I. INTRODUCTION The need for more accurate microstrip circuit simulations has become apparent with the recent interest in millimeter-wave and near millimeter-wave frequencies. The development of more accurate microstrip discontinuity models is very important in improving high frequency circuit simulations. In most applications, the circuits are enclosed in a shielding cavity as shown in Figure 1. This cavity may be considered as a section of a waveguide terminated at both ends. The presence of the shielding cavity affects the performance of the circuit (shielding effects) and has to be taken into consideration. It has been shown [1] - [2] that one condition where shielding effects are significant is when the frequency approaches the cut-off frequency of the waveguide's dominant mode. In most cases, microstrip circuits including active devices are printed on multilayer structures which consist of a combination of dielectric and semi-conducting materials. The existance of these conducting layers can affect the characteristicts of the loaded cavity and, therefore, of the printed circuits. As it has been pointed out by many authors [3] - [5], the propagation characteristics of higher-order shielded-microstrip modes are very similar to those of the shielding cavity. Consequently, a good understanding of how microstrip modes propagate may be gained by just studying the dielectric-slab loaded waveguide. In this report we consider the case of a single semi-conducting layer with a dopping density ND which varies from 1

1014 to 1016 and we study the effect of this variation on the cutoff frequencies of the waveguide modes. Conclusions drawn from this study show a very interesting behavior in the propagation characteristics of the modes and may be extended to the case of the shielded microstrip. II. THEORETICAL FORMULATION Figure 1 in Appendix I shows a basic description of the loaded waveguide. The modes excited in this structure are LSE and LSM and their characteristic equations which may be derived by applying the transverse resonance condition [6] are shown below: k k X1 2 -xtan (k h) - tan [k (a-h) LSM (1) - cot (kxl h) =- cot [k2 (a-h) LSE (2) gL1 g,2 The eigenvalues kxl and kx2 are given by 2 2 2 2 kxl + ky + kz = (0 ~3) 2 2 2 2 kx2 + ky + k = 22 (4) with nic k = Y b E1 = E1' (1 - jtan61) (5) 2 = E2' (1 - jtan82) (6) 2

and tan2 = 0 eND tans =. (7) wE In equation (7), e is the charge of an electron and ND is the doping density of the material. For the case a perfect dielectric layer, cut-off is defined by kz=0. However, when tans is different than zero, the cut-off condition is modified to the following Re (k) = 0 (8) This condition imposed on equations (3), (4) can give: 2 2 2 2 kx = 1' l (1 - jtan1) - a2 - k (9) Xi l ~~~z y and 2 2 2 2 k2 = ( 2 - aZ - k (10) By substracting equation (9) from (10) we can derive a relation between kxl and kx2 which does not include the eigenvalue ky and the attenuation constant at cut-off az: kx - k2 = 0)2 [1 (1 - jtanSl) - e (11) kX1 -k~2 = ~ 181111 ~. - ~ ~2~2J (11) The solution of the sets ((1),(11)) or ((2),(11)) can be performed only numerically and results in infinite many but descrete 3

eigenvalue pairs (kxl, kx2)m which vary with CO and n. The frequency which satisfies (8) for a given pair (kxl, kx2)m and ky= nc/b is the cut-off frequency of the mn mode. This procedure is rather complicated and requires extensive computation. To avoid this shortcoming the cut-off condition is modified to k = 0 (12) z Equation (12) together with (3) and (4) transforms the characteristic equation into a complex equation for (0 resulting in complex cut-off frequencies. The real part of this cut-off frequency will be exactly equal to the one that condition (8) would give. However, the imaginary part which, in general, is about an order of magnitude smaller than the real, compensates for the neglected attenuation at cut-off and is disregarded. The results derived during this study are based on the second condition. III. NUMERICAL SOLUTION Numerical solution of the characteristic equation was achieved with Muller's Method [7] which is iterative in nature and requires a good initial guess for fast convergence. Furthermore, when solving for cut-off frequencies there may be a number of solutions to the characteristic equation in a relatively narrow frequency space. To overcome this problem, a method was developed to track a given mode through increasting doping densities. That 4

