > P^. oupling

16 10 an also be expressed in terms of the voltage oupling ratio, or the voltage oupling oeffiient, k, as [4] g = -10 log^qlkjz (2) The ommon mirowave diretional ouplers, suh as the Bethe-hole waveguide oupler, or other aperture oupled devies behave suh that the oupled energy propagates in the same general diretion as the inident energy, so that energy physially enters the oupler on one side and both the oupled and diret waves exit on the other side. This is sometimes referred to as forward oupling. The oupled stripline struture behaves like a diretional oupler. It is a four-port devie that uses eletromagneti oupling to transfer energy from one ondutor to another. Mathing all the ports, it an be shown [4] that a portion of the inident power is transmitted, a portion of the inident power is oupled to a different port, and one port is isolated. Maximum oupling ours when the oupler is one quarter-wavelength long. This type of oupler is inherently different from the multihole waveguide ouplers and some transmission line oupling arrangements that depend on disrete oupling regions that utilize onstrutive wave interferene for the diretional effet. The parallel line oupler employs ontinuous oupling throughout its length

17 11 and the oupled wave is indued in suh a manner as to propagate bakward with respet to the inident wave. This struture is often alled a ontradiretional, or a bakward wave diretional, oupler. This feature sometimes an ause awkwardness if ouplers are to be asade onneted sine this auses the onneting lines to ross. Nonoplanar oupled strips, suh as seen in Figure 1, an alleviate this problem. A feature of parallel line ouplers is that they perform well over wide bandwidths [4, 29, pp. v-vii]. They are limited, however, to TEM type transmission systems and are not designed to arry the high power level of whih a waveguide oupler might be apable. The basi mode of operation of the oupled stripline diretional oupler is to apply a signal to the input port of one strip ondutor and to ouple energy to the seond ondutor. In general, this type of operation may be onsidered to be the superposition of two more fundamental modes of exitation. In the even mode of exitation equal voltages are applied to both strip ondutors. The harateristi impedane of either of the strip ondutors to ground is termed Odd mode exitation requires equal and opposite potentials on the strips. The harateristi impedane in this ase is A disussion of this mode

18 12 differentiation may be found in Jones and Bolljahn [19], among others. High frequeny TEM transmission lines whose harateristi impedanes are real (or nearly so) may also be haraterized by, the shunt apaity of the line per unit length. This apaitane is related to the harateristi impedane by the phase veloity of propagation assoiated with the transmission line. A disussion of this onept may be found in hipman [7], and many other texts. Therefore, the apaitane of the oupled stripline is not the same for even and odd exitations sine the harateristi impedanes are different. One distinguishes between even mode apaitane, and ^, the odd mode apaitane. Both modes propagate at the same veloity, the speed of light. The differene between the mode harateristi impedanes, Z oe _ and Z oo or the mode apaitanes, and ^, is a measure of the voltage oupling oeffiient, k. These relationships are enumerated in Equation (19) in hapter III. Note that in Equations (1) and (2), the oupling in db is always positive and the more loosely oupled the struture, the greater the value of ^. This may be slightly onfusing sine the oupled wave is atually smaller in magnitude than

19 13 the inident wave. Some would desribe the oupling in negative deibels (multiply Equations (1) and (2) by a minus one). The onvention of positive db oupling will be used throughout.. Desription of Analytial Tehnique The general analysis used to study the oupled stripline is that desribed by hestnut [6]. The analysis is valid, in general, for multiondutor lines of many shapes and orientations (provided that they are of onstant rosssetion), but it will be speialized to the ase of symmetrial shielded offset oupled strip ondutors, as seen in Figure 1. Eah ondutor an be haraterized by a potential, V, a harge, Q, and a bounding surfae, S. In general, it is possible to onsider N of suh ondutors. A general rosssetion is seen in Figure 2. The entire assembly is bounded by a surfae 1. The surfae Z inludes the upper and lower ground planes and extends infinitely outward on the sides and enloses a region R that exludes the ondutor rosssetions. The eletri potential funtion, ^(x,y) within R must satisfy Laplae's equation:

