parameters of microstrip transmission lines

7/30/2019 Parameters of Microstrip Transmission Lines

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-16, NO. 12, DECEMBER 1.968 10212) An effective cutoff frequency of 400 GHz is achieved

for an input power of +17 dBm at 48 GHz with varactordiodes whose cutoff frequency is around 150 GHz.3) This ratio decreases rather slowly when the input power

increases, showing the possibility of using these switches forhigher input-power levels.4) The switching time is less than 1 ns, which was one of

the basic requirements.5) The operational bandwidth is relatively wide.

[4][5]

[6]

[7]

[8]ACKNOWLEDGMENT

The author is grateful to Drs. T. Sekimoto, A, Saeki, andH. Kaneko for their encouragement during the course ofthe work.

REFERENCES[1] S. E. Miller, Waveguide as a communication medium, Bel l sy, r.

Tech. J., vol. 33, pp. 12091265, November 1954.[2] W. M. Hubbard, J. E. Goell, W. D. Warters, R. D. Standley, G.D. Mandeville, T. P. Lee, R. C. Shaw, and P. L. Clouser, Asol id-state regenerat ive repeater for guided mil limeter-wave com-munication systems: Bell Sys. Tech. J., vol. 46, pp. 1977-2018,November 1967.[3] S. Kita and S. Seki, Millimeter wave pulse generation by multi-

[9]

[10]

[11]

[12]

[13]

plier, Proc. IEEE (Letters), vol. 54, pp. 71-7:2, January 1966.C. A, Burrus, Mil li rnicrosecond pulses in the millimeter waveregion, Rev. Sci. Instr ., vol. 28, pp. 1062-1065, December 1957.F. Ishihara, T. Kanehori, and K. Kondo, Generation of milli-meter-wave pulses by 1N26 and their reshaping, 1966 ZECEJNat1 Conv. Rec. (Tokyo), p. 395, November 1966.R. V. Garver, E. G. Spencer, and R. C. L&raw, High-speedmicrowave switching of semiconductors, J. AppI. Phys., vol. 28,pp. 1336-1338, November 1957.M. R. Millet, Microwave switching by crystal diodes, IRETrans. Microwave Theory and Techniques, vol. MIT-6, pp. 284-290, July 1958.R. V. Garver, Theory of TEM diode switching, IRE Trans.Microwave Theory and Techniques, vol. MTT-9, pp. 224238,May 1961.R. V. Garver, Fundamental limitations in RF switching usingsemiconductor diodes, Proc. IEEE (Letters), vol. 52, pp. 13821383, November 1964.T. Misawa, Negative res is tance in p-n junction under avalanchebreakdown conditions, part I: IEEE Trans. Electron Devices,vol. ED-13, pp. 137143, January 1966.R. W. Dawson, and L. P. Marinaccio, High-Q microwave filtersemploying IMPATT active elements, IEEE Trans. MicrowaoeTheory and Techniques, vol. MTT-15, pp. 272-273, April 1967.S. Kita, Millimeter wave CW oscillation using silver-bondedgermanium diode, Proc. IEEE (Letters), vol. 54, p. 1992, Decem-ber 1966.S. Sugimoto, MiKlmeter-wave switching by avalanching varactordiodesj Proc. ZEEE (Letters), vol. 55, pp. 2068-2069, November1967.

Parameters of Microstrip Transmission Lines andof Coupled Pairs of Microstrip LinesTHOMAS G. BRYANT, STUDENT MEMBER, IEEE, AND JERALD A.

WEISS, SENIOR MEMBER, IEEEAbstractA theoretical analysis is presented of microwave propaga-

tion on rnicrostrip, with particular reference to the case of coupled pa irsof microstrip lines. Data on this type of transmiss ion line are needed forthe design of directional couplers, f il ters, and other components in microw-ave integrated ci rcoi ts. The inhomogeneous medium, consis ting of thedielectric substrate and the vacuum above it, is treated in a rigorous man-ner through the use of a dielectric Greens function wldch expressesthe disconthurity of the fields at the dielecti lc-vacuum interface. Resnltsare presented in graphical form for substrate dielectric constants of 1, 9,and 16, and a range of values of width and spacing of the sti lps. Numericaltables for these and other cases are also available. The tables presentcapacitance, character istic impedance, and velocity of propagation of the

