Unified user-oriented computation ofshielded, covered and open planar microwave

and millimeter-wave transmission-linecharacteristics

R.H. Jansen

Indexing terms: Striplines, Waveguide theory

Abstract: A unified, rigorous and efficient hybrid-mode solution to the general planar 3-layer transmission-line problem with shielded, covered and open cross-sectional geometry is achieved by the spectral-domainmethod. Stress is put on the minimisation of the computational expense required to obtain the frequency-dependent design data of planar line structures with realistic geometrical dimensions. This implies the modalpropagation constants, uniformly defined characteristic impedances and any field quantities. On the basis ofuser-oriented considerations a computer program has been developed which is applicable to fundamental,and higher, even and odd modes on single and coupled coplanar strips and slots. This computational approachincludes most of the cases which today are of technical interest, e.g. stripline and microstrip, microstrip withdielectric overlay, suspended substrate strips and slots, slotline, coplanar line and grounded or isolated finline. It combines analytical simplicity, rigorousness and high efficiency with the possibility of performingstudies on numerical accuracy and convergence.

1 Introduction

The spectral domain hybrid mode approach to the planartransmission-line problem1"25 can today be considered themost efficient rigorous approach in this field if it usescomplete expansions and, particularly, if it includes thesingular edge behaviour into its strip-current or slot-fieldrepresentations. Recently, this technique has even beenapplied to arbitrary shielded coplanar transmission lines ona multilayer substrate, and is reported to result in shortcomputing times.20 With a proper choice of expansionfunctions, it leads to accurate numerical solutions with verylow determinantal orders of the eigenvalue equations, notonly for the phase constants but even for the more critical,field sensitive, characteristic impedances.3'14'15>23~2S Fordesigners who frequently employ a computer for the deter-mination of reliable transmission-line data this is of greatimportance.

This is, at the same time, one of the crucial points inthe spectral-domain approach. For practical applications ofthis technique low determinantal orders are mandatorysince the computational expense required for the repeatedgeneration of the coefficients of the final eigenvalueequations is high. These coefficients are improper integralsin the case of an open or covered line cross-section (asdefined in Fig. 1, see e.g. Reference 6) and are infiniteseries in the shielded case.9 Their convergence with respectto the upper integration or summation limit can be shownto be no more than linear if expansion functions are usedwhich satisfy the edge condition. Otherwise, i.e. for non-singular basis functions, the coefficients converge morerapidly ,6>9>n but this is paid for expensively by an increaseof the determinantal order necessary to achieve the sameaccuracy. Some results concerning this point will bepresented here.

Paper T291 M, received 25th September 1978Dr. Jansen is with the Institut fur Hochfrequenztechnik, TechnicalUniversity of Aachen, Alte Maastrichter Strasse 25, D5100 Aachen,West Germany

Another user-oriented aspect in connection with thespectral-domain method is the proper choice of the type ofcross-sectional geometry to be employed in the compu-tations. Here, authors often argue that shielding is necessaryto model the inevitable packaging in practice, and allows astudy of its influence. This is certainly true for the conduct-ing ground and cover planes shown in Fig. 1, particularlysince these planes can be removed to infinity for a model-ling of the open case without causing numerical problemsor increasing computing times.24 It applies in some way toofor the lateral shielding, especially if the transmission linesconsidered are tightly encapsulated in a waveguide as, forexample, in millimeter-wave applications.21 However, forthe modelling of realistic planar microwave-line geometrieswhich can mostly be considered as laterally near-openstructures, the shielded approach is not ideally suited.

The reason for this is an increase of the spatial spectraldensity of the electromagnetic-field representation directlyproportional to the lateral shielding/line-width ratio. As aconsequence, the number of terms to be taken into accountin the series of coefficients also increases linearly.12'21 Forexample, modelling an open structure with a shielding/line-width ratio of 30 and a determinantal order of 20(Reference 21) requires the numerical generation ofapproximately 10s series contributions in each step of theeigenvalue search. Particularly if narrow strips or slots,tightly coupled lines, suspended substrate structures or slotconfigurations operating in the lower gigahertz region areanalysed, this becomes a problem.20'21 The unified approachto the covered, open and shielded geometries which will bedescribed here provides flexibility and avoids these dif-ficulties.