20 14 (f)(x,y) o 1 ^2 Vi,Qi Li # R ^2'^2 # # u i 1 Figure 2. The general multiondutor transmission line struture and V (p(x,y) =0 in R 4) (x,y) (j) (x,y) = Vj on Sj = 0 on Z (4) It is possible to apply Green's Theorem in two dimensions to this struture. jjov^g-gv^olds' = () - G ^)djl' R boundary of R (5) where G(x,y; x',y*) is a two-dimensional Green's funtion

21 15 that satisfies the equation and V^G(x,y;x',y') = -6(x-x')ô(y-y') for x',y' in R G(x,y;x',y') = 0 on I (6) The parameter n is the outward normal to the region R and, hene, is direted outward along Z and in toward the ondutors on the surfaes 6 is the Dira delta funtion. The Green's funtion G(%,y;x',y') will yield the potential at the point (x,y) due to a longitudinal line harge of unit magnitude per unit length at the point (x',y'). Applying the boundary onditions on G redues Equation (5) to -*(x,y) = - G^)d&' (7) boundary of R The integration vanishes on S, leaving (8) The normal n, points into the S^'s. Sine on any partiular ondutor surfae Sj, tp is fixed, onstant and equal to Vj, then the left hand integral redues to

22 16 <p Si's 9n d ' (9) This is an expression of Gauss' Law determining the total eletri flux emanating from the losed surfaes Sj^. Sine the line soure is loated at the point (x',y'),whih is not loated within the S^'s^the net flux is S^'s an d&' = 0 (10) Therefore, Equation (8) redues to (f) (x,y) =.i (11) S^'s

23 17 III. ANALYSIS OF THE ZERO-THIKNESS ASE A. Analytial Tehnique A ommon physial situation involving the oupled stripline is to have a oupled pair of strip ondutors whose thiknesses are small relative to their widths. This situation an, and lassially has been, approximated by the idealized ase of oupled strip ondutors that have width but no thikness. Many of the onformai mapping solutions mentioned in hapter II utilize this approximation and it is one ontributing fator for their regions of greatest auray being the regions of widest strips. Applying this onept to Equation (11) by letting T-»-0 in Figure 1 results in the following, = rb (a) and -b G ( x, o, x ', - a ) _ a*(x',-o+g),ax, (12) (b) ^ G(x,-a;x',-a) *") _ 3(f> (x',-g+e) (j)^ and ^2 the potentials on the upper and lower ondutors.

24 18 respetively, as shown in Figure 1 and are therefore onstants e represents a very, very small distane from y=a. hanging the variable in the seond integral in both equations from x' to -x' and designating a-t as results in (j)^ = ^ (a) + G(x,y;-x\-o) ' (13) '^2~ (b) + G(x,-a;-x',-a) }dx' (14) There are two fundamental modes of exitation of the oupled line pair. These are referred to as even and odd modes of exitation. In the even mode, both strips are plaed at the same potential, i.e., volt. Referring to Figure 1, the following even symmetry exists; and p (x,y) = (J) (-x,-y) 9*(X'y) = _ ^<t> (-x,-y) (14) 9y 57^ ^ '

25 19 In the ase of odd mode exitation, ({)^=-(j)2=+l volt. Hene, the symmetry is odd, resulting in and 0(x,y) = -(!){-x,-y) 3ti> (x,y) 9y 9(1) (-x,~y) 3y (15) Applying Equations (14) and (15) to Equation (13) results in the following pair of equations, fb 1 = [G(x,a;x',a) + G(x,a;-x'-a)][^'^ ^ _ Set» (x',a ) ji 3y (a) (16) +1 = [G(x,a;x',a) ± G (x,-a;-x',-a)][^'^ ^ _ 3*(x',o )]dx' 9y (b) The + sign implies even exitation when the + sign is used and odd exitation when the - sign is used. The braketed term f3(t) (x',a ) _ 90(x',o ) 1 ^ 3y 3y when multiplied by permittivity, is atually an expression

26 20 of the harge distribution that exists on the infinitely thin onduting strip. The harge distribution will be different for the two types of exitation and so the expression may be subsripted, if desired, to indiate the mode of exitation. Sine symmetry is either even or odd, depending on the exitation, a knowledge of the harge distribution on one strip ondutor automatially supplies knowledge of the distribution on the other ondutor. It is only neessary, therefore, to solve the equations represented in Equation (16a). The even exitation apaitane, ^, may be defined as the ratio of harge stored on ondutor 1 to the potential of ondutor 1 and an be written as Qi. fb 3* (x',0 ) a* (x',0*) The odd exitation apaitane, ^, may be expressed as, Qi ft> 3*. ' L E is the permittivity of the dieletri material filling the struture. These mode apaitanes may be related to other ommon transmission line parameters by the following well-known equations:

