LI E) R.ARY 
 
 OF THE 
 
 U N IVLR.SITY 
 
 Of ILLINOIS 
 
 510,84 
 
 no. 156-163 
 c p 6 2 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/characteristicim157guck 
 
lo. 84^ 
 
 :op. 2^ 
 
 DIGITAL COMPUTER LABORATORY 
 UNIVERSITY OF ILLINOIS 
 URBANA^ ILLINOIS 
 
 REPORT NO, 157 
 
 CHARACTERISTIC IMPEDANCES OF 
 GENERALIZED STRIP-TRANSMISSION LINES 
 
 H. Guckel 
 
 December 11 _, 19^3 
 
 This work was supported in part "by the 
 Office of Naval Research under Contract No, Nonr-l83U(l5 ) 
 
«. <• p ^ 
 
 ?r?LA y 
 
 A- 
 
 INTRODUCTION 
 
 In order to study the effects of layout on high-speed circuits it is 
 necessary to be able to calculate the characteristic impedance of a rectangular 
 conductor placed parallel to, but spaced arbitrarily from; two ground planes. 
 The situation is described in Fig. 1. 
 
 Figure 1. Generalized Strip Line 
 
 The difficulties withthis type of geometry^ as far as solutions to 
 Maxwell's equations are concerned^ is well known. However, approximate methods 
 may be used to calculate an upper and a lower bounds to the characteristic 
 impedance c The usefulness of the analysis is thus mieasurable » 
 
 -1- 
 
1. THE LOWER BOUND 
 
 cp = 
 
 y = d r 
 
 y = a + t — 
 
 © 
 
 
 y = a p- 
 
 © 
 
 © 
 
 ir n t 
 
 
 cp = 
 
 J, _ u » 
 
 
 
 X = 
 
 X = i 
 
 Figure 2. Boundary Values and Definition of Regions 
 
 The potentials in the three regions are given by; 
 
 V^y V 1 nKx . nny 
 
 0'' La cosh — — sm ■ — - 
 
 1 a n=l 
 
 (1.1) 
 
 0^"^ '' V .- , n«x . nJt(d - y) 
 
 cp^ = ■ — ^ — + L b cosh -T— sm — ^— r — '^ 
 
 2 b ^ n b b 
 n=l 
 
 (1.2) 
 
 cp^ = E c sm -~ e 
 3 T n d 
 
 -^ n=l 
 
 n?tx 
 
 d 
 
 (1.3) 
 
 since the potenials satisfy Laplace's equation v cp = and the 
 boundary conditions. They are the solutions to the problem of Fig, 2, The 
 continuity conditions are 
 
 ^2(^^y) = ^2^^'^^ a + t < y < d 
 
 (l.i^) 
 
 cp^(£,y) = cp2(i,y) 
 
 < y < a 
 
 -2- 
 
and 
 
 dcp dcp 
 
 5^ (^>y) = ^ (^.y) a + t < y < d 
 
 dcp dcp 
 
 5^ (^.y) = ^ (^.y) 0<y<a 
 
 (1.5) 
 
 These equations may be used to evaluate the coefficients in the potentials. 
 However^ this method will not be used here. Instead an argument based on energy 
 considerations will be used. The electrostatic energy between two conducting 
 surfaces is given by Eq. 'lo6). 
 
 "e '■ I ^ 
 
 dcp\2 ^ f^Vs 
 
 ^1 \^y/ 
 
 C - capacity per unit length 
 
 Vq = cp^ - 9^ 
 
 dxdy = I CV^ (1.6) 
 
 If the first-order variation in W due to a first-order variation in the 
 
 e 
 
 functional form cp^ where cp satisfies the boundary conditions^ is computed^ it 
 is found that the variation vanishes if cp satisfies Laplace's equation^ a 
 condition which is already satisfied. Then 
 
 c 1 J/ ^t^ ° ^t^'^^y 
 
 r — ' L 1 
 
 £ 
 
 (1.7) 
 
 The integral is positive so that the stationary value is a minimiim. It follows 
 that 
 
 Zq^Zq""^"^* (1.8) 
 
 The total energy is the sum of the energies in the separate regions 
 
 -3- 
 
n-l 
 
 Mil \2 
 
 . , nnx . n:^yV2 
 inh sin — '^ 1 
 
 a a 
 
 cbcdy 
 
 (1.9) 
 
