AN ADAPTABLE APPROACH TO THE ANALYSIS OF COPLANAR MULTICONDUC-TOR SLOW-WAVE STRUCTURES USING THE METHOD OF LINES

A. G. Keent, M. I. Sobhyt

ABSTRACT

The 'Method of Lines' provides the analytical framework for the construction of algorithmsthat enable a class of coplanar slow-wave structures to be analysed, with particular referenceto interconnects on MMIC's. A suite of computational routines has been constructed whichprovides a high degree of flexibility, and produces results with the minimum of user inter-vention. This is intended to be used as a engineering design tool.

SUTMMARY

The 'Method of Lines' (MOL) has been further developed to analyse the dispersive proper-ties of general shielded multiple asymmetric coplanar electrode structures located betweenlossy isotropic dielectric layers, in which the losses arise through polarisation or conductioneffects. Perfectly conducting electrodes are assumed. As the losses may be large, perturba-tion methods cannot be used and the complex propagation factors must arise directly from afull-wave hybrid-mode analysis. This implementation of MOL has few restrictions, andallows the analysis of structures with arbitrary numbers of asymmetrc coplanar electrodesand multiple lossy dielectric layers. Attenuation and slow-wave factors, electrodeimpedances and equivalent lossy TEM transmission line parameters for all the dominantmodes of propagation may be calculated. These may be produced either as functions of fre-quency for fixed dielectric conductivities or as functions of a dielectric conductivity for fixedfrequency.

METHOD

The hybrid-mode analysis uses a linear combination of TEy and TMy fields, Sherill andAlexopoulos [1]. Hertzian potentials JI (x,y) and II(x,y) are defined within the i th. dielec-tric layer as

lI.h(xIy) = piF,h(x,y)e1Z (1)

where y= cx+jp is the complex propagation factor. The electric and magnetic potentials Fp,hsatisfy the Helmholtz equations

&2Fp,h a2FP,h1a2 + a r2 +ko2(4i-4_) Flth = 0 (2)

t Electronic Engineering Laboratories, Tle University of Kent, Canterbury, Kent CT2 7NT, UK.

1340

The complex effective permitvity is 6e = - kot1 and the complex relative permittivity ofthe i th. dielectric layer is .j = i'-j(rii"tI+i(Wxo)-'), for i-1 , s,NLE 2.A finite difference scheme with non-equidistant line spacing, as reported in Keen et. al. [2],is implemented, wit Dirichlet boundary conditions in x on Fe and Neumann conditions onFp. Equations (2) are consequently nornalised, (due to unequal line spacing), andtransformed, (by virtue of the finite difference scheme chosen), producing a system of uncou-pled differential equations for the components of what are now transformed normalised elec-tric and magnetic potential vectors, V4N(y) and 44(y) respectively. If the coplanar electrodesare located at the dielectric imterfac? specifiedyb y =YNB, then imposing continuity condi-tions on the tangential electromagnetic field components at the dielectric interfaces for whichY*YN3 allows the construction of transformed normalised potentials 'i and C4N at thesedielectric interfaces, for i=l,...,NL-l, with i.NB, which are themselves continuous. Eachinterface potential obeys a recurrence relation, as in the quasi-TEM MOL analysis in [2].This property allows the components of the transforned normalised tangential electric field,Et,NB, and the components of the transformed normalised electrode current density, JNB, atthe interface at y = YNB to be expressed in tenns of the potentials there, jB and 4i41. Elimnation of these interface potentials gives

ENB =A[ZN(e,o())I]tB (3)Inverting the transformations and performing the usual reduction process yields the complexeigenvalue problem

[ZN (Se, C) ]red JN, P 0 (4)

For non-trivial solutions

det[ ZN (se ,W0) ]red =0 (5)

must be solved. For Ns electrodes it is necessary to look for Ns dominant mode roots ofequation (5), ei for i=l,...,Ns, in the complex plane with Re(ee) .0 and IM(se)

Stage (1) uses a modified Brents root finding method, Brent R. P. [3], while stages (2)and (3) rely on the complex Newton-Raphson method, with calculated derivative, anddet[ ZN (Sec,O ) ]r modified, as each root is found, to avoid finding the same root again.This scheme gives each 6ei as a function of frequency for fixed structural parameters. Ifthe SEj are required at a fixed frequency but as functions of the conductivity of the i th.dielectric layer in the range [Oi,min , Ci,max ], then the previous 3 stages are performedwith (Yi = ai,min, and the frequency increased to the required value. Then a 4 th. stage isperformed where

(4) cs; is incrementally increased from Giminr to Cyi,maX, and the Ns roots from stage (3)followed, again usig the complex Newton-Raphson method.

Complex electrode impedances as functions of frequency or conductivity may be foundusing a 'partial power' definition or a reciprocity related definition, Wiemer and Jansen[4]. Equivalent lossy TEM transmission line parameters are calculated as functions offrequency or conductivity usig equivalent complex modal electrode voltages found fromthe modal powers and electrode current distributions.

RESULTS

Consider the symmetric coupled narrow electrode structure of fig. 1.