27 21 (19) /ëy k Z o where k is the voltage oupling oeffiient and is the relative dieletri onstant. It is possible, then, to solve for the line apaitane, or impedane, under even and odd modes of exitation by solving the Fredholm type Equation (16a) for the harge distribution on the strip ondutors and then using the distribution to obtain apaitane or impedane. A suitable Green's funtion to use is given by hestnut [6] as. G(x,y;x',y') = - ln[ osh -^(x-xm-os ^(y-y*) 1 # ] (20) osh j(x-x')+os j(y+y') This is an expression akin to the potential at point (x,y) due to a line harge of unit strength and infinite length loated at (x,y). This hoie of a Green's funtion displays a disontinuity of the form

28 22 as G(x,y;x',y') -> -^In (x-x')^ + (y-y')^ (x,y) ^ (x',y') (21) B. Numerial Analysis Tehnique Equation (16a) annot be solved readily in a rigorous manner but may be approahed by an approximate numerial sheme utilizing the digital omputer. The following setion desribes a tehnique used by hestnut [6]. Equation (16a) is of the form 'b K(x,t)f(t)dt = g(x), a<x<b (22) a where x now represents the point (x,y) and t represents (x',y'). K(x,t) is the sum or differene of Green's funtions, depending on the exitation mode. It is desired to evaluate the integral in an approximate manner using a Gaussian quadrature formula. b N K(x,t)f(t)dt = E w. K(x,t.)f(t.) (23) a i=l ] J ^ Wj is an appropriate weighting funtion. Letting x=tj for eah of the N values of tj results in N linear equations for the N unknowns, f(tj), whih is the harge at point (x',y'). This proedure is idential to determining the potential at so alled field points (x,y) due to harges

29 23 loated at soure points (x',y'). The singularity ours when field and soure points oinide. Beause the kernel of the integral has a logarithmi singularity (see Equations (20) and (21)), hestnut [6] has modified the form of Equation (22) to obtain the following: fb -, [K(x,t) + ^ ln x-t ]f(t)dt a ^ a A fx) ln x-ti[f(t)-f(x)]dt (24) b ln x-t dt = g(x) = 1 a Applying the quadrature formula to the first two integrals results in the following ; N, Z^Wj[K(ti,tj) + ^ ln t^-tjl]f(tj) - A I w. [f(t.)-f (t.)]ln t.-tj (25) J=L J J ' J 1 fb - f(tj^) Injtj^-tldt = g(tj^) = 1 a where i = 1,2,3,...,N. The term in the first summation that results when i=j is interpreted as. = limit [K(t.,t) + ln t.-t ] (26) ^ t+t^ 1 ZTT 1

30 24 Therefore, Equation (25), when evaluated at eah of the N values of i, represents a matrix equation, Bf = g (27) where f and g are both olumn vetors of length N (g is a unit vetor). The oeffiient matrix B (not to be onfused with the groundplane spaing) has elements. r WjK{t^,tj), i^i 'ij =-\ ^ 1 ^ k^i 27r' 1=] (28) where the term e^ represents the integrable term in Equation (25). rb i = ln t^-t dt = a-b+(t^-a)ln(t^-a) + (b-tj^) In (b-t^) (29) The apaitane of the transmission line is therefore obtainable from Equations (17) and (18). e = o = E: N f (t)dt = Z w.f (t.) (a) a i=l G rb N fo(t)dt = E w^fo(t^) (b) a i=l (30)