Manuscript received May 28, 1968; revised August 21, 1968. Thiswork was performed under the sponsorship of the Array Radars Groupat M. LT. Lincoln Laboratory, in part through contracts with theUniversity of Maine and Worcester Polytechnic Insti tute.T. G. Bryant was with the University of Maine, Orono, Me. He isnow with the Array Radars Group, M.I.T. Lincoln Laboratory, Lexing-ton, Mass. (Operated with support from the U. S. Advanced ResearchProjects Agency.)J . A. Weiss is with the Department of Physics , Worcester Polytech-nic Institute, Worcester, Mass., and the Array Radars Group, M.I.T.Lincoln Laboratory , Lexington, Mass.

even and odd normal modes. The method lends itself to the treatment ofother geometries which are of practica l interest, such as th ick s trips ,presence of au unsymmetrically located upper ground plane, etc .

1. INTRODUCTION

FOR GUIDANCE in the design of integrated micro-wave circuit components, data are required on theparameters of symmetrical coupled pairs of microstrip

transmission lines. The parameters needed to characterizethis structure are the characteristic impedances and velocitiesof propagation of the two normal modes. In addition, forcertain purposes such as investigation of spurious coupling,peak power capability, and gyromagnetic interaction (in thecase of nonreciprocal substrate materials), information isalso required on the RF field configuration.The term microstrip is a nickname for a microwave

circuit configuration which is constructed lDy printed-circuittechniquesmodified where necessary to reduce loss, reflec-tions, and spurious coupling, but retaining advantages insize, simplicity, reliability, and cost which such productiontechniques afford. Microstrip shares some of the trouble-


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1022 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, DECEMBER 1968

some properties of dispersive waveguide, in that conductordimensions influence not only characteristic impedance,but also the velocity of propagation; also in that, on struc-tures which support propagation in more than one mode, thevelocities of the modes are in general unequal.We present a solution of the microstrip problem in the

quasi-static limit; i.e., for the frequency range in whichpropagation may be regarded as approximately TEM. Sucha solution is valid in the range extending into the low giga-hertz region, in the case of microstrip dimensions and sub-strate materials frequently used in integrated microwavecircuit technology. For higher frequencies, the solution pro-vides design guidance and can serve also as a basis for solu-tion of the full propagation problem. The method developedfor this work involves the determination of a dielectricGreens function which characterizes the effect on the fieldconfiguration due to the presence of the dielectric-vacuuminterface. Such a function has potentially wide applicabilityto a variety of problems involving inhomogeneous dielectricmedia.The physical construction of a coupled pair of microstrip

lines is shown in Fig. 1. The configuration is conventionallyspecified by the parameters W/H and S/H, together withK, where W is the strip width, His the substrate height, andS is the spacing between adjacent edges of the strips; K isthe relative dielectric permittivity of the substrate material.The problem of a single microstrip may be identified withthat of coupled strips in the limit S/H~ UJ. In the present re-port the strip thickness T is assumed to be negligibly small.This assumption is not intrinsic to the method, however, andapplication to the case of nonzero T/H will be presented ina future publication.From the symmetry of the structure, the normal modes of

propagation on a pair of parallel strips of equal dimensionshave even and odd symmetry, respectively, with respect toreflection in a central bisecting plane. In the quasi-staticlimit, where propagation is approximately TEM, the char-acteristic impedances of the two modes can be determinedfrom their respective dc capacitances and low-frequencyvelocities. The difference in impedances becomes large asthe coupling between the strips is increased by reducing thespacing between them. The analogous problem of coupledstrips in the case of balanced stripline has been treated byCohn [1]. The microstrip problem presents additional diffi-culties due to the lower symmetry and the presence of thedielectric-vacuum boundary. Following the first thoroughtreatment by Wheeler [2], there have been calculations ofcapacitance, velocity, and impedance of single strips, andalso some work on coupled pairs, using various approxi-mate methods, by Cristal [3], Welters and Clar [4], Polickyand Stover [5], and no doubt by others. During the courseof this study, we were apprised by a paper by Silvester [6]which treats the dielectricvacuum boundary by means of aGreens function, similar in some respects to that employedin the present paper. A recent paper by Yamashita andMittra [7] presents an analysis based on the variational prin-ciple.