Beyond this, it is assumed here that the consideration of3-layer structures with arbitrary dielectrics is sufficientlygeneral for most practical applications. No restrictions, suchas the symmetry condition used in Reference 21, areimposed concerning the positions of the upper and lowerground planes. The definition of characteristic impedanceadopted here is uniformly based on the power transportedalong the line and an integral over the localised physical

14

0308-6976/79/010014 + 09 $01-50/0

MICRO WA VES, OPTICS AND ACOUSTICS, JANUAR Y 1979, Vol. 3, No. 1

expansion quantities. The convergence of this design par-ameter in the spectral-domain method will be discussed.The numerical solutions show that first- or second-orderbasis functions do not provide accurate results for certainconfigurations. In addition, computed data are presentedwhich throw some light on the question of numericalefficiency. The correctness of the computer solutions isproved by comparison with previous authors and severaltest measurements on standard alumina substrates.

Fig. 1 Cross-sectional geometry of planar 3-layer transmission line

a Shielded typeb Covered typec Open type

<T=oo

<7=ooX=-S X=S

Fig. 2 Representative planar transmission-line geometry

2 Analysis and computational procedure

The setting up of the spectral domain field relations forplanar transmission-line structures can, by now, be con-sidered a well known analytical procedure. So, many stepsmay be omitted in the following mathematical derivation.This is based on the representative geometry of Fig. 2which consists of three isotropic homogeneous layers ofarbitrary thickness and relative permittivity. The metallicplanes involved and the zero-thickness metallisation patternare assumed to have infinite conductivity. For the formu-lation of the electromagnetic boundary-value problemvector potentials normal to the dielectric interfaces, i.e.LSE- and LSM- scalar potentials,26'27 are employed. Theseare particularly suited to simplify the necessary algebraicmanipulations from the beginning and can be written as

= +sin

cos

n\r2zn\

sin {kznl (z h2)}

h2)(1)

02 = + I WnFnsin

(kxnx) x

| kzn2 cos{kZfl2(z + h2

cos (kzn2h2) k\nX cos ikzn2h2)|

=+c o s l

\(kxnx) xsin J

s{kzri2(z + h2)} Tnie2 sin{kzn2(z+h2)}

cos(kzn2h2) exkzn2 cos(kzn2h2)

sin

COS

u2 fr2Kzn\ Kzn2

COS

sin {kzn3(z-h3)}

= -- X WnGn{ \(kxnx) xn = i I sin

with

Tni |

w

Tn2\ e3 |cos

Tn3)

(«-O-S)U

( « - l ) i s (la)

and

^ =^KZ, /txn=^(2fc-l+^)

n = {(k-l)L+l)

respectively, and

^ni = ^ m tan (/:znl) etc.

(Ib)

(lc)

The notation used here, and in the following, is one inwhich the upper terms refer to even modes (magnetic wallin the plane of symmetry x = 0) whereas the lower onesdescribe odd modes (electric wall at x = 0). 7 denotes thepurely imaginary or real propagation constant and k0 is thefree-space wavenumber. The longitudinal dependence ofexp(—jy) and the harmonic time factor have already beenextracted from the scalar potentials in eqn. 1. From thesethe hybrid-mode electromagnetic field is derived by super-position and in the standard fashion.26-27 Note that theelectric field associated with eqn. 1 satisfies all its continuityconditions at the two dielectric interfaces. Also, the con-tinuity of the magnetic field tangential to the lower di-electric interface is warranted and the boundary conditionsat the conducting ground and cover plane are incorporated.Therefore, only two sets of unknown spectral amplitudesare present in the above formulation, namely Fn and Gn.The boundary conditions on an eventual conducting lateralshielding at x = s are satisfied by the choice of the discretespectral variable kxn given in eqn. la. For the laterally opencase, eqn. \b is valid with likewise discrete values of kxn

which, however, are now related to an Z-th-order Gaussianintegration algorithm28 in intervals k of length TT/W, withVx as the weights and with the abscissas Xx. The quantityw is a normalisation width introduced for numerical reasonsand not necessarily one of the strip or slot widths.