31 25 The form of Gaussian quadrature used in the following: ^i = ^ ^i + ^ (31a) where is the i^^ zero of the Legendre polynomial, P^^x) and ^i = ^ ^ [P%'(Xi)]2 ^ (1-x.)^ 1 (31b) These values may be omputed diretly, or may be found tabulated up to twenty deimal plaes [1]. In summary. Equation (27) may be onstruted from the Green's funtion by use of Equation (28) and onsists of an N X N oeffiient matrix B, a 1 x N vetor g, and a 1 x N vetor of unknowns f(t^). f(t^) represents the values of eletri harge at eah of the N points of evaluation along the strip width and may be solved for by standard linear equation methods. apaitane and, therefore impedane, may be alulated from the harge distribution f. It should also be noted that many analyses of oupled striplines onsider the groundplane spaing, B(not to be onfused with the oeffiient matrix), to be unity. This is true of Shelton [27], Kammler [20], and others. hestnut [6], on the other hand, hose the groundplanes at y = +1 (see Figure 1) ausing B to have a value of 2. This onvention is followed throughout this work. The following

32 26 relations apply; ^ b-a B - (a)!!ç = b (b) (32) B 1 = 0 ( )

33 27 IV. ANALYSIS OF THE NONZERO THIKNESS ASE A. Analytial Tehnique Often stripline diretional ouplers are fabriated from metalli lad dieletri boards sandwihed together. It is quite ommon for opper lad boards to have a metalli layer up to 2 mils (.508mm) thik. In many ouplers, the dimensions of the strip width W, and the sandwih thikness B are suh that a one or two mil strip ondutor thikness beomes a signifiant fator in the analysis of the struture. The eletromagneti behavior of a diretional oupler an be shifted signifiantly from the behavior that ould be predited on the basis of idealized strip ondutors of zero thikness. In these ases it beomes neessary to analyze the oupled stripline onsidering the existene of nonzero thikness ondutors. This struture is the ase where T is greater than zero, as seen in Figure 1. A ondutor with thikness has four sides so that, in the same manner that Equation (11) beame Equation (16a) when T=0, Equation (11) beomes:

34 28 1= [G(x,y;x',a) + G(x,y;-x',-a)] dx' a ^ - j [G(x,y;x',a+T) + G(x,y;-x',-g-T) 3--^ dx' a+t, (33) [G(x,y;a,y') + G (x,y;-a,-y')] ^'^ ^dy' G dx [G(x,y;b,y') + G(x,y;-b,-y') J-^y' where the + sign indiates even or odd modes, respetively. The first integral in Equation (33) represents integration along the lower surfae of ondutor 1, as shown in Figure 1; the seond integral represents integration along the upper surfae; the third and fourth integrals are onerned with the left-hand and right-hand surfaes, respetively. The terms, and - when multiplied by permittivity, e, represent the harge distribution on the lower, upper, left, and right-hand surfaes of ondutor 1, respetively. These terms may be subsripted, if desired, to indiate the exitation type sine their values are mode dependent. The apaitanes of the oupled line pair are then given by f 9( ) e.o = J, r 1 for the even and odd modes. F^, is the path of integration

35 29 around the ross setion of ondutor 1. B. Numerial Analysis Tehnique The Green's funtion integral equation for strips with thikness (Equation (33)) has four integrals, eah of whih exhibit the form seen in Equation (22) fb K(X/t)f(t)dt = g(x), a<x<b (22) a As in the previous hapter, K(x,t) = [G(x,t) + G(x,-t)] and exhibits a logarithmi singularity as x+t. The method used in Equation (24) to overome the singularity must be applied to all four of the integrals in Equation (33). Eah integral may then be redued to a disrete sum by Gaussian quadrature. Equation (33) then beomes a matrix equation, ^lower ^lower ^ ^upper ^upper ^left ^left + Bright fright = ^(ti) - 1 (35) If N field points and N soure points are onsidered on eah side of the ondutor, then i = 1,2,...,4N. The B oeffiient matries have dimension 4N x N and the harge vetors are of length N. This matrix equation may be ombined into the form of Equation (27) Bf = g = 1 (27)

36 30 as is shown in Figure 3. The oeffiient matrix B is then of size 4N X 4N with elements b^j, where i refers to field points and j refers to soure points. With N points on any surfae, then if l i_<n and lf.j N, b^j refers to field and soure points on the lower surfae of ondutor 1. If l<i<n and N+l<j<2N, then b refers to field points on the lower surfae and soure points on the upper surfae, et. The harge vetor f has length 4N and ontains the harge distributions on all four sides of the ondutor. The individual elements b^j are obtainable from the following equation ; for l<i<4n and l_<j 4N, bij, N, 1.. 2l "k Inlti'tkl " 2tt ' where for l i 2N, k^i-mn, m an integer t^ = x^ + = t?, for lli<n t. = t^ for N+l<i<2N 1 i-n Wi=. [P'(x.)]^ = wf, for l<i<n (1-x^)" w. = w* for N+l<i<2N X X N

