ANALYSIS OF FIN-LINE WITH THE FINITE METALLIZATION THICKNESS

REPORT R-999 OCTOBER 1983 U ILU-EN G-83-2220 g COORDINATED SC IEN C E LABORATORY ANALYSIS OF FIN-LINE WITH THE FINITE METALLIZATION THICKNESS TOSHIHIDE KITAZAWA RAJ MITTRA APPROVED FOR PUBLIC RELEASE. DISTRIBUTION UNLIMITED. UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

S E C U R IT Y C L A S S IF IC A T IO N O F T H IS P A G E fwi«o D a ta Enterad) REPORT DOCUMENTATION PAGE R E A D IN ST R U C T IO N S B E F O R E CO M PLETING FORM 1. R E P O R T N U M B E R 2. G O V T A C C E S S IO N NO. 3. R E C IP IE N T 'S C A T A L O G N U M B E R 4. T I T L E (a n d Subtitle) 5. T Y P E O F R E P O R T 4 P E R IO O C O V E R E D Analysis of Fin-Line with the Finite Metallization Thickness 7. A U T H O R r*; Toshihide Kitazawa and Raj Mittra Technical Report 6. P E R F O R M IN G ORG. R E P O R T N U M B E R R-999; UILU-ENG 83-2220 8. C O N T R A C T O R G R A N T NUMBERf*.) N000-1479C-0424 9. P E R F O R M IN G O R G A N IZ A T IO N N A M E A N D A D D R E S S Coordinated Science Laboratory University of Illinois Urbana, IL 61801 11. C O N T R O L L IN G O F F I C E N A M E A N O A O O R E S S JSEP 10. P R O G R A M E L E M E N T. P R O J E C T. T A S K A R E A ft W ORK U N IT N U M B E R S 12. R E P O R T O A T E October 1983 13. N U M B E R O F P A G E S 18 14. M O N IT O R IN G A G E N C Y N A M E ft A O O R E S Sfi/ different from C o n tro llin g O ffice ) 15. S E C U R IT Y C L A SS, (o f this report) Unclassified 15*. O E C L A S S IF IC A T IO N /D O W N G R A D IN G S C H E D U L E 16. D IS T R IB U T IO N S T A T E M E N T Cot (h ie R eport) Approved for public release; distribution unlimited. 17. O IS T R I8 U T IO N S T A T E M E N T (o f the abstract entered in B lo c k 20, If different from Report) 18. S U P P L E M E N T A R Y N O T E S 19. K E Y W ORDS ( C o n tin u e on re v e rse sid e if n e c e s s a r y and id entify by b lo c k num ber) fin -lines finite metallization thickness millimeter waveguides propagation characteristics 20. A B S T R A C T ('Continu* on re v e rse sid e If n e c e s s a r y and identify by b lo c k num ber) In this paper, we present a method for analyzing fin-line structures with finite metallization thicknesses. The method, although it is based on a hybrid mode formulation, by-passes the lengthy process of formulating the determinantal equation for the unknown propagation constant. Some numerical results are presented to show the effect of the metallization thickness for unilateral and bilateral fin lines. DD ij SM73 1473 Unclassified S E C U R IT Y C L A S S IF IC A T IO N O F THIS P A G E fwhen D ata E n tered)

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ANALYSIS OF FIN-LINE WITH THE FINITE METALLIZATION THICKNESS Toshihide Kitazawa* and Raj Mittra Electrical Engineering Department University of Illinois Urbana, Illinois \ The work reported herein was sponaroed in part by the Joint Services Electronics Program, N00014-79-C-0424. k Dr. Kitazawa is on leave from Kitami Institute of Technology, Kitami, Japan.

"V. 2 ABSTRACT in this paper, we present a method for analyzing fin line structures with finite metallization thicknesses. The method, although it is based on a hybrid mode formulation, by-passes the lengthy process of formulating the determinantal equation for the unknown propagation constant. Some numerical results are presented to show the effect of the metallization thickness for unilateral and bilateral fin lines.

I. INTRODUCTION Fin-line structures have received considerable attention because of their usefulness as millimeter-wave integrated circuit components. Recently, two efficient numerical methods for analyzing the propagation characteristics of fin-line structures were presented. The first of these employs the spectral domain technique [1], [2], whereas the second utilizes network analytical methods for electromagnetic fields 3], Both of these methods are based on the hybrid mode formulation as opposed to the TE approximation [4], but they neglect the effect of the metallization thickness, which increases with higher operating frequencies and narrower gaps in the metallization. Hybrid analysis and the effect of the metallization thickness were given for the microwave planar transmission lines [5], 16]; however, the procedure for deriving the Green s functions is rather involved and lengthy. This paper presents an efficient method which circumvents the laborious formulation process for fin-line structures with a finite metallization thickness. Although the method is an extension of the treatment in 13], [5] and [6], it derives Green s functions using the conventional circuit theory rather than by directly solving the differential equations with boundary conditions. Some numerical results are presented and compared with available data.

