~IEEE
TRAN SACTI 0 N S
ON
MICROWAVE THEORY AND TECHNIQUES A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY
SEPTEMBER 1995
VOLUME 43
NU MBER 9
IETMAB
(ISSN 00189480)
PART I OF T\\'0 PARTS
PAPI'RS
TE Modes of an Axially MultipleGrooved Rectangular Waveguide ... . .. . ... . ..... . K. P. Ericksen and A. M. Ferendeci High Frequency Performance of Multilayer Capacitors .... . .. .. .. . ... . .......... . .......... A. T. Murphy ami F J. Young Coplanar Waveguides and Microwave Inductors on Si licon Substrates . ..................... . .. . ... .. ....... . . .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . A. C. Reyes, S. M. EIG/wza/y, S. J. Dom, M. Dydyk, D. K. Schroder, and 11. Patterson Method of Moments Analysis of Anisotropic Artificial Media Composed of Dielectric Wire Objects .......... . ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. E. Peters and E. H. Newman Contour Integral Method wi th Fringe Complex Images for the Rapid Solution of Patch Resonators of Arbitrary Shape .......................................... . ...... . .. . ...... . ...... . ... . . A. A. Omar, Y. L. Chow, and M. G. Stubbs Direct Extraction of Equi valent Circuit Parameters for IIeterojunction Bipolar Transistors . .. .. . . .. . . ... . .. .. .. . . .. .. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. 1. Wei and J. C. M. Hwang Scattering at the Junction of a Rectangular Waveguide and a Larger Circular Waveguide .. .. .. . ....... . ... .. ... . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. H. MacPhie and K.L. Wu A Full wave CAD Tool for Waveguide Components Using a High Speed Direct Optimizer ... . ... ..... ..... . . . . . . .. . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Alessandri, M. Dionigi, and R. Sorrentino A ll igh Acc uracy FDTD Algori thm to Solve Microwave Propagation and Scattering Problems on a Coarse Grid ... . . . . . .............. . ........... . .................................... . ........ . . . . .. ....... . . J. B. Cole Inverted Stripline Antennas Integrated with Passive and Acti ve SolidState Devices ... . . . . .J. A. Navarro and K. Chang Mode Conversion and LeakyWave Excitation at OpenEnd CoupledMicrostrip Discontinuities ...................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. L. Cina and L. Carin Development of Self Packaged Iligh Frequency Circuits Using Micromachining Techniques ....... .. ............ . ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. F. Drayton and L. P. B. Katehi Generalized T LM Algorith ms with Controlled Stability Margin and Their Equi valence with FiniteDifference Formulations for Modified Grids ............... . ........ . .. . ...... . . . .... . . . . . ... . .. . M. CeluchMarcysiak and W. K. Gwarek Examination, Clarification, and Simplification of Modal Decoupling Method for Multiconduetor Transmission Lines ............... . .. . .. . ........ . . .. . . .... . ...... . ...... . ... . . . ...... . . G.T. Lei, G.W. Pan, and B. K. Gilbert Dispersion Analysis of the Linear VaneType Waveguide Using the Generalized Scattering Matrix .... . ... . ... . ... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. S. Best, R. J. Riegert, and L. C. Goodrich ModeMatching Analysis of T E011 Mode Waveguide Bandpass Filters ...... . ... . .. A. Melloni, M. Politi, and G. G. Genti/i (Continued on hack col'er)
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IEEE M IC ROWAVE T II EORY AND TEC H NIQ UES SO C IETY ·1 he MICrll"'II"C 'l11cury .md J"cchni(JUC\ Society J\ an org:miniiiUII, within the fr.unc\HIIl. of the IEEE. uf rncmhcl"'\ with prmcipal pmfc!l,imutl 1111Crc\h in the hcltl uf lm cruw:1vc theory and tcc:hmquc\. All mcmbcn. ol the ll+h :lt'C eligible for rncrnbcl"\hit) in the .Society :anti will receive thi, TJ(ANSACTIONS upon r•:lyrncnt or the :11\IIUUI Society IIIC IIIhCNhip fcc nf S2K.tXl. Fur iufunntllion ill\ JOIIlllla;. wntc 10 the IEEE :11 the .&tlt.lrc'' hcluw. Ml'llrhrr f'"l'~~'l uf1tmnaelirmVJomnuiJ llrt" for prnmu1l ,,·r tmh·.