is, the solution for the cut-off for a given mode is determined first for no losses where zeros are spread further apart and this solution is used as the initial guess as the tans is slightly increased. The numerical solution for the lossless case proved to be much more simple than the lossy one. The characteristic equation was solved with the bistatic method [7]. IV. RESULTS AND CONCLUSION The results derived using the technique described above are plotted in figures (1) - (34) and are for the waveguide geometries of Table 1 in Appendix II. From these results it can be concluded that the effect of the conductivity in the dielectric layer can be tremendous. In some cases, as the doping density increases from 1014 to 1016 there seems to be a switching of dominant modes. That is, higher order modes tend to exhibit a lower cut-off than the mode which was dominant at lower ND resulting in much lower cut-off frequencies. In addition, for other geometries, increasing conductivity seems to have an opposite effect. Presently, we are trying to investigate the effect of the presence of semi-conducting materials on the modes of a shielded microstrip printed on single, as well as multi-layer substrates. 5

APPENDIX I Computation of the cut-off frequencies for the case of a non-conducting layer: I } Region II Xdim= a 1'^/^''Region I 1 El tan1 h 1<- Ydim = b d = a-h 2 2 2 2 kl + k + = C1 2 2 2 2 kx2 + k + k = z E292 yz For LSM, LSE kz is set to zero to determine the cut-off frequency. The The dominant LSM mode corresponds to TEo1.*. ky = K/b dominant LSE mode corresponds to TE0.. k A 0.0 10 y = 6

Determining the next higher order mode: LSM: Interation begins assuming a lower bound which was the cut-off frequency for the dominant mode. Two possibilities are tested: (i) M = 0, N = 2 (i.e., ky = 2C/b) (ii) M = 1, N = 1 (i.e., ky = C/b) whichever case yields the lowest fc is taken as the next higher mode. LSE: Similar to the above, using instead the following two cases: (i) M = 2, N = 0 (i.e., ky = 0.0) (ii) M = 1, N = 1 (i.e., ky= 7/b) Iteration for dominant mode solution: (Lossless dielectric) LSM: A lower bound is determined by the following: F = c/Zb A TE1 cut-off for air-filled WG. o 01 01 with c the velocity of light in free space. Fo Fd = - A TE0 cut-off for completely filled WG. 01 / Fd is used as the lower bound. 01 LSE: Similar to the above, using F = c/Za A TE1 cut-off for air-filled WG. o 10 01 7

APPENDIX II Group A Plots Symbol definition: * A + LSM1 LSM2 LSE1 LSE2 Waveguide Parameters: Plot # a (in) 1 2 3 4 5 6 7 8 9 10 11.305.305.305.305.305.25.25.25.305.305.305 b (in).305.305.305.305.305.305.305.305.25.25.25 i Substrate Parameters e riL.15.08.025.025.025.08.08.08.08.08.08 3.0 3.0 3.0 12.0 16.0 3.0 12.0 16.0 3.0 12.0 16.0 8

GROUP A "Cut-off frequencies vs. tanS for LSE and LSM modes" 9

'' H 1-' 620:1 1:15 CUTOFF FREQUENCY.VSo LOSS TRNGENT I1 35. 26.25 - C) ) — LLJ C3 Ll L_ 17.5 8.75 0. 0. 1.25 3.75 5. 6.25 7.5 8.75 10. LOSS TRNGENT

21-APR — 88 20: 11:34 CUTOFF (REFL RND IMRG).VS. LOSS TANGENT FOR LSM # I 35. 26.25 NJ - LLJ z LJ C) LL 17.5 8.75 0. 0. 1.25 2.5 3.75 5. 6.25 LOSS TANGENT 7.5 8.75 10.

21-5PB-88 n0:. -1:52 o,,/$ LS GEN FO R.\f5. LOSS0 ONo I MRG CTLOFF {REFL 35. 26.25 r-J I N-,-, t 1Zz C3 uj cc uj CC U-.5 10. 7.5 3. 2.5 OSS T Gc - Uoss ^ P 0. -f0.