37 soure points lower surfae upper left N N+1 2N 3N fiel, points i lower right 4N f (t^) lower N N+1 upper ^ij upper 2N = 1 U) left 3N r f (tgq,) left right right 4N Figure 3. onstrution of the matrix equation Bf=g for nonzero thikness ondutors

38 32 is given in Equation (29) j^ is determined in Appendix A for 2N+l<i<4N, ^i " l^i ^ l<i<n t^ = t^+2n' 2N+l<i<3N ti = t^+sn' for 3N+l<i<4N wy = Ç = [P'(x.)]^, for l<i<n ^ 2 (i_x.)2 N 1 ^i " WÏ+2N' 2N+l<i<3N ^i " Wi+3N' 3N+l<i<4N e^ = -T+(t^-a)ln(t^-o) + (a+t-t^)in(a+t-t^) is determined in Appendix A. If the elements of the matrix B are alulated, then it is possible to solve for the harge distribution f by ordinary linear equation methods. The apaitane is then alulable from an appliation of Equation (34). =e,o - ^ b 9( ) a _(x,a) ^ ' b 3( )_ (x,a+t) e, o dx 9y I-a+T 3(() (a,y) ro+t 9(j) (b,y) " ' j, % ^ (37)

39 33 or 4N (38) where w^ and are obtainable from Equ. Modified Numerial Analysis Tehnique If the thikness of the strip ondutor in Figure 1 is small, but not insignifiant, it is possible to modify the tehnique desribed in the preeding setion to obtain an approximate solution in a slightly different manner. This modifiation is based on the premise that the harge distributions on the wide sides (upper and lower sides of ondutor 1) of very thin strip ondutors must be very lose to the harge distribution that exists on the idealized zero thikness strip with the same geometry (W/B, W^/B, S/B, et.). In fat, ideally one would expet the harge distribution on the zero thikness strip to be divided equally between upper and lower surfaes on the nonzero thikness, but very thin, onduting strip. Mathematially, f(t )I = Af(t ) 1 l_. 1 IT>0 T=0 (39) where t^ is a soure point on the wide side of the strip and A might be expeted to equal to one half. The f's for the zero thikness ase an be easily and

40 34 quikly omputed and by means of Equation (39) one half of the f vetor in Figure 3 may be determined. This redues Equation (27) to. Bf 1-Bf g' (ti) (40) 2N+l i<4n 2N+1<j<4N l<i<2n l<i<2n This matrix equation represents 2N lines equations for 2N unknowns and is somewhat faster to solve for the remaining f's than before. The remaining f's represent the neessary harge distribution on the narrow sides of the ondutor. This modified tehnique is atually more aurate for analyzing very thin onduting strips than is the omplete tehnique as desribed by hestnut [6]. This will be disussed in more detail in the next hapter.

41 35 V. RESULTS A. Zero-Thikness ase The solutions of Equation (11) (shown below for onveniene) that have been disussed involve Gaussian quadrature numerial integration. (11) The purpose of this work has been to solve Equation (11) in a region ontaining strip ondutors that have real physial dimensions, inluding thikness. In this situation, Equation (11) beomes Equation (33) in hapter IV. The solution of Equation (33) requires the redution of eah of the integrals by numerial quadrature. This results in a solution for (e ^), whih is the surfae harge on distribution on the strip ondutors. Quadrature is then required a seond time to find the total harge and the apaitane. The question is then raised about the auray of the numerial quadrature formulae in approximating the integrals. What error is involved? How may it be alulated? Gaussian quadrature is a very powerful numerial integration tehnique. An effet of its power is to obsure the amount of error involved in its use [21]. The traditional