ADI\II NISTRATI VE C O I\I M I'ITEE R. W. BIERIG. Set·r,•tary M. A. MAURY. JR.• Vice l',rt'.lidt•llt T. KEMJ; RLEY M. GOLIO M. SCJIINDLER I{. I'OLLARD D. IIORNIJ UCKLE R. SUDIIURY E. A. RI;?.EK R. II. JANSEN G. T IIOI!EN
E. D. COIIbN. l'rt'•idelll J . T. BARR R. 1::. ARYAN S. J . FlbDZJUSn\0
/)i.rtilrguLrlu•d Ln·tun•rs W. CURTICE V. RIZZOLI P. GOLDSMITII J. R. WJIINNERY F. IVANEK
1/mwrary Uft' Mmrbrr.<
A. C llJTK S. ll. COliN T. ITOII
T. S. SMD K. TOMIYASU L. YOUNG
D. G. SWANSON. Trc•a 1'1 CI INIQUI:..\ i\ publi,hcd monthly hy the ln\titutc of ElcciJical and Electronic~ Engmccl"\, Inc. Rct~pon,ibality fur the comcnt'l reliitt~ upon the au tho~ and not upun the IH L, the Socatt)/Council, ur ~~ ~ rncmbcn. IEEE Coq mrutt Office: 3U J!a,t 47 Street. New VorL. NY 1001 72391. IEEE Optnallons Center: 44 5 lloc\ Lane. P.O. Bo~ 1331. PI\CJtoway. NJ OliS5511JI 'JJ TdCJJhonc: 'XlS981.()06() Prlee/Puhllcatlon lnformullon: lndivulual copiC\; IEEE Mcmbc" $ 10.00 (hn.t ropy only). nonmcrnhcN $2:0.00 per copy. (NOic: Add S.tOO pchl3j!C and h:andlmg th:trgc to omy urdcr lm111 S I.OO to S50.00. mcludmg prepaad ordcD.) Member ami nonmcrnhcr ~ub'iotription price\ :1\'tlllublc upon rt'tJUC'ol. A\oulnblc 111 micmhchc and rn~trohlrn. CopyriJthl und Ht print l•crmh\iun'\: Ah,twcung 1\ pcnmttcd w1tl1 crcd11 to the 'ourcc. Lll>rnnc\ nrc pcm1i11cd to phutocopy for pnv:uc u~ of p;llrOn\, prov1dcd the percopy fee indic:ucd 111 the code m the bonum uf the rif'\1 page '' poud throut:h the Cupynght ClcurJncc Center, 222 Ro"K:WCXKI Dri..c, Danvc~. M 1\ 0 1 92~. ror ull other copying. reprint, ur rcpublicution pcr111 1\\iun, write to Copynght~o ami l)crm•"'I OII" Dcp:1runcnt. IEEE Puhhc~ttonli Adm1111\lrat10n, ·ll$ llocs Lnnc, P.O . Box 133 1, f'1\Cntawny. NJ 08855 1331. Co1•yright (C) 1995 by "l11e 111\titutc of Elcctricnl nnd Electronic\ Enginc.."Crlrl, Inc. All riglu" n:\Cr\cd, Second cia!y ' 11 .1J ) ,
1m
( I 0) where m is an integer. Substi tu ting ( I 0) i nto (5)(7) gives the expressions for the fields i n the main body of the AGR waveguide. These arc
(4)
Here i 0. · · ·. N  1 and \li is the phase difference between 0. \If 0. the electri c field has successi ve grooves. For i the same magnitude and phase at the entrance to every groove. Al so. the value for EJ. at the groove entrance i s a max i mum . This results in maximum electron beam interaction when the wavegu ide i s the i nteraction reg ion of a high harmonic rectangular gyrotron. T his case is referred to as the 21r mode and the AG R cavity is designed so that it supports thi s mode. The other modes arc designated by their effect on 1]1 with the appropriate values for i and N substi tuted into (4). For example. w ith i = 1. N = 3. (4) gi ves \jt = 1r / 3. which is referred to as the 1r /3 mode 171. For !J = IJ. ( I ) gives a constant electric fie ld across the groove entrance. T his i s an approxi mation to the actual elec tric field. A more accurate assumption for the fields. w hich takes into account Ihe fringing fields at the entrance to the groove. is given in the Appendix. However, it is shown in the A ppendi x that if the groove depth to groove pi tch rati o rl/ 2 L is small. the constant field approximation siill gives very accura1c results. ext. it i s necessary to find the field expressions in Region I. For a waveguide in cutoff. the transverse components o f the magnetic field arc zero. Therefore. the T E field components in Region I arc E.,.. By, and H=. The field expressions in the AGR waveguide have Ihe same form as those found for a rectangular waveguide. The infin ite sum formul ati on for the three fie ld components arc 181
11
d
"co ...Jo
11 
COS
11
( ~ .)
(l)"'N sin
+
(. )sin (ky 11 .IJ) (OS h J..,  .1J  .   =  7r S l li(/>11 ,+1J) sin (/.:.,. 11  W ) ( )sin(k,111 cos k.,.ll  .1: . . II 7r S ill (/,; IJ) sin(k.c 11 .,. W).
+
'11.
y)}
( I I)
1111
111 = 
kJ.II + . ( . . cos(/>y 11 +.lJ) · { +s111 k.,.,, \1 )sllt (k.,., . .,;) . ( n 1r ky 11  S i ll ky 11  IJ)
k.,. 11 n 1r
•
+  _  Sill
II= =
!.:~oEo } Wf l
(
.)