MICROWAVES, OPTICS AND ACOUSTICS, JANUARY 1979, Vol. 3, No. 1 15

So, altogether, the formulation of eqn. 1 is a unifiedrepresentation for guided waves on shielded, covered oropen planar transmission lines. It naturally arises from thefact that for a real spectral variable kx the finite Fouriertransform29 approaches the infinite case if the lateralshielding is removed far away from the line. Its advantageis a great flexibility in the choice of the spectral abscissaskxn in the series of eqn. 1. As a consequence, the numericalefficiency of the shielded approach is preserved even for thelimit of infinite shielding width.

In the following, it also turns out that great algebraicsimplicity results from the use of eqn. 1. A surface-currentdensity representing the jump of the magnetic field tan-gential to the metallisation in the upper dielectric interfaceat z = 0 is introduced. The electric field e at z = 0 is evalu-ated and, after suitable normalisation,24 an analytical,compact relationship between the spectral componentsof e and the current density / can be set up as follows:

Jxn

\ hn= (Yn)

= (Zn)

'-xn

eyn

Jxn

Jyn

•yn

= (Yn)- l Ixn

Jyn

(2)

for an array of slots and, with an interchange of Y to Z ande to /, for a strip configuration. Note that the elements of(Aik) can be formulated as a symmetric product of diagonalmatrices (Yan) ... (Zcn) with the frequency-independentcolumn vectors (el

xn) ... (/yn) which is important to obtainnumerical efficiency in broadband computations.24 Thespectral components el

xn . . ./yn of the expansion functionsused are generated easily in an automated fashion by thecomputer algorithm for arbitrary strip or slot patterns ofreasonable complexity. At present up to four strips or slots(symmetrical pattern) can be handled by choosing a controlparameter in the queue of the input data.

In order that this is feasible with low computing times avery careful choice of the set of expansion functions has tobe made. Especially near modal,S>23"2S complete sets ofbasis functions satisfying the edge condition term by termhave to be chosen for the prevailing general kind ofproblem which will be demonstrated in Section 3. Ex-plicity, the choice made here is

cos{(/- l)7T(x-xm)/wm)

(4)

(Yn) ='an I bn

^<l2){klnYGn-l2YFn) + h\kxn(YGn-YFn)

(YGn-YFn) y2YGn-k2xnYFn

YFn =

l__nl_li± +ln2

lc2 \ k2 IT2 I T

]n2xn/\k2 ' k2

\zni

klm

Tm +Inl)l2-)k2o

This relationship is real and symmetric for both propagationand attenuation modes. It is also valid for 2-layer structureslike microstrip since the bottom layer can be eliminatedsimply by introducing h { = 0 (Tn i = 0) without causingnumerical difficulties. In the limit of large values of hi andh3 it should lead to the same results as those reported byKnorr and Kuchler15 if the spectral variable of eqn. \b isused in eqn. 2 and the expansion functions chosen for theGalerkin solution applied to eqn. 2 are the same. Extensivealgebraic manipulations15 are not necessary to achievecomputational efficiency. The matrix of coefficients of thefinal Galerkin equations (see for example Reference 9)takes a very simple form, namely

bn^yn^xn

y. e« ek

1 bncxncyY el ek

(3)

sin{/7r(s-sw)/ww}

[l-{2(x-xm)lwm-lf]— 1\21 1/2

i.e.

forCyn./xn

which applies to strips of slots which are located betweenxm and xm+wm. Bo denotes the zero-order Besselfunction of the first kind28 and Kxn = kxnw is the normal-ised spectral variable. Depending on the actual value of iand the symmetry condition considered at x = 0 the trigono-metric factors and the signs in the right half of eqn. 4 haveto be properly interchanged. By insertion of the spectralcomponents el

xn . . .fxn of eqn. 4 into the series of coef-ficients in eqn. 3 the eigenvalue equation of the trans-mission-line problem is generated. The modal propagationconstants y are found as the roots of the determinant of(Aik(y)). Finally, a process of back-substitution yields thefield quantities and the current-density distribution of theguided wave under consideration.