2 I1 PR-88 20:18:-56 F BEOUENC a.VS LOSS TRNGEN? CUTOFF 35. 26.25 I I.5 [7.5 l_ LOI CJ uclLl 8.75 10. 7.5 0. 4 — 0. 2.5 1.25 LOSS 1 NGENT

21-RPR-88 20:19:10 TRNGENT FOR LSM CUT OF REL N 35. 26.25 5I — O 17.5 Z I LUI CS3 LL o/Ss LOSS 8.75 0. 8.75 10. 7.5 q 7S J.' 0. LOSS TRNGENT

I -RFR-88 20'19:1'O NS. LOSS T ANLE F SE. V,0 LOSS R RNO i MEGA CUTOFF BRERL 35. 26.25:rC5 o 17 C3 LLJ g:2 LL. 8.75 I 1to. 7.5 3.' 2.5 5.C S G ^.25 LOSS ANGENT 0. -- O. 1.25

21 -RPR-88 20:21:45 CUTOFF FREOUENCY,VS5 LOSS TRNGENT *3 35. 26.25 -) 1[7.5 LU CD LL LL 8.75 - 0. - I i I 0. 1.25 2.5 3.75 6.25 7.5 10. LOSS TRNGENT

21-PPR-88 20:22:0] CUTOFF (RERL RND IMRG).VS. LOSS TFNGENT FOR LSM #3 35. 26.25 ----- -- - ---— ~~~~~~~~~~~~-~~ I~~ ~~~~ —I'-~~mom TLLJ z LU CD Ill 17.5 - 8.75 0. ---— cL-cc~ ""cc3" —`1~1" —~~~~~- -- — 14 01 00i 64. — --- — 0 i0 o 0. 1.25 2.5 3.75 S. 6.25 7.5 8.75 10. LOSS TRNGENT

21 -APR-88 20:22: 11 CUTOFF (REFL RND IMRFG.VS. LOSS TANGENT FOR LSE 35. 26.25 _LLJ C3 LL 17.5 IIXXM39 8.75 0. 0. 1.25 2.5 3.75 5. 6.25 7.5 8.75 10. LOSS TRNGENT

21-FAPR-88 20o:'-0 n CUTOFF FREQUENCY.VSo LOSS TRNGENT u4 35. 26.25 r-41J C-) LLJ Li Lij LL 17.5 7 i -—.a- wwowwww I - I - bd 0 0 & WOO "**** I!! I r., r —— r~i ~~* " * f~'tw - -LY Cn- &"-** —— s --- -,,,, -- 8.75 - Y~c~r~tl~rS~-~- ~ -~-ILLWIC~llr~~ ~f~~-***two 0. n1' 1 - 4 - 1 1 - -1 4 1 t U. 1.25 2.5 3.75 LOSS 5. 6.25 TRNGENT 7.5 8.75 10.

21 APR -88 20:25:32 CUTOFF (REFL RND IMRG).VS. LOSS TRNGENT FOR LSM 35. - 26.25 - - ---------- - 1- - - - - - - - - - - - - M- --- --. W. -- -- ---- ZI LJ ID UJ LL u 17.5 - 8.75 - 0. A - h.0 -—. —-.. - 6 --- i i O. 1.25 2.5 3.75 5. 6.25 7.5 8.75 10. LOSS TRNGENT

21 -APR-88 20: 25:4 CUTOFF (RERL AND IMHG).VSo LOSS TRNGENT FOR LSE:a4 35. 26.25 LC CD Ll rL Li_ 17.5 - I __,. —----- -, —--------- ------------------ -- - ---------- - - ------------- --------------- -------- 8.75 0. L4zw --- - fts-31M -- --- - mooo-z - r - - to I*$* 1618400004466idba- - - I —-------- r- I -— I I 4 i - i 1 1- - 1 - i - -, I -,, I -- - - a - up — I I I I I I I. - T-.;f - - Z-FZ=4 I 0. 1.25 2.5 3.75 5. 6.25 7.5 8.75 10. LOSS TANGENT