•
k.,. 11  ll 1 Slll (k.,.,  .r)
cos(k,,,  y )
. ky,  sin (ky 11  b) ( 12)
f (
1)'" N
lll= 
(ky 11  !J) } . { sin (kx11 + + W ).cos (.f..kll " ./.·)  cos  '  .",.'..,... n 1r ky 11 + Si ll (ky 11  b)  .'J) } + si11 (!.:J., _  W ) cos ( /.:.,. :r ) cos(ky . ( 13) 71 7r /,·!JII S ill ( k!JII b) 11
11
where
11
= 2mJV i.
k.,.,. (I
( 14)
The amplitude Eo is determined fro m the input power suppl ied to the waveguide.
ERICKSEN i\ND FERENDECI : TE
~ l OD ES
OF where the impedance becomes infini te. When col(wf),f[(') equals zero the series resonances characterized by zero input impedance occur. T hen
(7) where i l. :3. 5. . . . Parallel resonances occur when tan wf../[C = 0. Then
(2)
(8)
(3)
(4)
In the expression for total inductance ( I ) the mutual term subtracts because the currents in the two plates o f the capacitor llow in opposite direc ti ons. In the circuit of the top plate current !lowing in the top plate cau cs an inductive voltage drop whereas current !lowing the oppo. itc direction in the bottom plate causes an inductive voltage ri se. A ll electrode currents. dielectri c displacement currents. and ground plane cu rrents form a closed loop i n which a magnetic field is generated. An altern ating current in the top electrode and ground loop produces a timevarying magnetic fi eld. a field which induces a current in the bottom elcctr·odc. The induced current in the bottom electrode increases as a functi on of frequency and tends to be in the opposite direction to the currcm in the top electrode. These induced currents cause mutual resistance which i s zero at zero frequency. T he mutual resistances compri se the ofTdiagonal term s i n the resistance matrix. The diagonal terms arc the ac resi stances of each indi vidual electrode. At zero frequency they arc the de electrode resistances and they increase as frequency increases. Although the mutu al resistance i s due to induced currents it docs not behave the same way as mutual inductance. Due to the proximi ty effect resistance. increases in cases where the currents now in opposite directions. Thu s in (3) the mutual
where i = 1. 2. J .... To obtai n a l umped equivalent circuit ror the transmission line we usc the power series expression ror . S. J. Dorn. and 11. Pallcr>on arc with S PS. M otorola. Inc .. Tempe. AZ 85281 USA. S. i\ I. ElGhaLaly and D. K. Schroder arc with the Department of Elec trical Engineering. Ari;ona State Uni versity. Te mpe. AZ 852875706 USA. ,\ J. Dydyk i> wi th GSTG. Motorola. Inc .. Sco11sdalc. AZ 85252 USA. IEEE Log Number 9.J 13427.
maximum Si resisti vity was about 1600 ncm. Si processing was poor (by today' s standards). some aspec ts o f semiconductor phys ics were not as we ll understood a they arc now . microelectronics interconnects i n a producti on environment were in the form of w ire bond, and electrical interconnects were microstrip lines. Si substrates w ith resistivities greater than 1600 ncm have been used in both theoretical 111. [91 and experi mental investigati ons 1101. In 111. 131 1101. the struclllre used for analysi s was the microstri p line. In most cases. the loss analysi s was done by considering metal and bulk semiconductor contributions. In none of the cases were Schottky (metal semiconductor) junctions nor metalinsulatorsemiconductor j unct ions considered. The microwave loss analys is was performed by considering the natband vol tage and the depletion w idth to be zero. Currently, both microwave and semiconductor technolog ies have matured. Several fabrication techn iques have been developed. incl uding submicron gate length transistor technologies. and flipchip. and bump interconnects 1111. The semiconductor physics concepts arc now well establi shed 1121. Moreover. two and th reedimensional electromagnet ic analysis tools arc well developed and can be obtained from several sources. Circui ts i n coplanar waveguide configuration s arc in products [ 131. Uncompensated highresi sti vi ty ( HR ) Si wafers arc available at low cost. T his certainly suggests that the potential o f Si as a microwave substrate should be reassessed. Previou sly. research has been performed on eloped semico nductor substrates to realize slowwave structures II+ 111 61 . T he aim was to control the substrate eloping and carrier distri butions to enhance the slowwave phenomena. One should distinguish between thi s previous research and the work presented here. In the past. doped substrates. which slow the wave at the expense of increasing losses, were utili zed. On the other hand, the ai m of this paper is to demonstrate the potenti al of using HR sil icon as a lowcost. lowloss microwave substrate. In the next sections. experimental results wi ll be presented to assess the performance o f HR Si substrates in the microwave band. Coplanar waveguides fabricated on HR Si. semii nsulating (S I) GaAs, and quart z substrates arc tested and thei r characteristics will be compared. M oreover. microwave i nductive struclllrcs arc also fabricated. and their performance wi ll be analyzed as well. II . S EM ICON DUCTOR TIIEORETICAL B ACKGROUND T he behavior of the semiconductor surface needs to be taken into account when realizi ng passi ve structures li ke
00 189.J80/95SOUJO © 1995 IEEE
REYES eta/.: COPLANAR WAVEGUIDES AND 1\IICROWAVE INDUCTORS ON SILICON SUBSTRATES
tran mi ion lines and inductors on emiconductor ub trates,
20 17
;\lclul Thlcknc\.~ " I''
"tl "
in panicular, the metalsemiconductor and the mctalinsulatorcmiconductor interfaces. When a metal is brought i nto contact with a semiconducti ng material, the surface of the semiconductor is depleted of carrier . The rc i ti vity of the depleted region is very high. The depth of the depletion region width " I )!'' and is given by Fig. I.