DIELECTRICSUBSTRATE%

MICROSTRIPS

I Illt~w++-w+ 1 GROUND PLANEFig. 1. A coupled pair of inicrostrip transmission lines.

II. THE DIELECTRIC GREENS FUNCTIONWe perform a rigorous solution of the electrostatic prob-

lem; namely, the determination of capacitance for a singlestrip, and for coupled strips, on a dielectric substrate. Wethen apply this solution to the propagation problem, incor-porating the quasi-static approximation in which the longi-tudinal components of the fields are neglected.

Consider an arrangement of conducting boundaries,denoted collectively by S, and a harmonic function (poten-tial) %(r) which satisfies given Dirichlet boundary condi-tions on S, and thus solves the electrostatic problem for thecase of a vacuum or homogeneous dielectric medium. Forthe microstrip problem, S is simply the conducting groundplane, assumed to be infinite in extent. The potential iscreated by an elementary source, which we might take to bea uniformly charged line lying parallel to the plane at theposition ro. A more suitable source is a narrow elementarystrip, uniforndy charged, since its potential is finite every-where; the element actually used in this work is of that form.Dielectric material, of relative permittivity K, is now intro-

duced into this field, so as to occupy a region bounded by thesurface T. In the following, we shall term interface thatpart of T which does not coincide with S. We seek the solu-tion @(r) of the new electrostatic problem in the presence ofthe inhomogeneous dielectric medium,Suppose we were to assume, as a first approximation, that

the solution @o(r) of the vacuum problem solves the dielec-tric problem also. This function is, indeed, the unique solu-tionnot of the desired problem, but of a related one:namely, that in which, in addition to the charge distributionsrequired to establish the specified values of potential on S,there is also a certain distribution of free charge aO(rT) onthe dielectricvacuum interface. For, since @oand grad @oare continuous at T, the surface divergence of dielectricdisplacement D is

CO(K l)n~. grad @O= ao(r~) (1)where n~ is the outward unit normal vector (directed fromT into the vacuum).Intuitively, we may imagine placing a compensating dis-

tribution of bound charge u(r~) on the interface so as tocancel ui)(rT). Let @(r) denote the new potential, includingthe effect of this additional charge. Such bound charge,corresponding to a surface divergence of electric field E,


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BRYANT AND WEISS : PARAMETERS OF MICROSTRIP TRANSMISSION LINES 1023creates a discontinuity in E such that

~an~. [grad @(rT_) grad @(r~+) ] = a(rT) (2)where r~+ and rT_ refer to the + (vacuum) and (dielectric)sides of T, respectively.Assuming that the correct bound charge distribution u(rT)

has been found, its effect is to annihilate the free charge dis-tribution uo(r~):

conT. [K grad @(rT-) - grad @(rT+)] = O. (3)To evaluate the contribution of u(r~) to the potential, wemay employ the (vacuum) Greens function G(r I r) appro-priate to the conducting boundary S. Let G satisfy

vG(r I r) = 4d(r r), (4)with the boundary condition

G(r~ ] r) = O. (5)Then the total potential @(r/ ro) due to the source at ro,including the contribution due to u(rT), is

1@(r I ro) = @o(r) + s u(r~,)G(r I r~)da. (6)4~~0 TAs will be shown, (6) is the dielectric Greens function whichplays the role of kernel for the solution of the quasi-staticrnicrostrip problem. In terms of this potential, the boundarycondition (3) becomes

1uO(rT) + s1r Jcr(rT)nT o [~ grad G(rT I rT) (7)

grad G(rT+ [ rT) ]da = O.This constitutes an integral equation for the unknown boundcharge distribution u(r~). By making use of the propertiesof G(rl r) we can put the relation (7) into a form suitablefor computation.We decompose the factor in brackets in (7) according to

K grad G(rT_ I rT) grad G(rT+ \ rr)= *(K+ 1) [grad G(r~- I rT) grad G(r~+ \ r~)] (8)