21 -PPR-88 19:50:42 _I C) )_z Z LlJ iD LU ffr I CUTOFF FREQUENCY 35. 26.25 - 17.5 8.75 - 0. I I I I I I I I I I I I.VSo LOSS TRNGENT u5 t0. LOSS TRNGENT

21 -APR-88 19:5C:27 CUTOFF (RERL AND IMRG).VS. LOSS TANGENT FOR LSM 45 35. 26.25 LLJ LU LL 17.5 8.75 0. 0. 1.25 2.5 s. 6.25 7.5 8.75 10. LOSS TANGENT

21 -PPR-88 19:58:26 CUTOFF (REFL RND IMRG).VS. LOSS TFNGENT FOR LSE 35. - 26.25 - ----— --- —- --- -----—` - -- -- I z LU CD LL rI 17.5 - riir ri 8.75 - 0. I a,I, 1 1 1 1 1 1 I 1 1 F m r..: T r.-..1._I 0. 1.25 2.5 3.75 5. 6.25 7.5 8.75 10. LOSS TRNGENT

23:24:32 TRNGEaf. So. LOSS FBEQUE~jcY C,O-OFF 35. 26.2S t I \ z t7.6' - 17.5 U3 O3 UL~C 10. S. 3.75 2.5 XpNGENW 0. 40. \.25

21 -FPR-88 23:.:'4 4 CUTOFF (RERL RND IMRG).VS. LOSS TRNGENT FOR LSM tG 35. 26.25 TJ >C.) z LL C3 LJ c:I 17.5 8.75 0. 0. 1.25 2.5 3.75 5. 6.25 7.5 LOSS TANGENT 8.75 10.

21-RFR-88 23-.24:56 LOSS TRNGENT FOR LSE CUTOFF RBERL PNO MVGI *VS0 35. 26.25 - >z LU LU LL 17.5 8.75 1 0. 8.75 10. 6.25 7.5 5. 3.75 1.25 2.5 w 0. LOSS T NGENT

21 -PPRF-8 23:32:42 CUTOFF FREQUENCY 35. 26.25 17.5 8.75 -.VSo LOSS TRNGENT #7 Uz LUJ C3 rr ID LL 0. 0. 1.25 2.5 3.75 5. 6.25 7.5 8.75 10. LOSS TRNGENT

21 — PR- 88 23:32:53 CUTOFF (RERL RND IMRG) 35. 26.25 oVSo LOSS TANGENT FOR LSM C-) CD LL UI i 17.5 8.75 0. 0. 1.25 2.5 5. 6.25 8.75 10. LOSS TRNGENT

21 -APR-88 23:33:04 CUTOFF (RERL RND IMRG).VSo LOSS TRNGENT FOR LSE'-/ CID C-) I i LLJ LUI GC 35. 26.25 - 17.5 8.75 0. 0. 1.25 3.75 56.5 —T7 —r. I,.. 75. S. G.25 7.5 8.75 lO. LOSS TANGENT

21-H -'-88 23:43:56 CUTOFF FREQUENCY.VSo LOSS TRNGENT 35. T 26.25 + N_ Cz LJ 03 LLJ rf' LL_ 17.5 + 8.75 - N% --- i offimam - PM 0. 0. 1.25 2.5 3.75 5. 6.25 7.5 8.75 10. LOSS TRNGENT

21 -APR-88 23:44:07 CUTOFF (REFL IND IMFG).VS. LOSS TFNGENT FOR LSM 48 35. T 26.25 - z LlJ CD LL Ml 17.5 - 8.75 0. - --------- 0. 1.25 2.5 3.75 5. 6.25 7.5 8.75 10. LOSS TRNGENT

I I -ne88 -23:44:6 293..44-..1.S LOSS G L-SE PNO I Oq} CuoOFF REP L 35. 26.25 t^c 5LO C3 ccL 8.75 7.5?4 S.''6.S25 LS 1 ~,GENT 0. - O.