II' =
( I)
where. e,. is the dielectric constant of the semiconducti ng material , e0 i s the pcrmi uivity of vacuum, lj1" i s the bui ltin potential which is a function of the metallization system and of the semiconductor to w hich the metal system makes contact to. t f i the magnitude of electron's charge. and Nb is the dopant concentration. The key parameter in ( I ) is the dopant concentration. which for standard Si is approximately 5 x 10 16 (cm 3 ) and for HR Si is approx imately 10 13 (cm  3 ). On the other hand, when a metali nsulator system i s brought into contact with a semiconducting materi al, the surface of the . cmiconductor is fi lled with charges "Q_.." These charge at the urface can be: I ) accumul ation charges (same as bulk type): 2) depletion charges; or. 3) inver ion (opposite to bulk type) w ith depleti on charges ( mall compared to inversion charges). The resi sti vity at the surface of the emiconductor dccrca es due to the presence of such charges. T he semiconductor charges due to inversion are given by
.
.
Q.• (znvC'rsw n);::::
e,.e0 kTe¢·f2 L
(2)
Dif/
where. k is Boltzmann· s constant. T i s the temperature, s is the surface potential. and L o; is the intri nsic Debye length. The urface potential is given by
_ V _ \1 s 
•
_ {; e,.e 0 kTF (U.• , UF)
FD
g
CoJ·qL o,
·•
( ) 3
Us
U~=IU,I .
us Up =
F(U., . UF)
q., kT. qp
IT
(4)
= J cu,. (cU• +Us 1) + cU,. (eU• Us 1) (5)
where. I ~ i~ the gate voltage, F is the Ferm i potential, and \ ·FD i the flat band voltage which is gi ven by
vFBIo, .• CJJ 0
 I
0 o.r
Q;,(.• C
= 0)
OX
OJ'
l .r"(::1:) Q,.(.r)d.r. o
Cross scc1ion of lhc CP\V.
(6)
.l o
In (.f), ¢>,,.., is the metal emiconductor work function due to the metal semiconductor energy band difference. CJJ is the fixed charge density due primarily to structural defects in the insulator, Q, is the mobile charge density due pri mari ly to ionic impurities in the insulator, Q; 1 is the interface trapped
charge density due to structural defects at the insulator semiconductor inter face, and C'0 ". is the insulator(s) capacitance. T he key parameter is the insulator capacitance Cu . wh ich is inversely proportional to the insulator thi ckness. For an MOS gate. the insulator thickness is about I 50 A. For the transmission lines and inductors in the MMIC technology under study. the insulator thickness i s about 20.000 A. Further definition of parameters and explanation of the usage and restrictions of ( I )(6) can be found i n 17 HI 91.
r
Ill. Si. GaAs. QUARTZ SUBSTRATES A D TilE REALIZED STRUCTURES The topology of the structures analyzed in this study can be divided into two groups. The first group consists of coplanar waveguides (CPW) on several substrates. A cross ·ccti on of this structure is shown in Fig. I . The second group consists of inducti ve structures, also on variou s substrates. T heir cross section is shown in Fig. 2. T he CPW and the induct ive structure arc chosen for the study due to their strong field interaction with the propagation medium. The cross ·ection of the CPW and the inductors can be divided into two groups: metalsemiconductor and metalinsu latorsemiconductor. Three mai n substrate materials arc used in thi s comparative study, Si. GaAs, and quartz. The quartz substrate is u ·ed as a standard. T hi n insulati ng layers arc i ntroduced i n some structures between the substrate and the metalli zati on layers. as shown in Fig. 2. These thin insulat ing layers are common ly used in MM IC process for the realization of M IM capaci tors and the electric i olation between two metal layers. A ll the structures were processed under the same conditions I I 31 w ith the exception of SUB#6 which went first through a gate MOS oxide step. The geometries of interest arc: metal th ickness '·t" (2.5 JLm); substrate thickness "h" (quartz = 525 11m. Si = 400 JLm, GaAs = 625 11 m ); and insulator thickness .., , •· as shown in Fig. 2. The structures arc in a coplanar configuration. The CPW' s line w idth .. w·· is 10 Jtm and line spacing ..s .. is 30 tt m. Table I summarizes other properties of the structures including ground separation. A I 2.7 Jt lll diameter gold wire bond was used to connect the center of the single metal layer spiral inductor to its output. T he spi ra l inductors have an outer diameter of 300 11m. T he HR N type Si substrates have a ( I I I ) crystal orientation. The resisti vity of the substrates were measured [201 before and after processi ng (not including the wa fers where MOS gate oxide wa grown). The resisti vi ty remai ns constant in the range of 30007000 Hcm before and after processi ng wh ich is in agreement wi th 12 11. and i t is maintained as a function of substrate thickness.