+ *(K 1) [grad G(rr- I r~) i- grad G(rT+ I rT)].In substituting in (7) according to (8), we note that the twoterms on the right side of (8) have the following interpreta-tions. The first term (the one with coefficient K+ 1) yields ineffect an integral of the Green s-function field over the sur-face of a small region enclosing an element of T; with theaid of Gausss theorem we replace it by a volume integral ofV2G which, according to (4), becomes 4d(r-r). Thesecond term (with coefficient K 1) represents an average ofthe Greens-function fields on the two sides of T. It is there-fore continuous at T; in fact, it is the part of the field con-tributed by the images of the elementary source ti(r-r);or, put another way, it is the field due to the charge distribu-tions induced on S by this source. We denote by GI(r] r)this image part of the Greens function.With these interpretations, (7) becomes

. grad GI(rT \ r~)da = O.Equation (9) is an inhomogeneous Fredholms integral equa-tion of the second kind [8] for the unknown bound chargedistribution u(rT), in terms of the known initial free chargedistribution u@T), given by (l), and the image part of theGreens function for S, which is known, or in any case isderivable with more or less difficulty depending on the shapeof the conducting boundary S. For the case of the flat micro-strip ground plane, GI takes a simple form. Of the variousmethods available for solution of such equations, one whichwe have used successfully in the present work is an iterativemethod based on a slight rearrangement of (9):

2 I. K-lu(r~) = K+l uo(r~) s7r K+l ~(rT)nT (lo).grad GI(I-T I r~)da

As a first approximation, set the function tT(rT) appearingunder the integral sign equal to zero; (10) gives the secondapproximation as simply 2/(K+ 1) times the negative of theinitial free charge uO.Using this in the integral, we obtainthe third approximation of u, and so on. The method is notguaranteed to converge under all conditions, but in any casewe can directly determine the success of the process by eval-uating the left side of (9) after each stage of iteration. Forthe microstrip problem contemplated here, convergence hasbeen shown to take place without incident. (In experimentswith other surfaces S, we have acquired some experiencewhich enables us often to anticipate convergence difficulties.)With u known, the desired dielectric Greens function

@(r (ro) may now be generated according to (6). An illustra-tion of the result is shown in Fig. 2. The graph refers tosubstrate K= 16; it shows the potential due to a strip ele-ment of W/H= 0.025 carrying unit charge. Also shown inthe figure are the distributions of bound charge u and ofsurface charge on the ground plane. The vacuum imageGreens function G1 used fol this calculation is obtained byelementary methods. The potential due to an element ofwidth 2& located at height His() (z $) -+ (y H)G(z, vIO, H)=+--f--l in (z $) -+ (y + H)

lx()

(Z+ g)+ (y H) +1 ln(x+g)~+(v+-H)~T~ (11)y+H

(X+i xt

+7 arc tan )arc tan Y+H u+HuH

(X+t x.$_ arc tan )arc tan .f yH yH

For use in (10), we require the component of field normal toT, due to the image part of (1 1):


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1024 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, DECEMBER 1968

i580 X 1 0-4L POTENTIAL AT y/H = i

\ t BOUND CHARGE AT y/H = I

! i, } SURFACE CHARGE AT Y/H = O _ ~~

~ 10 -u \~ 40K=i61-Z ELEMENT WIDTH: 00250n. 20

: _________Io i 2 3 4 5

Fig. 2. The potential and charge distributionsof the dielectric Greens function.

awn.wgrv

( x 5EV(X, H) = ~ arc tan X+f ) arc tan 2H 2H (12)I II . CALCULATION OFNORMAL MODE PARAMETERS

Consider a coupled-strip configuration composed of twostr ips, spacing S/H and equal widths W,/H. One of thesestrips is at potential V; the other is at potential + V for theeven or odd mode, respectively. We divide the strips andthe space between them into elements of width 2~ (12); letW/H= 2&4, S/H= 2$N, and let x,.i= 2&(j i). The poten-tials on the strips are to be produced by the placement ofcharge ki on the ith element, i= 1, . . ., M, and i k, onthe symmetrically located element in the case of the even orodd mode, respectively. We determine the charges ki. Re-ferring to the dielectric Greens function (6), let