With the electromagnetic field information available,strip and slot characteristic impedances can be computed.However, owing to some arbitrariness in the definition ofcharacteristic impedance for hybrid modes (see for exampleReference 14), die criteria which form the basis of such acomputation have first to be discussed. Here a dominant

16 MICROWA VES, OPTICS AND ACOUSTICS, JANUARY 1979, Vol. 3, No. 1

requirement is the usefulness of the quantity ZL for designpurposes. This means that the approximate solution ofcertain planar circuit problems, like for example that of aT-junction or a coupler,1Sl23 should be made possible byemploying the value of ZL in simple transmission-lineformulas instead of solving a new boundary-value problem.So, this is equivalent to requiring the applicability of anetwork approach. Therefore, the physical quantitiesconstituting the characteristic impedance should, as far aspossible, be lumped, i.e. localised tightly compared to wave-length, like longitudinal strip current and transverse slotvoltage. Transported power is also a relatively localisedquantity since its lateral decay away from a planar line ismuch faster than that of the field itself. The use ofstrip-to-ground voltages or of the current parallel to a slotcan be justified for quasi-TEM modes with strongly con-centrated field distribution, e.g. for microstrip23"25 or forfairly narrow slots.21 Certainly it is not an optimal choicefor wide slots, coupled slots and suspended substrate lineswith large ground-plane separation.

Consequently a definition of characteristic impedancebased on the power transported along the planar linepromises maximum usefulness for design purposes, which isin agreement with previous authors.15 In addition, it allowsa unified computation for strips and slots. Note that theslot voltage Ux and the strip current Iy both result from a1-dimensional integration over the unknowns ex, j y ,respectively, by means of which the problem is formulated.So, finally, the definition adopted here for the rath slot orstrip in a planar configuration is

and

1

' ym

ulRe f f (

J Jem xh*)ydxdz, h=

' y mRe (e x htn)ydxdz,

(Sb)

In this definition eqn. Sb results from eqn. 5a by an inter-change of dual quantities analogous to that performed ineqn. 3. By the utilisation of partial fields em, hm each ofwhich is associated only with the mth portion of the totalof the expansion functions, the characteristic impedance(eqn. 5) is a generalisation which applies even if more thantwo slots or strips are present in a problem. It includes thespecial cases treated by Knorr and Kuchler.15 In the limitof very loosely coupled structures it automaticallyapproaches the single-line value.

For an easier numerical evaluation of line impedances,an explicit detailed expression of the power transportedalong a planar 3-layer configuration has been derived fromthe formulation of eqn. 1. This can be accomplished in arelatively compact form and will therefore by providedhere. With the abbreviations

4 "/, — (kln-l2Wnkzn2\2

and

j K ( k l n4 /we0

w \y\

Cnl = \lcos2(kznlh1)

Un\ = TnJklni etC.

Snl =

in which PFn and PGn represent the LSE- and the LSM-field contributions, whereas PFGn is the mixed-field term,the total transported power is

p = --Me

u2Kzn'nl /vzn2

1

4-c -2 T-2

0 2 T TI 2 nX

- 2 — Tnl Un2 — — - 5 —e, e\ k\n2

rK

e2 JTn3

e2--TnlUnl-TnlUn2 H zn2

zn\

^n3

(6b)

The partial transported-power terms Pm of eqn. 5 arecomputed in complete analogy, except that partialamplitudes Fmn and Gmn have to be used in eqn. 6a. Theseare linearly related to the mth terms em,jm of the solutionin the same way as Fn, Gn, respectively, are to the totalsolution. In the case of symmetrically coupled linesP\ =Pz =0-5P is valid, in equivalence to eqn. 5. So, in-dependent of an interest in a general-case solution accordingto eqn. 5, the above power expressions are useful for single-and coupled-line computations.

3 Results

Without any exception, the numerical results presented inthis paper have been obtained by means of a singlecomputer program. The storage required by this programamounts to a total of 560008 words of memory on aControl Data Cyber 175 (computer centre of the TechnicalUniversity of Aachen) if solutions up to the twelfth order(M= 6) and up to 600 series contributions (Af=600) pet-matrix coefficient shall be considered in a manner providingmaximum computing speed. In principle, values of TV

MICROWA VES, OPTICS AND ACOUSTICS, JANUARY 1979, Vol. 3, No. 1 17

greater than 600 can also be handled, which is importantwhen studying the limits of the widely shielded case;however there are no time-saving capabilities for the termsexceeding TV = 600. This version is particularly suited forhigh-speed broadband computations. If storage require-ments are a critical point, about 200008 words of memorycan be saved by a slight reorganisation of this version of theprogram but, of course, at the cost of computer time. Sincethe numerical stability of the computations is excellent thewordlength of 60 bits at present (Cyber 175) can bereduced drastically and the implementation of the programon a comfortable desk-top computer seems feasible.