22-PR-88 00oo-.:l3-.16 E C so SSS TRNGEN FBEQUENI.^ L0C5 CUTOFF 35. 26.25 r -, 17.5 > — U:1 LU IlI 8.75 d 0. 10. 3. 1 I I - 7.5 5. 6.25 75 5. L0SS X NOEN1 2.5 1.25.O

22-Hi -0-88 00 13'26 LOSS T NGENT F OR LSM iREPL NDNO IMFRG CUTOFF IREL g 35. 26.25' L0 t7.5 0 LU C3.- AvS. 8.75 0. 8.75 to. 7.5 3.' C..575S LOSS T NGEN 0.

2'2-PR-88 o: 1 3:35 R. 55. SS RHGEN FSOR LSE RNO i MRG) -.V. L 0 cu7 OF 35. r — CC) zZ`Z: 0G uj cc-_ 17.5 to. 8.75 7.5 -'G.25 5. i NGEWN 2.5 0. -F 0. 1.25 LOSS

22-APB-db 00:18:32 CUTOFF FREQUENCY.VSo LOSS TRNGENT 35. - 26.25 t10 T-J >z LU 03 LJ 1:: cr I I 17.5 8.75 0. 0. 3.75 5. 6.25 7.5 8.75 10. LOSS TRNGENT

22-APR-88 00:18:39 CUTOFF (REFL RND IMRG) 35. 26.25 -,VS5 LOSS TRNGENT FOR LSM itI0 I -j (_) z LLJ ID1 LLJ fi 17.5 8.75 0. 0. 2.5 3.75 5. 6.25 7.5 8.75 10. LOSS TRNGENT

22-A -88 00:-18:4 ANO IMRG) oVSo LOSS T RNGENT FOR LSE CUTOFF 35. 26.25 () 17.' zL C3 CC: LL 5% + 8.75 0. 10. 5. 6.25 L6 5 T.G 7.5 375 _.., 1.25,j. 0. LOSS TANGENT

22- HH- 88 00:21:02 -IJ r-41 z Lli LUJ CL LLJ CUTOFF FREQUENCY 35. 26.25 - 17.5 - 8.75 O, I i I 1 1 1 1 I i 1.VS. LOSS TRNGENT n11 0. 1.25 2.5 3.75 7.5 8.75 10. LOSS TANGENT

22-PPR-88 00:21:09 CUTOFF (REFL FND I MGI oVSo LOSS TFNGENT FOR LSM 35. 26.25 I',, -) z LJ CD Li 17.5 8.75 0. 0. 1.25 2.5 3.75 S. 6.25 8.75 10. LOSS TRNGENT

22-PPR-88 0021-2'7 CUTOI3FF ERL AN\D A 5 L55 T E Hh 35. 26.25 I I () 17.5 Z LU CE L t 87 8.75 0. 7.5 6.25 3.75 5. 2.5 ql t 1.25 0. LOSS FTNGENT

Comments In this group there are two plots which served as a motivation for generating the plots of groups C-F. These plots demonstrate the interchange of the mode order (Figure B.1) and the sensitivity of Muller's method to initial guesses (xi, Xi.1, xi_2) (Figure B.2). 10

17-32PR-88 21 -.:32:-.36,O IRCH^ VS iOSS RNO M M R, TFRNGEN CUTOFF 35. 26.25 ItS >. 17.5 uJ C3 LGi I to10. 8.75 7.5 2.5 3.'. 75 LOSS 1NGENT 0. — V 0. 1.25

INI3 NUli SS31 SE'9'S S/'E'OI SZ'8 -T Du z F -!2D m S'/I n — < rj IN3NUHI SSS1 ~SA i JloinL e:90:E8 899-HdiU-81

Specs: Type of Mode = LSM ky = 7/b h = 0.025" a = 0.305" b = 0.305" e -= 16.0 Observations: * It is obvious that there are two LSM modes very close together. * As tand increases, one can see that the original 2nd order mode has moved to become the dominant. 11