IEEE TRANSACTIONS 0:1 1\IICROWAVE TIIEORY AND TECIINIQUI:S. VOL.
2018
~3.
NO. i>ti\·ity. Si. P. (I= 0.33 ncm   ; Si . .\". fl = 11. ncm  : Si . .\". p = 10 k 0cm    : Gai\s . .\'. fl = 10 ~ ~ !!ern    .
The surface poten tial is a function of the Oat band voltage which in turn depends on many parameters (3) includi ng the metalsemicon ductor work function di fference, fixed charges. mobile charges, interface charges, and the insulator th ickness. Even though the bulk resisti vity of SU B#2 is betwee n 3000 and 7000 nem, the surface resistivity (several microns into the substratc l i almost infi nite. T he majority of the field penetration of the CPW under study is withi n the first several micron . Fig. 7 shows the measured losses for the CPW on SUB#5 which i rcahzcd on Si substrates with two microns of insulator. Here. the effect o f the drop in the surface resistivity is clearly shown. T he emiconductori nsulator interface is inverted or accumu lated. The CPW on SUB#5 has almost an order of magnitude higher losses than the CPW on SU B #~ (GaAs substrate covered w ith insulators). One should noti ce th at the surface states density in Sl GaAs is orders of magnitude higher than Si. The surface states trap the inversion/accumulation charges from goi ng to the cond uction/valence band; therefore, the effecti ve surface resisti vity of Si i s much lower than th at of GaAs. The CPW on SU B#6 di ffers from the CPW on SUB#5 in that the si licon wafer was fir t procc. sed through an MOS gate ox ide growth (about 150 A). The gate ox ide growth reduces the mobi le charges (Q,,) and the interface charges (Q ;1 ). The charge reducti on translates into a reduction of the Oatband voltage according to (6) which further translate. in to the reduction of semiconductor charges at the surface (2 and 3). Even though two or the sources of Oat band voltage arc reduced. there is only a slight reduction in los e . T hi s i s due to the relationship between the charges and the metal semiconductor capacitance w hich is reduced by addi ng the MOS gate ox ide thickness which increases the effect of all the charges. Fig. 7 al so indicates that the losses due to the semiconductor charges increased as the square root of freq uency.
T he measured results of CPW structures on HR Si substrate moti vated further investigation in inducti ve structures. The cross sections arc shown in Fig. 2. A s expected. the inductors. shown in Figs. 8 and 9, fabricated on H R Si exhibited characteristics very similar to those on GaAs. and quartz substrates. The measured real and imaginary parts, show n in Figs. I 0 and I I. respectively. of the unloaded input impedance of the meander inductor on SUB# I. SUB#2. and SUB#3 are comparable. A t low frequencies. both the real and imaginary parts of the unloaded i nput impedance arc indisti nguishable from each other. Around 5 GHz the curves start to separate. The separation i s due to the resonant characteristics of the structu re. T he di fferences in the curves in Figs. I 0 and I I at high frequency arc attributed to the difTcrcncc in the dielec tri c constant of the substrates. The inductors on GaAs resonate before th ose on Si and quartz due to the higher dielectri c constant of the GaAs. T here arc no noticeable differences due to losses. Figs. 1214 show the measured unloaded Q for the meander. the three turn spiral. and the four turn spiral inductors, respecti vely. As mentioned before. the Q is only calculated at the frequencies at which the structure can be used as an inductor and that is reason why the curves extend only to 3 GHz. The maximum value of Q. twice of the maximum shown. i n the i nducti ve structure under study is achieved in the frequency region of resonance. H owever. these arc not reported because the inducti ve structures arc not used at these frequencies due to the strong frequency dependence. The inducti ve structures on Si present un loaded Q factors that arc very close to those of the other strucwrcs. Thi s interesting result can be explained as foll ows: I ) The metalsemiconductor i nterface depletes the surface o f the Si substrates or carriers: 2) the higher the resisti vity. the wider Isec Fig. 5. and ( I ) I the depletion region: and 3) the bul k o f the electromagnetic field of the inductors is ncar the surface of the subMrate. The effect o f a thin insulator layer between the metal and the substrate were also investigated i n the inductive structures. Fig. 15 shows the real part of the input shunt admittance o f the 71'nctwork of the sixlllrn spi ral inductor realized on SU B#4. SU B#5 . and SUB#6. T he cross sectional views o f these struclllres arc shown in Fig. 2. Fig. 15 shows a similar pattern to th at of Fig. 7. Here, as in the case of the CPW. a sharp difference between the GaAs and Si substrates is observed. When the insulator is introduced. a strong i nversi on or accumu lation layer or charges is formed at the surface (sec
IEEE TRANSACTIONS ON .\JJCROW;\VE THEORY AND TECHNIQUES. VOL. 13. NO. '!.