250

OO2h2/Ao 0 04 006 008

Fig. 3A Covered slot-line characteristics

. 3ht =h3 = 20 h2

\ / \ 0

zL no o oo test measurementsh.m. First higher mode

The correctness of the results reported here has beentested in several ways. For example, the papers by Conn,30

Mariani etal.,31 Knorr era/.,14'15 Smith,32 and Hofmann,21

and recent computations performed by the authorhimself,24> 2S have served as valuable sources of comparison.Reasonable agreement with all these sources has beenachieved and some data taken from them have been in-corporated into the Figures below. In addition, testmeasurements of the wavelengths of single and coupledslots on standard alumina substrates (h2 =0-64 mm, e2 =9-7) have been performed. These have been obtained bya modification of Deutsch's and Jung's method33 to looselycoupled short-circuited resonating slots. Fig. 3A containssome of them and shows an average coincidence of about1% with the computed characteristics. This method ofmeasuring is not well suited for very narrow or very wideslots because, in the first case, the coupling probes have tobe positioned extremely near the slots, whereas in the othercase the Q-factors of the resonances are rather low. As tothe impedance values computed in Fig. 3A these agree wellwith the results given in Fig. 2 of the paper by Marianiet a/.31 Nevertheless, in the present approach, thecomputed curves all run down to a normalised frequency

of h2l\0 ^0-004 (2GHz for h2 = 0-64mm) which mightbe of interest for the application of narrow slotlines in thelower GHz-region, for example in mixers. Note, however,that in such low-frequency design applications the influenceof the packaging must be considered. This is clearly visiblein Fig. 3B even for a relatively wide slot of width w/h2 = 1.Here, the influence of the lateral shielding on the slot wave-length is still noticeable for an aspect ratio of s/w= 100.The influence of s/w on the low-frequency characteristicimpedance is of the same order of magnitude. Besides that,the approximate relative time factor t.f. depicted in Fig. 3Bshows that at low-to-moderate frequencies the covered oropen approaches are much faster than the shielded one ifa wide shielding is used to model the open case (s/w chosenfor less than 1% deviation from the equivalent coveredsolution).

0-8

0 7 -

0-6-

0-5-

0 4

s / w /

U-50 /- L*

1/100 /

\ 1

vVT /

s/w=10 ' ^ V

. ^ S m-

' ^ W p

• €2=97

w/h2=1

1-5

05

0 0 2 h2/A0004 006 008

Fig. 3B Influence of lateral shielding on the fundamental slotmode wavelength

T.F. is factor relating the computing time of the covered (s/w — °°)to that of the shielded approach

t.f.

In the following, now, some general remarks have to bemade concerning the numerical accuracy of the unifiedspectral-domain approach under discussion. For practicalapplications, usually, second-order single slot or stripsolutions (M = 1, matrix order 2) and fourth-order solutions(M—2, matrix order 4) for the symmetrically coupledversions provide sufficient accuracy, i.e. about 1% for boththe phase constant and the characteristic impedance ZL.Fig. 4 reveals the degree of inaccuracy which comes alongwith the use of a first-order only expansion as employed byKnorr and Kuchlerls in the computation of coplanar stripcharacteristics (consider a factor of 2 in the definition ofZL). Especially, for tightly coupled line structures theerror introduced in this way cannot be tolerated. In suchcases, at least one additional longitudinal expansion termis indespensable for a reasonable description of the trueproximity distorted current or field distribution.