Sr4 1, 8r4 4= 04 10 mm

Fr3-'3.6 r3=1"=3 =0 1.2 pm5mmn 5 gml.5pml.5 pm 5mm

42'-3= 9 4r2"= T2 =0 0.6 pm

erj`=12, r"=0,Oa1 400 pmFig. 1 Symmetric coupled narrow electrode structure

The even and odd mode attenuation factors ci (db/mm), slow-wave factors Xg.X6 and thereal and imaginary parts of the electrode impedances, based on the partial power per elec-trode definition, as functions of frequencies up to 20 GHz, for a1=0.04 S/cm, aredisplayed in figs. 2a to 2d respectively. Figs 3a to 3d show corresponding data where oyvaries in the approximate range [0.005, 1000] S/cm for frequencies of 1 GHz, 10 GHzand 20 GHz.

1342

I(c/os) vs. FrquenCY UGz)0.5

0,5

5 0.351

u 0.3" 0,3

v

ao 0.2

r

0.15

/

0.1

0.OS

Frsqmc (I*1

Slow Wave Factor vs. Frequency (6Hz)

reMDk MiLEven -Od d -

__

is its

Fr,tpcy (F51

Fig. 2a Attenuation of even andodd modes (db/mm) vs. frequency(GHz), where ol=0.04.

Fig. 2b Slow-wave factors of evenand odd modes vs. frequency(GHz), where ol=0.04.

Re(Electrode lrpedamces) (Uns) vs. Frcq.nq (6Hz)

rode aOm -

I~

60

50

40

30

20

50 t _ _ -__-

Lo -a

10

II III|I I I I -it1s

FruqAc0 19*)

It I

(Electrode mpedaxcesi (Iis) vs. Freqenrg(6Hz)

Evn -Osd -

Le-I it I 10 s

Fr""w If*J

Fig. 2c Real pans of even and oddmode electrode impedances(Ohms) vs. frequency (GHz), whereaY=0.04.

Fig. 2d Imaginary parts of even andodd mode electrode impedances(Ohms) vs. frequency (GHz), where01=0.04.

1343

r mlEar -Om - --

10

la I

t

u I* -.a

et

is -.l -. laI

300 .

25awwc

P 200d

n

7 1500ISO

* 100

0 . .I I I

r II

I

9da

0h

F

oI

*0 Atteauotion Factcr (dt/rn) vs. Caxnxtivity (S/cm)

A S%

-

Is

_ f

-rEaI

0ixi

'J,S

la.

I o /-

I*' 1s a'eCuxtivCtC(Stci

'.s

F

o 0.

*04

r 0.2

0.1

1it 10 I lo

rSlow Wave Factor vs. Conrcbtivity (S/cm)

Fig. 3a Attenuation of even andodd modes (db/mm) vs. conduc-tivity a1 (S/cm) at frequencies of 1GHz, 10 GHz and 20 GHz.

Fig. 3b Slow-wave factors of evenand odd modes vs. conductivity a,(S/cm) at frequencies of I GHz, 10GHz and 20 GHz.

:lWedaces (Gus) vs. CatxtivItH (S/cal

Ev'Od --

o otI Sb

aS-a0 igs -t5 1

Cststivlit (Ysd)

Fig. 3c Real parts of even and oddmode electrode impedances(Ohms) vs. conductivity a1 (S/cm)at frequencies of 1 GHz, 10 GHzand 20 GHz.

Fig. 3d Imaginary parts of even andodd mode electrode impedances(Ohms) vs. conductivity a1 (S/cm)at frequencies of 1 GHz, 10 GHzand 20 GHz.

1344

t

t1

dh/

10 -1le -4

lo -6

as - is- a1 i itIs t I

CwdxtCditC IS/sd

Be

O 2S*

d Z2n

1 150o

10

50

I

nc

h1 20

s0~~~~~Cstvt CS1^eC4 -4 t;l it' tS 1o

14111 I I itill I 11 .1 11

A

The attenuation factor of the even mode is larger than tat of the odd mode by about afactor of 10 throughout the frequency range considered where a1l=0.04. This has beenobserved experimentally to cause unexpected and unwanted odd mode coupling in asimilar lossy structure. Fig. 3a shows much larger differences appearing between theeven and odd mode attenuation factors as the conductivity al is varied, especially at thelower frequencies, and could have significant bearing on the design of such components.

CONCLUSION

MOL has been shown to be capable of analysing coplanar electrode slow-wave structuresby including the loss mechanisms in a hybrid-mode analysis. In particular, although theresults shown here deal with a symmetric two electrode structure, results can be gen-erated for structures with multiple asymmetric electrodes and multiple lossy (semi-conducting) dielectric layers.

ACKNOWLEDGEMENTS

The authors would like to thank Dr. A. J. Holden and Dr. R. G. Amold of PlesseyResearch (Caswell) Ltd. for many useful discussions. This work has been funded by anSERC collaborative award.

REFERENCES

[1] Sherrill B.M. and Alexopoulos N.G."The Method of Lines Applied to a Finline/Strip Configuration on an AnisotropicSubstrate"1987 IEEE Trans. Microwave Theory Techn., Vol. MTT-35, pp.568-575

[2] Keen A.G., Wale MiJ., Sobhy M.I. and Holden AJ."Quasi-Static Analysis of Electro-Optic Modulators by the Method of Lines"1990 IEEE J. Lightwave Technology, Vol. 8, pp.42-50

[3] Brent R.P."Algorithms for Minimisation Without Derivatives"Englewood Cliffs. N.J.: Prentice-Hall, Chs. 3-4

[4] Wiemer L. and Jansen R.H."Reciprocity Related Definition of Strip Characteristic Impedance for Multiconduc-tor Hybrid-Mode Transmission Lines"1988 Microwave and Optical Tech. Lett., Vol 1, pp.22-25

1345