 ■V, 
 
 / n-^N / , nTtx nny\ 
 
 - + I a — I cosh cos — ^ 
 
 a Vna/V a a/ 
 
 dxdy 
 
 where 
 
 < X < i < y < a 
 
 This la ly be integrated to yield; 
 
 r^i 
 
 w ^ - e 
 
 1 2 
 
 r L an sinh — 
 
 ^ T n a 
 
 n=l 
 
 (1«10) 
 
 The energy in region 2 may he obtained from Eq, (1,10) by replacing a by b, 
 
 2 2 I b 
 
 r Z-. b n smh — r— 
 u , n b 
 
 n-l 
 
 (1,11) 
 
 In region 3 
 
 a p 00 
 
 W. = 
 
 l^l(^<J J {- 
 
 2nTCx 
 
 2 nny . 2 nny\ d 
 
 — — + sm --^ £ 
 d d / 
 
 dxdy 
 
 _ 2nn£\ 
 1 / nn „2 d \ 
 
 (1.12) 
 
 The total energy is thus 
 
 T 2 
 
 ,2g 9g / 2njxi? 
 
 ^^ V Ti ; / 2 . ^ 2n7T^ P . , 2nn^ o - -"d" 
 
 -T— + ~ + r Z: n a smh — — - + b smn — r~ + 2C € 
 
 _b a4__\n a n b n 
 
 (1^13) 
 
 -k- 
 
The coefficients may be evaluated in terms of the unknown potential distribution 
 at X - i. The distribution is assumed to have the form: 
 
 \ I ^ g(y) 
 
 G(y) - V, 
 
 
 
 < y < a 
 C < y < t 
 
 (1»1^) 
 
 Vo H^ - ^(^) 
 
 d - b < y < d 
 
 The equating of Eq ^ 1 l^J- ' to respective potentials at x. = H yields 
 
 , nrti 2 r ... nny , 
 
 a cosh -^ ■■- — / gi,y,' sm ~- dy 
 
 n a a J °^'' a 
 
 
 
 (1„15) 
 
 C € 
 n 
 
 d 
 
 G(y) sin ^ dy 
 
 (1.16) 
 
 ■u 1 nrti 
 
 b cosh —rr- 
 
 n d 
 
 f(y) sin 
 
 nn(d - y. 
 
 dy 
 
 d-b 
 
 (1.17) 
 
 The approximations are introduced at this point. It will be assumed that to a 
 first approximation g(y; -■ f(y> == Oo In this case only c has a nonzero value 
 
 c G 
 n 
 
 njtf 
 d 
 
 a-b 
 
 „ , nny 
 V„ sm -~ dy t 
 d "^ 
 
 ,, y . nny 
 V^ - sm —~- dy + 
 C a d 
 
 .., d - y . nny ^ 
 '^0 b ^^^ d ^^ 
 
 d-b 
 
 This may be integrated by standard methods to yield 
 
 {i.is;. 
 
 nniJ 
 
 2V^d / fA^\A 
 
 / . nna a . njr - d - b j \ d 
 
 ^rr l^^^ "d" ^ b ""^ "^d — ) ^ 
 
 n j: a \ / 
 
 (1„19) 
 
 -5= 
 
The total energy becomes then; 
 
 ^T = I ^^ 
 
 H H 2d V 1 ^ . nna 
 
 I b a 3 2 - 3 \ d b 
 *- n a n^l n 
 
 a . nTi(d - b) 
 
 + r- sin — ^' — 
 
 (1,20) 
 
 This is the total energy in the three regions of Figo 2, The total energy in 
 the entire cross-section is thus twice that amount. 
 
 Hence 
 
 ^W„ 
 
 C = 
 
 V? 
 