This interpretation, also, is supported by the con-vergence curves given in Fig. 5. These show indeed that afourth-order edge-type solution is capable of producing anoverall accuracy of 1% whereas a solution with M'= 1

18 MICROWAVES, OPTICS AND ACOUSTICS, JANUARY 1979, Vol. 3, No. 1

(KXN - IOCTA/) may still be far from the goal. In contrastto this, ordinary Fourier expansions, which have beenstudied in Fig. 5 by simply substituting Si-functions28 forthe zero-order Bessel terms in eqn. 4, exhibit relatively poorconvergence properties. Particularly, the accuracy of thefield-sensitive characteristic impedance is affected severelyif the singular edge behaviour of the field is not incorpor-ated explicitly into the numerical solution. In terms of thecomputational expense required to achieve the 1% margin,this means a roughly estimated increase of c.p. time by atleast a factor of 25 compared to the edge-type solution. Toillustrate this further, a relative, approximate measure ofnumerical expense, namely

M:M' = 40M+2(2M+ \)M2 and

M" = 20 M2 (7)

shall be given and discussed here. M' applies if a largernumber of frequency points is considered and the frequency-independent structural information on the problem hasbeen evaluated and stored away once, at the beginning ofthe computations. If the amount of storage necessary totake up the spectral contributions of the expansionfunctions cannot be spent the increased quantity M" applies.The total number of point operations or accesses whichhave to be performed during each step of the eigenvaluesearch is then obtained by multiplication of M with afactor ranging between 40 and 100. So, even for a very lowdeterminantal order in the spectral-domain method, somethousands of point operations per step cannot be avoided,which puts an extreme stress on the proper choice of theexpansion functions.

0-60

055

0-50

045

0-40

9-6"/.

g/h2=2 \

g/h2=1/5

2277,

75

70

60

50

qsi

40

30

002 004 006 00825

Fig. 4 Coplanar-strip characteristics (fourth-order solution) incomparison with References 15 and 21A Reference 21 (without edge terms)o Reference 1S

/ 0zL, n

e, = e3 = 1hx = / j 3 = 2 0 / 1 ,w/h^ = 1.5

The situation is somewhat different for tightly shieldedplanar lines, as for example the fin lines of Fig. 6. Here, theaspect ratio and the selection of the variables in terms ofwhich the problem is formulated determine the duration ofa program run.12'13>21 Except for small values of the linewidth w, the factor by which eqn. 7 has to be multipliednow is lower than for open problems. Besides this,the physical behaviour of the results depicted in Fig. 6again confirms the correctness of the computations. As isexpected, the corresponding electrical parameters of theisolated and the grounded fin line approach each other inthe case of small slotwidths (consider the factor of 2 as aconsequence of the definition in eqn. 5). That the lineimpedances calculated by Hofmann21 on a voltage-per-current basis differ from the values resulting in this papershould not astonish very much. This is quite clear in viewof the non-TEM character of the field of grounded fin lines.

In the next Figs, la—lc once more a laterally open, i.e.covered, configuration, is investigated. Specifically, thephysical properties of the fundamental even mode (co-planar line quasi-TEM mode) and those of the odd modeon coupled slots are discussed. The Figures show that thephase velocity of the coplanar-type transmission-line modedepends only weakly on the width of the coupled slots. Thesame prevails for the low-frequency characteristic im-pedance of the odd mode. Moreover, due to its quasi-TEMcharacter, the even mode exhibits only a few percent offrequency dispersion in its parameters and reacts fairlyinsensitively even to a rapid decrease of the ground-planeseparation h x. The results for both of the modes are againchecked by some test measurements. Furthermore, theconvergence speed of these two types of coupled-slotsolutions is graphically represented in Fig. 7c. Here, the oddmode is similarly well behaved as for a single slot and couldsatisfactorily be described by a determinantal order of 2( M = l ) . On the other hand, the coplanar transmission-line mode poses the same accuracy problems as thecoplanar strips in Fig. 5.

1000

750

Fig. 5 Convergence behaviour of coplanar strip solutions

• with edge-type expansion functionso_without edge-type expansion functionsM is a relative measure of numerical expense

A ( /

w/h7 = 1g/h2 = 1/8

MICROWAVES, OPTICS AND ACOUSTICS, JANUARY 1979, Vol. 3, No. 1 19

As a final interesting example of microwave andmillimetre-wave transmission lines, the electrical propertiesof suspended substrate strips shall be analysed and inter-preted. For the frequency-dependent characteristics of thisclass of lines merely a few specialised results21 have beenpublished to the present day. These may now serve forcomparison and have, for this reason, been introducedinto Fig. 8. Excellent agreement can be established there asfar as the effective dielectric constant eeff = (A0A)2 of thevarious strips is concerned. Naturally, the high-frequencyvalues of the line impedances ZL do not coincide becausethe definitions applied are different. However, since thefundamental strip mode is a quasi-TEM wave this differencevanishes in the quasistatic limit.