4 II. THE NETWORK FORMULATION OF THE PROBLEM The unilateral fin line shown in Fig. 1 is used to illustrate the formulation procedure, but the method itself is quite general. Application to other structures will be discussed later in this section. As a first step, we express the transverse (to z) fields in each region by the following spectral representation: E ^ (x,y,z)' i i =1 n=0 \ -jf30y (1) (x,y,z)' i i 2 > w 4 n )(x> i = 1,2,3,4 where the vector mode functions f ^ ^ and g ^ ^ in each region are given as: A) region (1), (3) and (4) fln} = i ^ i l A {x0aa Cos(aAx) ** ytf&q sin <aax)} TCi) 1_ r~~. q f2n K J 2A ^X0J^0 cos(aax) - yqaa sin Ca^x)} r(i) _ t - 7 a > In "0 A An zn x fo' O' = 1.2) ( 2) a, n^r * KA / a + e2q n { 1 (n=0) 2 (n^o)

5 B) region (2) f(2) zl/3a /=- In = v/2w {W os(awx) - y02 60sin(oiyX)} r(2) 1 :2n = *W {x0j cosco^x) - yq O tysinco^) } (3) ~C2) _ - T (2) g n z0 x f n a = i,2), 0,, = ^, is, = 7 4 + where 3Q is the propagation constant, and xq, yq, and ~zq are the x-, y-, and z-directed unit vectors, respectively. It should be noted that the vector mode functions f ^, g ^ satisfy the boundary conditions at x = jjtf and +A with the following orthonormal properties: J (i) * Ci) «'a' 0 0 * z0 x fin «= «U. 6, (i = 1,2,3,4) (A) Substituting expression (1) into Maxwell's field equations, we obtain the differential equations for the modal voltages and currents: i y U ). (1) (il.ul dz V - 2K zi V 4 xa). 1Ka ) wa ) v a ) dz in JK» v»" h V (5) where C l) f >/io2 ( i) -yq - K (i)-2 en (region (3)) Cl) f r 0 ^ (otherwise) q K (i) Ky (region (2)) { (otherwise)

6 p - K (i) z ( i) (x) (IT z2 %,(i) ( 6 ) C Q 1 H = (: a = 1,2) The boundary conditions to be satisfied are expressed as follows: v f a < dl> = 0 (7) V in)(t+0) = Wf (8) «r (91 v i > > - yr (10) v t o t - > = v\ ~ (11) ^ (-d2+0> = V 2n (-d2- > (12a) 4 n (-d2+0> 4 n <-d 2- > (12b) Vi n (-d 2-d3) ' 0 (13) and H^1) (t-h>) = h 2) Ct-O) ( x <W) (14) h 2)(+o ) = h 3)(-o ) ( x <W) (15)

7 where voltage sources v a+ Z b+, are given as: e (xf) CL dx ' (16a) t (2)* Jin Cx1) efe(x') d x 1 Cl6b) f(3)* Jin Cx ) and where e^ and e^ are the transverse electric fields at z = t and z = 0, respectively, = xne 0 ax + Yne y0 ay (17) Equivalent circuits in the z-direction can be derived by considering the differential equations (5) together with the boundary conditions (7)-(13) (Figure 2). The modal voltages V ^ and currents in each region can be obtained by using conventional circuit theory. Electromagnetic fields in each region can then be derived by substituting and 1 ^ & Jin Jin into (1). Finally, the application of the remaining boundary conditions (14) and (15) results in the following set of equations for the unknown electric fields ea and e^ at z = t and z = 0, and for the unknown propagation constant 3^:

8 I l / Y ^ C t + O l t ) g ^ C x ) f }*(x ) e_(x') dx* Jin =1 n=0 -W = 2 oo w \ ( 2 ),, 7 ( 2 ) *,, l l / i v f ( t - 0 t ) I ^ ( x ) f ^ " ( x ' ) e (x') =1 n=0 -W 1111 * a (18a) + Y<2)(t-0 0) g ^ ( x ) f 2)*(x') e ^ x ' ) } dx A J o L {Y^ )( K t) S*n>(x> *?*<* > * ea(x > (2) + Y^'(+0 0) g[2)(x) 4 n )X(x,) * eb (x')} dx' (18b) 2 00 W f l l J =1 n=0 -w,(3) I(3) SJln T O )* Jin (x') eb (x') dx' where (i) (z z ) are the Green s functions which relate the modal currents (z) to the voltage source in each of the three regions. The set of equations (18) is rigorous. Numerical solution of the equations is discussed in the next section. Before concluding this section, we explain why the present formulation is efficient and why it is convenient to analyze various types of fin-line structures using the present approach. The equivalent circuit for a unilateral fin-line was derived during the formulation process of (18). However, these circuits could have been obtained directly by inspection, without going through the intermediate steps represented by Equations (1) - (13) regardless of the number of gaps and/or dielectric layers in the fin- line. For example, Fig. 3b shows the^equivalent circuits for the bilateral fin line whose geometry is given in Fig. 3a. Once the equivalent circuits are obtained, we can derive the set of equations in (18) by using the

9 conventional circuit theory approach rather than by solving a set of differential equations, together with appropriate boundary conditions.