42 36 formula for error evaluation requires a knowledge of the derivative of the integrand (N being the number of sample points in the quadrature formula). In the ase of the stripline solution, the funtion itself (- ^) is not known, muh dn less its derivative. Lanzos [21] has developed an upper bound estimate of the error that is muh more pratial to apply, but even this formula is less than satisfatory. The approah used, therefore, to test the auray of the omputational method used is to ompare resulting solutions with known rigorous solutions that exist in speifi ases. Exat rigorous solutions are available through onformai mapping proedures for the entered single ondutor stripline of infinite thinness. These solutions are well known in terms of ellipti integrals and may be found in Kammler [20], as well as elsewhere. 1. Single entered strip In order to obtain a solution for the single entered stripline, with whih to ompare to an exat solution, it is neessary to allow the strip overlap, W^, to equal the strip width, W. The strip spaing S must be set to zero. Zero thikness is assumed (T=0). This means that soure points exist only on one ondutor instead of two. The Green's funtion term K(x,t) that appears in Equation (22), and

43 37 similar equations, must be redued to a single terra rather than the sum or differene of two terms as in the double strip ase. Only one ondutor ontributes to the integral. This requires that ^2 = 0 in Appendix A. There an be no distintion made between even and odd modes, so there is but one transmission line apaitane to alulate. Implementing the analysis tehnique of hapter III for the zero-thikness single strip transmission line on the digital omputer yielded values of transmission line apaitane. It has been found, invariably, that if (N) is the value of apaitane omputed when onsidering N quadrature points along the strip width, then When (N^) < fnj) < djj) < (41) %1 < ^2 < H3 It has not been proven that (N) is a lower bound on the exat value of apaitane, but in all ases examined numerially, it has been the ase. This has also been observed by hestnut [6]. In order to improve upon the auray of the alulations of (N), the extrapolation formula developed by Kammler [20] and used by hestnut [6] was employed. This expression eliminates seond order errors of the type

44 38 Bl 32 n = -4 + "4 (42) N where 6^ and $2 ^re onstants. This extrapolation expression is ( N 3 "N ) (N 3 )+N 3 ^ (N2-Nj)(N3) {N3-N2)-N2^(N3-N3^) +N3^ (Ng-Ni) and is disussed in greater detail in Appendix B. Use of this extrapolation formula does improve the auray of the final result in that when (N,) < tn;) < (N,, < ^^trapolated <"> "1 < N2 < N3 A omputer program was devised to implement the analysis tehnique, along with extrapolation of the apaitane values. N was hosen to be 8, 12 and 24. Table 1 presents the results that show that the extrapolated apaitanes omputed were within a very small fration of a perent of the exat values that were obtained through onformai mapping. Strip widths varied from O.OOOIB to B. 2. oplanar zero-thikness oupled strips If one allows the parameters in Figure 1 to attain the values S=T=0 and < 0, then the resulting geometry is referred to as a oplanar pair of oupled strip ondutors.

45 39 Table 1. Results from analysis omputer program single strip stripline^ GAUSS for W/B /e, Exat /e, extrapolated S Ï differene ^differene = extrapolated -, Exat ^ VmK y ili^âo In this ase, eletromagneti oupling ours primarily at the inner edges of the two strips. oupling is therefore limited to weaker interations than are possible if the strips overlap. For instane, it is diffiult to build 3 db diretional ouplers of the oplanar type sine extremely small gaps, W^, are required. This type of oupler onstrution is popular, however, when pratial due to the simpliity of its sandwihed onstrution. The oplanar stripline is amenable to solution by onformai mapping and, hene, a rigorous solution exists.

46 40 This solution is given by ohn [9] and others in terms of omplete ellipti integrals of the first kind. hestnut [6] has published so alled exat values of even and odd mode apaitanes: g/e = q/s = for the speifi ase of W/B = 0.2 and W^/B = The units of g/e and ^/e are meters. A omputer program was written (referred to as program GAUSS in Appendix D) implementing the hestnut analytial tehniques desribed in hapter III. Again N was hosen as 8, 12, and 24 and the apaitanes were extrapolated by Equation (43). The resulting apaitanes that were alulated, when rounded to five signifiant figures,ompared exatly in every digit to the exat values seen above. In an attempt to generate additional exat values of apaitane for the oplanar strips, a omputer program was written utilizing subroutine DELl from the IBM Sientifi Subroutine Pakage (SSP) to evaluate the neessary omplete ellipti integrals of the first kind. Although the auray of the subroutine is predited to be of almost the double preision auray of the omputer, it was found that