200

100

095

0-90

085

080

</zL=Ux/ly .(reference 21),250

200

150

100

50

0-2 0-i,w.rnm

06 08

Fig. 6 Characteristics of grounded and isolated fin lines in a WR28 waveguide

c, = e, = 1hx = hy = 20 h2; h2 = 0125 mm

h — 0-125 mmd =• 0-500 mm/ = 34 GHz

0-6

0-55

05

(K5

1 U 1/8 1/16 w/h 2

150

100

50

1/2

0 02 006 008

1000

• 750

•500

250

Fig. 7 Even and odd mode solutions

a Dispersion characteristicshi = 20 h2

/,, =h2o test measurements«i = e3 = 1g/h2 = 1 / 2(i) even mode(ii) Odd modeh3 = 20 h2

b Characteristic impedances for coupled slots ofdifferent widths/i, = 20 h2

«i = e3 = 1(i) Even mode(ii) Odd modeh3 = 20 h2

Convergence of solutions for coupled-slot configurations• with edge-type expansion functionso_without edge-type expansion functionsMis a measure of numerical expense

002 0-04h2/A0

006 0 0 8

/ 2g/h2 = 1/16h2/X0 =0-064(i) Even mode(ii) Odd mode

20 MICROWA VES, OPTICS AND ACOUSTICS, JANUARY 1979, Vol. 3, No. 1

Fig. 9 gives additional information on the behaviour ofsuspended strips. It features the line characteristics as afunction of the normalised frequency ft2Mo and the ground-plane separation hx. Also, the influence of a lateralshielding is indicated. The parameter hx acts here as aneffective means of designing transmission lines with animpedance level which is high compared to the microstripcase hx = 0. Therefore, shallow grooves in a metal plateattached at the substrate bottom (hx = 0-5 . . .2h2) canbe utilised to create microstrip-compatible high-impedancelines without causing sensitivity problems.25'34 Similarly,the effect of the ground-plane separation on the even andodd mode characteristics of coupled suspended strips, asplotted in Fig. 10, can be employed with advantage. In thiscase the odd mode depends only modestly on the value ofhx and the reason for this is physically obvious. So, theseparation parameter can be adjusted for an equalisationof the even- and odd-mode phase velocities and highdirectivity couplers can be constructed25 which seems tobe practicable at least up to X-band frequencies. As ageneral rule for all of these suspended-strip geometries,flat packaging, i.e. especially a low ground-plane separation,reduces the sensitivity with respect to the lateral neighbour-hood of metallic objects and thus enables higher circuit-integration densities.

125

5

3 •

ZL=UZ/I

1/2"N

///

1 "•*

w.mm

• . 1 /2 -

1

y .(reference 21)

> o-^if- \ \' \' \

\ \

\ \

%(

©(D '

— o^A

> </

°~3

- n^_r«XSi»8»668Se2=9 6

~^ / / .° / / // / // / // / /

r ° / // /

/ /

• 100

- 75

- 50

002 0O625

008

Fig. 8 Electrical parameters of suspended substrate lines in a WR28 waveguide compared to Reference 21

o Reference 21

2h1 — 0-640 mm

4 Conclusion

A unified, highly efficient approach to shielded, coveredand open planar transmission-line configurations has beenpresented. The corresponding computer program is believedto be a valuable tool in the design of microwave and milli-metre wave circuits. Its results throw some light on theproblems of efficiency, accuracy and speed of convergenceassociated with the spectral-domain method. In addition,some of the physical aspects treated here may not havebeen published before. Due to space restrictions, the

computed and measured results already obtained forstructures with more than two strips or slots will bereserved for a later publication. It is intended, also, toimplement the line losses into the computations.

10

\

. \\A

• 2 0\

" °'\J-~—

1̂/2 Jh,/h2

0

1/2JC

2 0 ,

(2)—W

w\\%

-*

j€2=97 \

S/W=0O ^ "*"

^-s/w-20

\

~-—s/w = 20

250

150

- 100

W^.-=.^ 50

002 00A 006 008

Fig. 9 Characteristics of a laterally open suspended substrate linefor different ground-plane separations /J,

€effZL,a

w//i2 = lh, = 20 h.