I L I CJ m e\ (Q

Equation for Epsr O9405 3o 3

-I~s=A 4 A09 (g) 7, -1

g H CO' jr4 F& &I -1 r I _ = I ( N b"I /w / *4 /V

DO 1 r -I -V CO. fo' k C / *' /I; I c9

Equation for Epsr= L O 31.4 27.4 23.4 F-real 19.4 15.4 11.4

4' -, ) 4 c V A /'A; Itt.4 3 F, A (-JO 1d

I O O B ^^ * — - _ _ / t <b', I c~ co~ < b* F s CO' ( S IV

0 cci O 0F4 &I\ la

Type of Mode = LSE ky= 0.0 a = 0.305" b = 0.305" ~r = 16.0 h = 0.025" Observations: * At first glance, one might sense a problem with these plots. The specs are nearly the same with those of Group A, #5 which shows no mode crossover. Yet it is clear from these plots that the dominant LSE mode is replaced as tand is increased. * The difference between this curve and the corresponding one in Group A is in ky. 12

Co 0\ A, S. ^ -A) lb AC 4% ) r rIj L e)4 H o C L -.Cb',. co'r

A) <o Ij r0 Ln ^'\ J.V V )., co 11.1

I 4N. t IV 0i r l 9 E 0 f-p 9 1. K <<', ~Q "le

C~ I ~ F-T A.' o. v' IV

Equation for tand=LO3 (LSE) -19L 1 EB-' 31.4 27.4 23.4 F-real 19.4 15.4

Equaitiorn for tand=0.5 11.4 F-real

Eqiuator for t-anld0I V F-real

!b A.o L j 0 1 L~ Oq 01

k ) L A L I I-. r t \J ~ 0

GROUP E Type of Mode = LSE ky = K/b a = 0.305" b = 0.305" Sr = 16.0 h = 0.025" Observations: * Following the progression of these plots, it is clear that there is a mode which remains nearly fixed for increasing tand. This would be the LSE2 mode which appears in #5 from Group A. It also appears that another higher order mode is cutting across. 13

L. 6).s D' L A, E C ) L I I A co I \ C I S er

Equation for tand=0.5 (LSE) V V 11.4 F-real

t-1 -- 1 ~-'.v o t 42

Equation for tand=i.5 (LSE) A i t>_ \ v V, C. /L7 e`4.If <9 —

Equation for tand20 (LSE).Z. %7,~~,o. / 44^

GROUP F Type of mode = LSM ky = K/b a = 0.305" b = 0.305" h = 0.025" Er = 12.0 Observations: * By changing Er from 16.0 to 12.0 we can observe much greater stability in the relationship between LSM1 and LSM2. Note, however, that a higher order mode is still seen to be moving to the dominant position. 14

Equation for tand=0.5 (LSMi Vil I F-real

Equatiaon for tand=ILO (LSM) I 31.4 27.4 23.4 F-real 19.4 15.4 11.4

Equation for tarndL,5 (LSM I 1 1 F-real

"I.r m -7 ff —tl r-?-I I L 6 -A C -.jA.-j -IL k-l/ LI/ A- j 15.4 11.4 F-real

1. L.P. Dunleavy and P.B. Katehi, "Shielding Effects in Microstrip Discontinuities - Part I: Theory." It has been submitted for publication in IEEE Trans. on Microwave Theory and Techniques. 2. L.P. Dunleavy and P.B. Katehi, "Shielding Effects in Microstrip Discontinuities - Part II: Applications." It has been submitted for publication in IEEE Trans. on Microwave Theory and Techniques. 3. E. Yamashita and K. Atsuki, "Analysis of Microstrip-Like Transmission Lines by Nonuniform Discretization of Integral Equations." IEEE Trans. Microwave Theory and Techniques, vol. MTT-24, No. 4, pp. 195-200, April 1976. 4 M. Hashimoto, "A Rigorous Solution for Dispersive Microstrip." IEEE Trans. on Microwave Theory and Techniques, vol. MTT-33, pp. 1131-1137, Nov. 1985. 5. C.J. Railton and T. Rozzi, "Complex Modes in Boxed Microstrip." IEEE Trans. on Microwave Theory and Techniques, vol. MTT-366, pp. 865-874, May 1988. 15