 7L _ '0 
 ^0 ~ 4W, 
 
 tie 
 
 or: 
 
 ^^ 
 
 lb a 32 , 
 \ 71 a n-1 n 
 
 njta 
 d 
 
 L — r: I sm —rr- + (-1; — sm 
 
 vn+1 a . mtb\^ 
 b ^^^ ~d-j 
 
 (1.21) 
 
 where 
 
 £ =- €qK 
 
 ''■■R 
 
 -6- 
 
Two limiting cases are of interest » 
 (a) Symmetric." a = b 
 
 Th = 5 -— (l„21a) 
 
 ° , /^ i.d" Z 1.2 nna\ 
 
 \ n a n=l;,3^5,o , „ n / 
 
 (b) Thin center conductors t-O^ d=a+b 
 
 ZL ^ -— ^0-^ ________ (1^21^) 
 
 ^ i a 2d r^ 1 . 2 n7ta\ 
 
 \ a b 3 2 2 3 d / 
 
 \ nab n^l n / 
 
 -7- 
 
2. THE UPPER BOUND 
 
 It is possible to calculate an upper bound of Z„ by making use of 
 the Green's function approach. 
 
 The removal of a charged conductor from a system of conductors is 
 permitted if the conductor is replaced by an equivalent surface charge., 
 
 dcp 
 
 p ^ eE - -e ;t^ 
 
 8 n On 
 
 The potential function which has to be found is then the solution of 
 
 V^cp . ^- i p(x',y') 
 
 where,, for the problem at hand; 
 
 cp(S^) = pCS^) - p(xSy') 
 
 This problem may now be solved in terms of a unit line charge at position 
 (x'^y'} in the presence of So Thus^, if 
 
 ^cp(x,y; - ^ i &(x - x')5(y - y' ) cp(s^) = 
 
 has the solution G(x^yjx* ^y' )^ then the potential due to the charge distribution 
 PixSy') is 
 
 cp^x.y) - j) J(x,y|x',y' )p(xSy')di 
 
 ^2 
 
 The charge distribution is determined from the following equation^ where use 
 was made of the fact that ^\x{S^).y[S )) = V . 
 
 Vq - ji G(x(S2),y(S2)|x%y') p(xSy')dii 
 
 ^^2 
 
 -8- 
 
but 
 
 VqQ-Vq J p(x,y)di5 = |- = J^G(x(S2),y(S2)|x',y')p(x,y)p(x',y')didi 
 
 Sg S2 Sg 
 
 Hence s 
 
 f j G(x,y!x^y' )p(x,y)p(x ' , y' )d/.di 
 
 1 ^2 ^2 
 
 oi p(x,y)di 
 
 P 
 
 S£ 
 
 This expression is stationary for arbitrary first-order changes in the function 
 
 ^0' 
 
 p(x;,y)= It is a minimum and therefore yields an upper bound on Z^„ Hence 
 
 U ^ exact 
 - 
 
 .In order to use this method the Green's function for a line charge 
 between conducting ground planes must be available „ It is the solution of 
 
 ^ + ~ = - - 8(x - X' )&(y - y' ) G(x,0) = G(x,d) - (2,l) 
 dx- d/ ^ 
 
 Since^ ifxj^x% y r y\ G must satisfy Laplace's equation,, which is separable 
 in rectangular coordinates^, G must have the forms 
 
 G = Z f^(x)f^(x' )g^(y)g^(y' ) (2,2) 
 
 n 
 
 The y part may be determined by expanding the function &(y - y')° 
 
 ~9- 
 
6(y- y') = Ea„ sin -^ (2.3) 
 
 n d 
 
 n 
 
 Hy - y' ) = -J E sin —J- sm -^ 
 n 
 
 Hence^ if g (y)g (y' ) = sin -r^- sin -^^i> tbe differential equation for the 
 X dependence iDecomes:; 
 
 f U. X yX ) I \ '> r\ \ 
 
 Since 3 is a potential it Is con'inuous at x - x'^, y - y' o The discontinuity 
 in the derivative is found by integrating Eq, (2„4) about x' + A„ This yields 
 
 df \^) df (x") 
 n^ '^ n ■ 
 
 dx dx f (x' jcd 
 
 n^ ' 
 
 Since G must remain finite as x -■* <» suitable solutions are 
 
 njtx 
 
 a e X < X 
 
 n 
 
 (2o5) 
 
 f Jx) = (2„6) 
 
 mtd 
 
 b e '^' X > x' 
 n 
 
 The coefficients may be obtained from Eq^ (2o5) and the continuity of 3o This 
 yields^ after some manipulations : 
 
 V 1 . nny . njty d^ '^ ^ t 
 
 •—r L - sm — ~ sm ~- € x < x 
 
 Tte n d d 
 
 n 
 
 G- (2o7) 
 