250

200

; 150

• 100

50008

Fig. 10 Even and odd mode parameters of coupled suspendedsubstrate lines for different ground-plane separations ht

ZL, aw//i2 = 1 /28/h2 = 1 /4f. = *3 = 1ZL = 4 0 . " - - . 4 4 ^ for the odd modefc3 = 20 h2

MICROWA VES, OPTICS AND ACOUSTICS, JANUARY 1979, Vol. 3, No. 1 21

5 References

1 DEN LINGER, E.J.: 'A frequency dependent solution for micro-strip transmission lines', IEEE Trans., 1971, MTT-19, pp. 30-39

2 ITOH, T., and MITTRA, R.: 'Dispersion characteristics of slotlines', Electron. Lett., 1971, 7, pp. 364-365

3 KRAGE, M.K., and HADDAD, G.I.: 'Frequency-dependentcharacteristics of microstrip transmission lines', IEEE Trans.,1972, MTT-20, pp. 678-688

4 KOWALSKI, G., and PREGLA, R.: 'Dispersion characteristics ofsingle and coupled microstrips', Arch. Elektron. & Uebertragu-ngstech., 1972, 26, pp. 276-280

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7 KOWALSKI, G., and PREGLA, R.: 'Dispersion characteristics ofsingle and coupled microstrips with double-layer substrates',Arch. Elektron. & Uebertragungstech., 1973, 27, pp. 125-130

8 JANSEN, R.H.: 'A modified least-squares boundary residualmethod and its application to the problem of shielded micro-strip dispersion', ibid., 1974, 28, pp. 275-277

9 ITOH, T., and MITTRA, R.: 'A technique for computingdispersion characteristics of shielded microstrip lines', IEEETrans., 1974, MTT-22, pp. 896-898

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11 JANSEN, R.H.: 'A moment method for covered microstripdispersion', Arch. Electron. & Uebertragungstech., 1975, 29, pp.17-20

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13 JANSEN, R.H.: 'Numerical computation of the eigenfrequenciesand eigenfunctions of arbitrarily shaped microstrip structures'.Ph.D. thesis, Technical University of Aachen, 1975

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20 DAVIES, J.B. and MIRSHEKAR-SYAHKAL, D.: 'Spectraldomain solution of arbitrary coplanar transimission line withmultilayer substrate, ibid., 1977, MTT-25, pp. 143-146

21 HOFMANN, H.: 'Dispersion of planar waveguides for millimeter-wave application', Arch. Elektron. & Uebertragungstech., 1977,31, pp. 40-44

22 BORBURGH, J.: 'The behaviour of guided modes on the ferrite-filled microstrip line with the magnetization perpendicular to theground plane', ibid., 1977, 31, pp. 73-77

23 JANSEN, R.H.: 'Fast accurate hybrid mode computation ofnonsymmetrical coupled microstrip characteristics'. Presentedat the 7th european microwave conference, Copenhagen, 1977

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26 COLLIN, R.E.: 'Field theory of guided waves' (McGraw-Hill,New York, 1960)

27 HARRINGTON, R.F.: 'Time-harmonic electromagnetic fields'(McGraw-Hill, New York, 1961)

28 ABRAMOWITZ, M., and STEGUN, I.A.: 'Handbook of math-ematical functions' (Dover Publications, New York, 1970)

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31 MARIANI, E.A. et al.\ 'Slot line characteristics', ibid., 1969,MTT-17, pp. 1091-1096

32 SMITH, J.I.: 'The even- and odd-mode capacitance parametersfor coupled lines in suspended substrate', ibid., 1971, MTT-19.pp. 424-431

33 DEUTSCH, J., and JUNG, H.J.: 'Messung der effektiven Dielek-trizitatszahl von Mikrostrip-Leitungen im Frequenzbereich von2 bis 12 GHz', Machrichtentech. Z., 1970, 23, pp. 620-624

34 DELFS, H.: 'Versatile construction of thin-film circuits forrf-applications'.ATG-FacMer., 1977,60, pp. 201-206

22 MICROWAVES, OPTICS AND ACOUSTICS, JANUARY 1979, Vol. 3, No. 1