47 41 the resulting apaitanes were approximately 0.1% lower than the above published values when W/B =0.2 and W^/B = Table 2 displays the exat values of even and odd mode apaitanes as alulated with SSP subroutine DELl versus the approximate values omputed by program GAUSS. A range of strip widths from W =.0001 B to B was examined while maintaining W^/B = The approximate values are a maximum of 0.07% above the exat values. The differenes between the exat values of voltage oupling oeffiients and the approximate values are not shown but are also quite small (about %). Regardless of the small disrepany in determining the exat values of apaitane, the approximate values obtained by implementing the analysis tehnique desribed in hapter III are very lose to the exat values of apaitane. The numerial approximation tehnique seems quite good. 3. Overlapped oupled strips of zero-thikness The major work that has been applied to overlapped oupled strip transmission lines in the past has been that of Shelton [27]. In general, Shelton's work was restrited to > (45) Two models were employed to desribe a wide range of oupling

48 Table 2. omparison of approximate and exat values of even and odd mode apaitanes of oplanar oupled strip ondutors (W^/B=-0.1)^ Exat Values Approximate Values W/B g/e q/e ^/E A% Q/E A% ^AQ. _ approximate value - exat value i nna A% - exat value 1 %'

49 43 values. For loose oupling: 2 (W-W ) I+S^ (46) For tight oupling: ^ ^ > 0.7 (47) b These two models are approximate and the solutions for loose oupling do not math the solutions for tight oupling at the boundary between the two models. This boundary region where the two models do not math is a region of maximum inauray within the limits presribed by Equation (45). Shelton developed his equations for stripline design. The resulting geometrial parameters W/B and W^/B are produed for given impedanes and oupling oeffiients k. As suh, it is awkward to ompare his results with the values omputed by the hestnut numerial tehnique desribed in hapter III. Mosko [22, in 29, 51-54] generated graphs from Shelton's equations showing values of W/B and W^/B versus the oupling oeffiient k for fixed values of harateristi impedane and dieletri onstant e^. Shelton has defined a region of solutions deemed the "best range of parameters." This region is defined by values of and S/B. Within this region, the imperfet

50 44 math between loose and tight oupling is minimized. Table 3 presents the parameters for three 50SÎ diretional ouplers with = 2.65 and S/B = 1/7. These ouplers lie outside, but very near the "best range of parameters." Theoretially, Shelton's values for these ouplers should be quite good. The three diretional ouplers were hosen for their onvenient values of k (0.5, 0.3, and 0.2) and the resulting values of W/B and W^/B were graphially obtained from Mosko's urves [in 29, pp ]. The ase of k = 0.5 represents tight oupling, the ase of k = 0.2 represents loose oupling, and the ase of k = 0.3 lies within the region of mismath. These values of W/B and W^/B were supplied as input data to program GAUSS for analysis. The alulated values of k are shown in Table 3. The differenes between these values of k from program GAUSS and the original values of 0.5, 0.3, and 0.2 is less than 1% in all ases, indiating good orrelation with Shelton's results within a region in whih his solutions are deemed aurate. The maximum perent differene ours at k = 0.3 whih lies within the region of model mismath. Some allowane must also be made for inherent error in translating graphial data. The design problem, rather than the analysis problem, is usually of more pratial interest. It is possible to

51 45 Table 3. Results of analysis program GAUSS for overlapped, oupled stripline ompared to Mosko's urves [22,in 29, pp ] s/b = 1/7, = 500, = 2.65% W^/B m ^ ^ k % differene differene k = '^GAUSS-'^MOSKO ^ MOSKO use the zero-thikness analysis sheme in an iterative omputer program and ahieve design results. Suh a program has been written and tested with some suess [24] utilizing the iteration funtion [28] f(xi) *i+l *i f [x^,x^_^]+f [%i-l'xi-2] where f[a,bl = for finding the roots of the equation f(x) = 0. f{x) was hosen as f(x) = - (49) desired extrapolated was obtained from given values of Z and k. Seemdesired ingly good results were obtained in many ases. Table 4 shows design results for 50ÎÎ diretional ouplers when

52 Table 4. Results of design program SYNW for oupled, offset stripline ompared to values omputed from Shelton's [27] equations^ Shelton's Values Design Values k W^/B W/B W^/B A% W/B A% o , 0003 design value - Shelton's value, TTT. 1 * XUU«, m = son, = 2.22, S/B =