@ii = l+,, H I z,, H) = @(xi-;, H ) O, H) (13)denote the potential at xi due to the element at Xj carryingunit charge. Then the resultant potential at the center of theith element is

This system is to be solved for the M values of charge Ai.The solution yields the charge distribution (and thereforealso the current distribution in the propagation problem) andthe total charge

(15)j= 1

from which the capacitance C~ per strip follows as CK = Q/V.The method of calculation outlined above is obviously notlimited to symmetrical pairs of strips, but maybe applied toany strip configuration.To determine the effective relative permittivity K,~~, the

calculation must be performed for the vacuum K= 1 as wellas for the value of substrate K contemplated. The vacuumproblem can, of course, be solved without iteration, sincethere is no bound charge in that case. Denoting the resultingcapacitance per strip by Cl, we have

K,i~ = cK/cI. (16)The velocity of propagation v is

(17)

where c= 2.997925X 108 m. Sl is the characteristic velocityof the vacuum. Within the quasi-static approximation, wemay calculate the characteristic impedance per strip from

1 120= -= .ttK cCldKeff (18)

The wavelength ratio, which is occasionally convenient indesign work, is

A (K.Jv

KI(K) = m (19)

IV. DATA FORSINGLE AND COUPLED STRIPSThe method outlined above has been carried out for

various values of K; in particular, for K= 9, appropriate foran alumina substrate, and K= 16, for the magnetic substratematerial yttrium iron garnet. Fig. 3 shows the characteristicimpedance of a single strip as a function of W/H for valuesof K from unity to 16; Fig. 4 shows the correspondingvelocities.The computational procedure contains its own means for

determining accuracy, through the assessment of errors inthe fulfillment of the boundary conditions, as will be dis-cussed in Section VI. In addition, it is illuminating to com-pare these results wit h those which Wheeler [2] obtainedanalytically by means of a conformal transformation. Thecomparison is shown in Fig. 5, where the differences be-tween our values of ZOand those of Wheeler are shown on agreatly expanded scale. Wheeler offers two approximations,appropriate for wide (Wheelw) and narrow (Wheeln) strips,respectively. Agreement between those two is best at aboutW/H= 1. The figure shows that our results are intermediatebetween the two, the disagreement being well within onepercent everywhere except for W/H


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BRYANT AND WEISS : PARAMETERS OF MICROSTRIP TRANSMISSION LINES 1025~ -1l--r--t7 r:3 x 108 K=1oL=52 x 108G 40 y1 901x.& ](1 50)90> 1 x 108 1(1 00)160 1[075),WjH

Fig. 4. Velocity of propagation on microstri]>-single strip.Fig. 3. Characteristic impedance of microstripsingle strip.!

UPPER: (wHEELN) (w and B)

l LOWER: (WHEELW) - (W and B) ]---- j-, %Z. (ohm) 51~

I35!0+ 26.874i -, \ ~. =.-_&i~__

o/ ------------ - -1%

,/-t ! I I I 1 I 1 I I 10 t 2 3W/H

Fig. 5. Comparison of three values of characteristicimpedance of microstrip.

I 1 I 1 {

\CHARACTERISTIC IMPEDANCE

z ~, = EVEN MODE4 z ~0 = ODD MOOE 1

: oJ- ioo~

W/HFig. 7. Characteristic impedance of coupled pairs ofmicrostr ip transmission l inesK= 9.0.

: 300

l\

CHARACTERISTIC IMPEDANCEz~ 280 ~e = EVEN MOOE

E z ~0 = ODD MODE~ 260 \

W/HFig. 6. Characteristic impedance of coupled pairs ofmicrostrip transmission l ines K:= 1.

: 120 1 1 1CHARACTERISTIC IMPEDANCE 4z ., = EVEN MODEz ~ 0 = OOD MODE

~

Fig. 8. Characteristic impedance of coupled pairs ofrnicrostrip transmission l ines-K=, 16.0.