2020
SEPTE~ IBER
1995
fi g. 12. Measured unloaded Q (meander). SUI3111 (quart!)    : SUB112 (Si) _ _ : SU I3 113 (GaAs)  · IS
llo
fig. 8.
Thrcclllrn 'piral intluctor.
(J
f ::

I! Ill
·
s (o
J
! II II. I
U.(•
1.1
J.(,
2. 1
!.t.
.1.1
Fig. 13. i'VIcasu red unloaded Q (lhrcclllrn spiral). SUB# I (quart/ )    : SUI3112 (Si) _ _ ; SUBIIJ (Gai\s) · IX 
Fig. 9.
Meander inductor.
1~11
j
J! ~
c•.• , ...
;
1:!11
fi g. 1l. Measured unloadctl Q (rourlllrn >piral). SUB# I (quart;.): SU I3 112 (Si) _ _ : SUBII:l (GaAs)   .
IIIII XII
l'o I
fill
'
~II
1H II ' II. I
Fig. I0. 1\lca>ured real pan or unlocatetl input impedance (meander). SU B# I (quant )    : SUI3#2 (Si) _ _ : SUBIIJ (Gat\;).
~~~~ , ~
(,
/
.11111
~;
z:;u
_§ §
11111
~~

.. "
J:11
1:'11 IIIII :'II
 •.

I
II. I
!
ljll.lrll
1. 1
~. I
(t.l
X. I
Fig. 15. Measured real pan or " nct\\'ork input '>hunt admiuance (si\ turn spi ral). SUB Ill (Gat\s cover \\'ilh insulators)   : SUI3#5 (Si cover'' ilh insulators) _ _ : SUB116 (Si cover \\'ilh insul:nor; and i\ IOS gale o\ide)
Ill. I
J n:ciUl'lll".' 1(;11 11
Fig. II. ~lca,ured imaginary pan of unloaded input i111petlancc (lllcander). · SUB# I (quart!)   : SUB#2 (Si) _ _ ; SUBII3 (GaA>) 
Fig. 6) of the Si substrates. The MOS gate oxide process reduces the interface and mobil e charges and decreases the metalinsulatorsemicond uctor capacitance. The end eiTect i s a slight reducti on i n the accumulation/ i nversion charges at the surface: therefore. the input shunt losses arc reduced. The losses due to the substrate in version/accum ulation charges
increase as the square root of frequency. The subst rate losses start to i mpact the device around I GHz. A t 2 G Hz. the unloaded Q. shown i n Fig. 16. o f the si x spiral induc to r~ on SUB#5 and SUB#6 ( HR Si with i nsulators) is half that or SU8#4 (GaAs substrate cove red with insulator). Figs. 15 and 16 explain why standard (i.e .. below 30 S2cm) Si has been used to fabri cate MMIC designs up to 2 Gllz frequency region. The lower Q of the si x turn spiral i nductor compared to the three and fourturn ones is attribu ted to I ) the ground pl ane is closer to the conduc tor, which leads to higher cu rrent density at the edges. 2) the losses of the under bridge metalli zation arc higher than that of the wire bond. and 3) presence of inversion/accumulation charges.
REYES eta/.: COPLANAR WAVEGUIDES AND MICROWAVE INDUCTORS ON SILICON SUBSTRATES
\lf lai·!'IO:''I l •.a ·( :a h


0 ~~ !1. 1
0.6
1.1
1.6
2. 1
2.6
3.1
FrC. 0
~ 3 >.
~
Z II' I
8
P'i
< 1 . 2 ~

I·U·! Ellipse
~!:: ·g
1J
lll Ill .~:
60
"'
a;
6=?r opaga tion Ang l e (degrees)
;

0 L....~0 0 .5
1. 5
1.0
2 .0
2. 5
Frequency (GHz ) Fig. 3. Dbpcr~ ion c urve for a 3 D array of PEC wire crosses and for vcrlical and horiw n1al dipoles. ~
29
r
r
r
i ~ola1cd
,
Ni r e Crosses
.......
;
28 r·. hori • z:  root   .
J
,
1
vr~~tt.