53 47 = Eah oupler was designed using about 17 seonds of omputer time on an IBM 360/370 tandem ombination. A listing of the omputer program, SYNW, appears in Appendix D. This design sheme does not perform uniformly well for all ases. Great dépendane is plaed upon the initial values of width and overlap hosen with whih to begin iteration. Initial values poorly hosen an result in a design with the proper even mode apaitane but not the desired odd mode apaitane. The resulting and k may be in error. This diffiulty seems orretable, but at the expense of eonomy. The testing of two variables in the iteration proess, suh as g and ^, or and k, instead of merely would eliminate many of the diffiulties enountered. This has not been attempted. B. Nonzero Strip ondutor Thiknesses As disussed at the beginning of this hapter, onsideration of strip ondutors with real physial thiknesses involves the solution of Equation (33) in hapter IV. The resulting matrix equation. Equation (36), has dimensions 4N X 4N. The zero-thikness analysis involved an N x N matrix equation. N is the number of points used in the Gaussian quadrature deomposition of eah integral. For

54 48 eonomi reasons, N has been hosen to be smaller than it was in the zero-thikness ases. A FORTRAN omputer program, AST, (Listed in Appendix D), was written to implement the tehnique desribed by Equation (36). N was allowed to take on the values of 4, 8, and 12. The extrapolation proedure, used so suessfully in the zero-thikness ases, was used. Again, it has been invariably found that the resulting apaitanes are lower bounds in that Equation (44) is obeyed. 1. Single entered strip Many single strip transmission lines were examined on the basis of zero ondutor thikness in Table 1. Two of these ases were re-examined on the basis of finite thinness by omputer program GAST. The first ase dealt with a strip of width ratio W/B = This is a relatively narrow strip. The seond ase was a wider strip with a width ratio W/B = 0.3. The transmission line apaitanes were alulated by program GAST (some modifiations must be made on GAST to aommodate a single strip) for inreasingly small ondutor thiknesses. The results have been plotted in Figures 4 and 5. The exat rigorous solution (T=0) is marked for omparison. The apaitane was also omputed using ohn's [8] approximate tehnique. ohn onsiders his tehnique to have a maximum error of 1.2%. It an easily be

55 SINGLE STRIP LINE HESTNUT. AST O GAUM ta=0. S) & GAUM IA=0.1192S1 + OHN m EXAT SOLUTION. T=0 X t/ = ) W/B=0.001 îxïô^ b ' k b'\xio- ' k ' k"k'^xlo-^' b ' k' STRIP THIKNESS, T/B k'uiq-' k ' n Figure 4. apaitane of the single entered strip transmission line for a range of thiknesses

56 o (n~ SINGLE STRIP LINE HESTNUT. GfiST GAUM tr=0.5j GAUM [R= OHN EXAT SOLUTION. T [/s= ) W/B=0.30 I M I I o UJO n <P \tm~ Q_ Œ O in to ÎM ûl M 11 III 1x10 I M I I III I I I 11 III I I I I I III,1,1 I Mini TTl STRIP THIKNESS, T/B Figure 5. apaitane of the single entered strip transmission line for a range of thiknesses

57 51 seen from either graph that the results from the hestnut [6] method of program AST do not onverge asymptotially to the rigorous zero-thikness solution. The problem is intensified for the wider strip. 2. Modifiation of the hestnut method Although the explanation for this great error as T^O is not understood with great ertainty, it seems apparent that the problem lies in the numerial analysis proess. As T beomes small, the range of integration beomes small in the two integrals that deal with the thin sides of the ondutor. There seems to be signifiant error assoiated with allowing the N Gaussian quadrature points to beome too losely paked. The error ould lie in the omputation of the elements B^j of the B matrix when soure and field points beome too losely spaed. Physially, one would expet the harge density on the strip to peak higher and higher at the orners as the thikness is redued. It is well-known that harge tends to aumulate in greater densities at orners, edges, or sharp regions on a ondutor. This sort of behavior is onsistent with the harge distributions alulated by program AST as T is made smaller and smaller. A threshold is reahed, however, beyond whih the harge distribution behaves in an unrealisti manner. In this region, the error in the apaitane alulation beomes