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1026 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, DECEMBER 1968TABLE I

PARAMETERSOF MICROSTRIPS~H=O.2, K=1O.O

W/H L(KefJ/(p~m) IGf (o;%) (lOVm/s) L(K)

0.100.200.300.400.500.600.700.800.901.001.101.201.301.401,501.601.701.801.902.00

0.100.200.300.400.500.600.700.800.901.001.101.201.301!401.501.601.701.801.902.00

Coupled StripsEven Mode54.51266.46376.81386.55896.009105.298114.490123.621132.709141.767150.803159.822168.826177.819186.801195.775204.742213.701222.656231.604

6.25066.41066.54296.66116.76956.87006.96357.05087.13267.20947.28157.34957.41357.47417.53137.58567.63727.68617.73287.7772

152.984127.072111.07999.45990.39583.03076.88271.64967.12863.17659.68756.58153.79651.28449.00446.92645.02343.27441.65940.165

Coupled StripsOdd Mode120.499150.072170.603186.988201.035213.624225.252236.220246.724256.892266.817276.560286.167295.671305.095314.457323.769333.042342.281351.494

5.49935.51195.52755.54585.56665.58965.61465.64125.66935.69865.72895.76005.79175.82385.85635.88895.92175.95445.98706.0195

64.91652.18345.96842.01039.14836.91735.08933.53932.19130.99729.92328.94728.05227.22526.45825.74225.07124.44023.84523.283

1.19911.18411.17201.16161.15221.14381.13611.12901.12251.11651.11101.10581.10111.09661.09241.08851.08481.08141.07811.0750

1.27841.27691.27511.27301.27061.26801.26521.26221.25911.25581.25251.24911.24571.24231.23881.23541.23201.22861.22521.2219

1.26481.24901.23631.22531.21541.20651.19841.19091,18411.17771.17191.16651.16141.15671.15231.14821.14431.14061.13721.1339

1.34851.34691.34501.34281.34031.33751.33461.33141.32811.32471.32121.31761.31401.31041.30671.30311.29951.29591.29241.2889

acteristic impedance Zo, velocity of propagation v and thewavelength ratio (19). The table refers to the even and oddmodes and covers a range of the width parameter W/H.Complete tables are available covering the following ranges:W/H= O.1 (0.1)2.0, single strip and even and odd modes forS/H= 0.2(0.2)1 .0, for substrate dielectric constant K= 9,

1Tables have been deposited as Document No. NAPS-00087 withADI Auxi liary Publications Service, American Society for informa-tion Sciences,c/o CCM Information Sciences,Inc., 22 West 34Street,New York, N. Y. 10001. Microfiche copies are available at $1.00 or8* by 1l-inch photocopies at $3.00. To order, include the document number and the advance payment.

II K=16s/H = 0,350 .rl L I I W/H= i.O

tot

ODD MOOE >+

49OLi4%4%w#z4%j /jj+#/, /? //v /////~/f

DIELECTRIC HsUBSTRATE

ljljjjjillj 1111111 ////,///y# y, ,7,,,

Fig. 9. Charge distributions for the even andodd modes on coupled strips.

10, 15, 16; also, a table for wide spacing: S/H= 1.0(0.5)4.0,for K= 9.6.The programs also furnish other useful information, in-

cluding the determination of errors and the values of charge,potential, fields over the complete cross-section of the trans-mission line. An example of the charge distributions on thestrips is shown in Fig. 9.The data points in Figs. 7 and 8 are values measured by

MacFarland [9].V. pROGRAMS

The computations were performed at the computer facilityat the University of Maine and on the CP/CMS system atM.I.T. Lincoln Laboratory. In the present form, the programis composed of two parts: the first computes the dielectricGreens function according to (10) for any specified K andstores these data. This computation has been used to gener-ate data for a sequence of seven values of K in the range from6 to 18. The dielectric Greens fuction for intermediatevalues of K is determined by interpolation. The second pro-gram uses these data to compute the single- and coupled-strip parameters as illustrated in Table I, according to (14).