]
26 :.__ '''
PEC \Vire Crosses
The ccond set of data is for a PEC wire cross, since it is one of the simplest geometries wi th two distinct roots (different from k, k0 ) . A s shown in the insert to Fig. 3. the wire cro ses of radiu. a = I mm have a vertical member of length L 5 em and a hori zontal member of length L / 2 2.5 em located L / 4 1.25 em from the top of the verti cal member. The wire crosses are arranged in a 3D laHicc with d 1 = 3.75 em, d 2 = 7.5 em, and d:1 = I em. and the directi on of propagation is along the zax is. This artificial dielec tric i s uniaxial with nonunity values for both c~~ and c:,~ . The solid lines in Fig. 3 show dispersion curves for c~~ and c~~ for the wire cross. while the dashed lines are for the i. olated vertical or horizontal members of the cross. ote that a vert ical or iJ polari zed wave will directl y induce currents on the vertical member of the cross. and then through mutual coupling currents will be induced on the horizontal member of the cross. For thi s reason Fig. 3 shows different values of c~;; for the cross as compared to the isolated verti cal dipole. By contra t. a hori zontal or :i; polari zed wave wi ll directly induce current on the horizontal member of the cross. However, since the vertical member is symmetri cally located with respec t to the horizontal member, the horizontal currents will not induce currents in the verti cal member. For thi s reason. Fig. 3 shows the same values of c~~ for the cross as compared to the isolated hori zontal dipole. This effect can also be observed in Fig. 4, which shows the magnitude of the determinant of the impedance matri x JZ (k, )I versus the normalized effecti ve wavenumber kef ko for the array of PEC wire crosses (top figure). as well as for the arrays of isolated vertical (middle figure) and horizontal (bollom figure) dipoles, at the frequency of 2 GHz. The wire cross has two roots at k, f ko ~ 1.08 and 1.58, corresponding to hori zontal and vertical polari zati on.
~
=
=
yy
c~; ~~~i!te~0 ~or!~ ~f~~fesand
27
=
~/ Err
lmm
isolated vert. dipoles .
where k,. ~ 1.54k0 (at fJ = 0) and k y = k 0 (at 0 = 90°) . k,.(fJ ) computed by ( I 0) i shown by the dashed li ne in the bouom of Fig. 2. and is cen to be in very close agreement with /.,., computed by the PMM from (3).
B.
L=Scm a
I
..... Q)
90
Fig. 2. The normalit.ed effective wavenumbe r (I.·, / ko) and I he relmive cffccli vc pcrrni11ivi1y versu~ propaga1ion angle 8. for an array of pcrfcc1Iy conducting shon dipole>.
3 . 75cm
dt
cm dd 7l l . 5cm 3
.~''
~~~~'
30
r
a.
z
________
l . o ~·_.
vert. YY pol. (Y) cross
(hor iz.)
t
f
21
X
  err
Q)
IIIII :
0 · 2 >o ~
L/ 4 3L/ 4
I
c
1
tI
I
u z
en
1. 6 :.         . . . . , . .  : ......._ •1=0. 001A 0 °0 I• J 1 "'
i
Eo
u
1
1. 5 !
L/2
I
(vert. )
~ 4
a:
"<
I y
"'g
20 19
r .
18
el7 r c:~ 16
1

r·
Vertical Dipoles
1j
111
r
I I
I
vre;o\·
~ 1 5 '~~
=
9 ~~~
'
:I 6 1. 0
Horizonta l Dipol es
,.I
1
ho riz. r oot 1.2
IZI
1
1.4 1.6 k, ko
1.8
2.0
Fig. 4 . 1\lagnilude o f versm, norrnali£cd cffcclive wavenumber for an array of PI::C wire crosses. and fo r lhc isolalcd vcn ical and horitonlal dipole me mbe rs.
respec ti vely. By contrast the isolated vertical and hori zontal dipoles show only one root (the other i s a free space root k, = ko ).
C.
Bent PEC Wires
The next set of data w il l illustrate a simple bent w ire clement geometry which results in non zero offdiagonal clements in the effecti ve pcrmiui vity tensor. A s shown in the insert to Fig. 5. the geometry consists of a bent PEC wi re of total length :JL = 30 em and radi us n = I mm. The bent wires arc arranged in a 3D Janice wi th spacings d 1 = 12.5 em, d 2 = 22.5 em, and d:1 = 2.5 em. An polari zed electric field incident upon the bent wire will induce directed currents on the horizontal secti on and iJ directed current on the two vertical sections. These currents will rad iate both and f; polarized
x
x
x
2026
IEEE T RANSACTIONS 0
M ICROWAVE TII EORY AND TECHN IQUES. VOL. 43. NO. 9. SEPTE~ IBER 1995
.j ~
4 .5 i 4.0 ~
~
PrupJ£JUOn Dn~r..:11on u = 1
3 e
u

~
Root 2
2 ..1 1:
u
B
::: !.:.l
~
5f
.jr 3r
P
~........_
l
_ _ _ _ ___,
\
t~)
I
""' f
~!
I ~
0.0 (~~
f
?oo
200
=
300
~
10
(~~
400
500
100 1000 lOK lO OK tan 6 1 = Dipole Loss Tangent
Fig. 7. Re lative e ffecti ve permittiv ity and loss ta ngent for a 3 D array of lossy dielectric dipoles.
f ( MHz)
Dispersion c urves for a bent PEC wire geometry.
lauice w ith spacings d 1 = 0.2:3.Ao and d2 = d 3 = 0.03,\o. Propagati on is in the iL direction. and polarization is i n the x direction. The dipoles have relative dielectric constant c 11. = 1. and the problem is to choose the dipole loss tangent. Lan 81 . to maximize the effecti ve loss tangent of the artificial medium. Fig. 7 shows the relati ve effective permini vi ty c~.~ and effecti ve loss tangent Lan o;_.r of the artificial medium versus dipole loss tangent for 10 ::; tan b 1 ::; 100. 000. Note th at as the dipole loss tangent increases, the arti ficial dielectric effective loss tangent i ni tially also increases, and reaches a maximum of Lan o;_J. : : : : 0.75 for a dipole loss tangent of tan 8 1 ::::::: 600.