10 III. NUMERICAL COMPUTATIONS The numerical procedure for solving (18) is analogous to that used in [3], [5], [6]; therefore, only a summary of the steps will be given below. The first step is to expand the unknown electric fields e and e, at a D z = t and 0, in terms of an appropriate set of basis functions, e.g., (19) where a ^ and are the unknown coefficients. The second step is to apply y y ' the Galerkin's procedure to Eq. (18), which results in the determinantal equation for the propagation constant $q. Finally, the determinantal equation is solved for the propagation constant $ q in the fin line structure. Accurate solutions can be obtained with only a small number of basis functions, if these functions incorporate the edge effect. The following basis functions were used for the numerical results presented in this paper: ( 20) where T^(y) and U^(y) are the Chebyshev's polynomials of the first and second kind, respectively.

11 Preliminary computations show that N = N = 2 in (19) is sufficient x y for deriving accurate results for finite metallization thickness just as in the case of zero metallization thickness [3]. Figure 4 shows the effect of the metallization thickness t on the effective dielectric constant and the characteristic impedance in a unilateral fin line. The effective dielectric constant and the characteristic impedance Zq are defined as <V<sTO2 ave where Vq is the voltage between the fins and P& is the average power flow along the y-direction. The finite thickness reduces the propagation constant in the higherfrequency range as it does in the open slot line I5j, because in these frequency ranges the fields are concentrated near the gap in the fin-line and it acts similar to an open slot line. In contrast, a fin line behaves as a ridged waveguide near the cutoff frequency, and consequently, the thicker its diaphragm, the lower its cutoff frequency [7]. Figure 5 shows the effect of the metallization thickness of a bilateral fin-lin e» The results for the limiting case of t 0, i.e., zero metallization thickness, are compared with those published by Schmidt and Itoh [1], and the agreement is quite good.

12 IV. CONCLUSIONS In this paper, the hybrid mode formulation was used to analyze fin- line structures with finite metallization thicknesses. This formulation used in conjunction with the equivalent circuit analysis is considerably simpler than the conventional Green s function approach. This method itself is quite general and can be applied to different types of fin-line structures by a simple modification of equivalent circuits which can be obtained easily. Numerical results are presented to show the effects of finite metalr lization thickness on the propagation characteristics of unilateral and bilateral fin-line structures.

13 REFERENCES [1] L. P. Schmidt and T. Itoh, "Spectral domain analysis of dominant and higher order modes in fin-lines," IEEE Trans. Microwave Theory Tech.«vol. MTT-28, pp. 981-985, September 1980. [2] L. P. Schmidt, T. Itoh, and H. Hofman, "Characteristics of unilateral fin-line structures with arbitrary located slots," IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 352-355, April 1981. [3] Y. Hayashi, E. Farr, S. Wilson and R. Mittra, "Analysis of dominant and higher-order modes in unilateral fin lines," A E U, vol. 37, no. 3/4, pp. 117-122, March-April 1983. [4] A. Beyer, "Analysis of characteristics of an earthed fin line," IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 676-680, July 1981. [5] T. Kitazawa, Y. Hayashi, and M. Suzuki, "Analysis of the dispersion characteristic of slot line with thick metal coating," IEEE Trans. Microwave Theory Tech., vol. MTT-28, pp. 387-392, April 1980. [6] T. Kitazawa, Y. Hayashi and M. Suzuki, "A coplanar waveguide with thick metal coating," IEEE Trans. Microwave Theory Tech., vol. MTT-24, pp. 604-608, September 1976. [7] J. R. Pyle, "The cutoff wavelength of the TE-^q mode in ridged rectangular waveguide of any aspect ratio," IEEE Trans. Microwave Theory Tech., vol. MTT- 14, pp. 175-183, April 1966.

REGION (I) d,-t (2) ][ (3) d2 (4) d3 Figure 1. Unilateral fin line.

15 SHORT CIRCUIT. Z= d, /C(l), ZjI'* Z=t + 0 Z = t - 0 k. z '2) Z= + 0 (3) (3) K. Zf,-(4) (4) K ZX Z.= - 0 9 Z = -d SHORT CIRCUIT Z = -{ d 2+ d3) Figur.e 2. Equivalent circuits for transverse section of fin line.

16 (a) SHORT CIRCUIT e e SHORT CIRCUIT (b) Figure 3 Bilateral fin line (Metalization thickness is neglected in this figure).

(AA,) Figure 4. Propagation characteristics of unilateral fin line.

18 </vu f(g H z) r- 3.0 W = 0.075 (mm) A = 1.78 (mm) d =d3= 3.4925 (mm) d2= 0.0625 (mm) Figure 5. Propagation characteristics of bilateral fin line.