 -lEisin^^sin^e^ ■ x.>x' 
 
 ne n n d d 
 
 -10- 
 
As a first approximation it will be assumed that the function p is 
 given by: 
 
 p(xj,y) = constant (2.8) 
 
 Since the quantity — depends only on the functional form of p it is permissible 
 
 ^ 1 
 
 to let the constant be unity,, The denominator of - becomes thus simply:- 
 
 p(x,y)di)' = k(t + Sif (2.9) 
 
 ^2 
 
 The problem is then reduced to the calculation of 
 
 J= i i G(x,y|x%y' )dfd<^-' (2„10) 
 
 ^2 ^2 
 
 The primed integration yields^ 
 
 nit . , X 
 
 nny / / . nna . njTi.d - b)\ d- \ , 
 
 1 d J \ d d / 
 
 -£ ' 
 
 + sxn--- sm -^ . sm —i-^— U dx ' 
 
 d-b , /^vx-^) - ^^'x+^A 
 
 sm -^ sm —^ ( e + e )dy' (2.11) 
 
 a 
 
 This may be integrated to yield 
 
 -11- 
 
nn^ 
 
 kd , mty , nn(2a + t) I njtt 
 
 J, = — sin —7^ sin — ^— ^r^ *■ 
 
 1 nn 
 
 2d 
 
 , , ^ d , nnx 
 
 cos — - 1 - e cosh — 
 
 njtt d , nnx 
 
 + sin -7- J e cosh — r- 
 
 2d d 
 
 (2.12) 
 
 Equation (2>,12) is next integrated in the xy planes 
 
 +£ 
 
 d-b 
 
 J = 
 
 -£ 
 
 J ^^x_,a)dx - / J (x^a + t )dx + 
 
 a 
 
 d-b 
 
 The final answer was found to be 
 
 (2ol3) 
 
 2 
 n n 
 
 Sd" . 2 nit, 2a -^ t '/ 2inn 2 n«t . nitt nut 
 
 J - ' ,_^ sm ~--;^^;— — - 1 --.:;— cos -;;^ + sm —T~ - cos 
 
 2d L d 
 
 ?d 
 
 2n_ni 
 
 ^ . njtt\^ d 
 
 1 - sm ~r"i& 
 
 (2. 14) 
 
 The upper bound of Z_ is then given bys 
 
 ^0= ^. 1~37~~:7T2 "^^ 2d 
 
 2 nKr2a + t)\ / 2in 2 nnt 
 
 n=l y^Tf^it + 2i') 
 
 -2: ^°^^ 2d 
 n d 
 
 2njti 
 
 1 / . njtt 
 + -3 (^sxn — 
 
 - COS 
 
 njtt 
 
 n' \ 
 
 njtt \ ^ 
 .m -- € 
 
 (2.15) 
 
 The limiting case of a thin-center conductor is described by: 
 
 ^0 
 
 o 
 d'^Z 
 
 ^ . 2 nita 
 
 sm — : 
 
 y 
 
 n \ n 3«2 3 ""' d 
 n^l \2^n^ n-^ 
 
 2n«i 
 
 1 + € 
 
 2mt^ 
 d 
 
 (2.16) 
 
 -12- 
 
Equations (l.21b) and (2ol6) have been evaluated for different 
 
 geometries,. The answers are satisfactory for many applications.. The arithmetic 
 
 T y^ 
 means of Z and Z is within eight per cent of the actual value for all 
 
 geometries tested. For the low impedance lines of interest here this deviation 
 
 is halved. This results in designable line impedances with uncertainties of 
 
 the saire order as those in the resistors. 
 
 -13- 
 
.300 
 
 t 
 
 .200 
 
 .100 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 2-2' 
 3-3' 
 
 w/d = .250 
 
 w/d - .375 
 w/d = .500 
 
 w/d = .625 
 
 1 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-5' 
 
 
 
 
 
 
 
 
 
 
 
 
 c 
 
 1 ] 
 
 - • 3T7 
 
 ' 1 ,., 
 
 
 a/d 
 
 Figure 3. Upper and Lover Bounds of Characteristic Impedance 
 for Thin-Center Conductor. 
 
 „li,_ 
 
m 2 s nm