W. DJiWERMINATIONOF ACCURACYThe calculation of the dielectric Greens function includes

means for verification of its accuracy, through determinationof errors in the fulfillment of the required boundary condi-tions. Using (9), we evaluate the surface divergence of elec-tric displacement on the dielectricvacuum interface. Typi-cally, the residual free charge is less than one part in 104 or10r relative to the bound charge at all points near the source,remaining roughly constant in magnitudebut thereforerising, relatively, to a few parts in 102at distances about 10Hfrom the source. We can assess the effect of this error bydetermining the total amount of missing bound charge.This estimate includes also the error resulting from the fact


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BRYANT AND WEISS : PARAMETERS OF MICROSTRIP TRANSMISSION LINES 1027that the calculation was not normally carried to pointsbeyond + 10H. As discussed below, this missing chargeamounted to less than five percent of the required boundcharge. If all the missing charge were lumped at the closestpossible position, namely at + 10H, its contribution to thepotential at points near the source would be less than onepart in 104. This value is, of course, a great overestimate ofthe error, since in fact the missing charge would have beendistributed over the entire infinite breadth of the interface.All of the electric field originating on the source must, ofcourse, terminate somewhere. The three locations of theterminating charge are 1) on the dielectric surface directlyin contact with the source strip, 2) on the exposed area ofthe interface, and 3) on the ground plane. The accounting ofthese charges is as follows, for unit charge on the source:under the source, (1 K)/(l +K)-leaving net charge 2/(1+K) at the position of the strip; on the exposed interface,(1 K)/K(l+K); on the ground plane, I/K. We find thatthe calculated total of each of these agrees with requirementsto within 1.6 percent at worst, and generally to within 0.7percent or better, depending on the value of K. The greatesterror is in the missing bound charge at the interface, asdiscussed above. It is noteworthy that the value of K forwhich the greatest deficiency is found to occur is in thevicinity of K= 2.5, consistent with the fact that the maximumvalue of (K 1)/K(K+ 1) occurs at K= 1+ @~2.4.The conclusion from our various tests is that our calcula-

tion of the dielectric Greens function leads to errors in themicrostrip parameters which are of the order of one partin 1(Y or less; completely adequate for all practical pur-poses.As pointed out in Section IV, there is another type of error

which enters in calculations involving values of W/H orS/Hin the range less than 0.2. This is the consequence of thefinite width of the discrete elementary strip, namely W/H

=0.025 in the data reported here. As Fig. 5 shows, the effectis that the characteristic impedance is falsely high in thatrange: about one percent high at W/H= 0.2 and becomingworse fast. As previously mentioned, this error can be easilycorrected when the need arises.

ACKNOWLEDGMENTWe wish to acknowledge the contributions of Prof. F.

Irons of the University of Maine and N. W. Cook of WPI.We are grateful to C. BIake and D. H. Telmme of LincolnLaboratory for their support and encouragement, and toH. T. McFarland and J. D. Welch for the benefit of manydiscussions and other assistance. We also thank McFarlandfor permission to quote his experimental results in advanceof publication. Finally, we thank Mrs. J, Reid for her val-uable instruction and assistance in the use of the CP/CMStime-sharing computer system.

REFERENCES[1] S.B. Cohn, Shielded coupled-strip transmissic,nline: IRE Tram.Microwave Theory and Techniques, vol. MTT-3, pp. 29-38, October

1955.[2] H. A. Wheeler, Transmission-line properties of parallel stripsseparatedby dielectric sheet: IEEE Trans. Microwave Theory andTechniques, vol. MIT-1 3, pp. 172-185, March 1965.[3] E. G. Cristal, Tech. Rept. USAEL Contract DA-28-043 AMC-02266 (E).[4] K. C. Welters and P. L. Clar, Proc. 1967 G-MTT Znfernatt Micro-wave Symp., paper V-2.[5] G. Policky and H. L. Stover, Parallel-coupled lines on rnicro-strip~ TexasInstruments, Inc., Rept. 03-67-61.[6] P. Silvester, TEM wave properties of rnicrostrip transmissionlines; Proc, ZEE(London), vol. 115,pp. 43-48, January 1968.[7] E. Yamashita and R. Mittra, Variational method for the analysisof rnicrostrip lines, IEEE Trans. Microwave Theory and Techniques,vol. MTT-16, pp. 251256, April 1968.[8] See, for example, L. P. Smith, Mathematical Methodsjor Scientistsand Engineers. Englewood Cliffs, N. J.: Prentice-Hall, 1953.[9] H. T. McFarland, private communication.

parameters of microstrip transmission lines

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