=z
v.
9
$
0.5
5 !:
(j ;;
"'
0.5
u
zc
Freq
=300 MHz
1.0
5
10
15
20
25
30
I =Distance Along Wire (em) Fig. 6.
IIIII
~
L
;; ?
Fig. 5.
l
'dt= """" 12.5cm' " ' mm ,_ d2= 22.5 ern x d 3=2.5cm
'
~
j
=1 I ·i a =0. 001>. 0 j d 1 = 0 .23 >o d 2=d3 =0. 03 ),o (1<
0.8
~ ~L
f u
I
I
1.5 !
1. 0
0 6,
/~
!· E~ P II III ~0 · 2A o ; : ~ l "IIIII z
3. 5
2.o
Root I
~
tX 0
T he two c urre nt mod es for a be nt PEC wire geometry.
electric fields, thus produci ng a nonunity c~.~ and a nonzero c~~  For propagation in the iL = z direction, the top curve in Fig. 5 shows the norm ali zed effecti ve wavenumber l.:c / 1.:0 versus frequency for the two disti nct nonfree space roots. The freq uency is swept from I 00500 MH z, correspondi ng 10 total wire lengths of 0. L\ 0 ::; 3L ::; 0.5.Ao . Note that the Root I effecti ve wavenumber i s nearly constant across the frequency range, whereas the Root 2 ef fecti ve wavenumber i ncreases rapidl y as the total w ire length approaches 3L = 0.5.\ 0 . Fig. 6 shows the two current modes induced on the bent w ire at a frequency of 300 MHz. The Root I current mode is essenti ally and  iJ directed, whereas the Root 2 current mode i s and +iJ di rected, similar to the current mode on a straight dipole. The electromagnetic field s of these two modes wi ll be plane waves propagating through the artificial medi a in the same directi on , but wi th orthogonal polarizations. The bonom curve in Fig. 5 shows the (nonzero and non un ity) relati ve effective permi!li vity tensor components versus frequency. The result computed by the Maxwell 's Eq uations method and the polari zation method are identical. A lso, the numeri cal results showed that r~.'~ and E~~;. are nearl y identical, thu s illustrating a similarity between real and artificial media 19, ch. 14.11.
+x
+x
£.
T he data in thi s secti on show the dispersion characteristics of the effective permiuivity of a dieleclric weave, a geometry which has current that nows between adjacent lallice cells. As shown in the insert to Fig. 8, the geometry consists of stacked or layered square gri ds of dielectric rods. T he dielectric rods have rel ati ve permilli vi ty c1 ,. = 10 and radius a = 2 mm. T he grid dimensions are d 1 = d2 = L = 5 em, and are 6 mm apart. Curves are shown spaced a distance of d 3 for lossless dielectric rod s, and also for lossy rods of lo tangent Lan 8 1 l. Propagation is along the zaxis, and clue to sy mmetry considerations, the medium is uniaxial with  0 and radius a = 0.001.> 0 . arranged in a 3 D
Dieleclric Weave
GraphileEpoxy 2D Composile Medium
This secti on considers a modern composite material consisting of very thi n graphite fibers embedded in an epoxy host binding material. The graphite fi bers arc modeled as material w ires of infi nite length in the .1; direction with radius a = 3.2 fW l. spaced in a square 2D Iau ice w ith rl2 d3 7.5,un. The conductivi ty of the graphite fibers is 7 1.4 KU/meter and
=
=
PETERS A 'D NEWJ\IAN: ANALYSIS OF AN ISOTROPIC ARTIFICIAL J\IEDIA
propagation changes. the propagation wavenu mber is typically a strong fu nction of angle. By contrast, for a real medi um the clements o f the tensor constituti ve parameters are independent of angle. and for an art ificial medi um they arc (typically) almost independent of angle.
2 . 0: 1. 8 ~~
tan
1.6
o1 = 0
'
1 .4
2027
1.2 1.0 1.0
R EFERE CES I
··=
0.8
"' ''
0.6
c
c
0 .2
"'
center ,f 1=d2= I. = Scm 13 =6mm (,· ltd 17 8  8:r i)I"' imt•( .~:r ) +jw1toh 11(~ > ( k,d'~ .'11)./;'n,Jn} ri/
2
(6)
where t{;;,.. is the equivalen t magnetic current on the perimeter 2 of the interior region, I!} > represen ts the Hankel function of the second kind of order one. and C is the contour of the patch. Rearranging (6). w ith the electric current and edge magnetic current to different sides of the equal sign. gives the following matrix equation
(7)
I EEE T RANSACTIONS ON ~JI C ROWAVE TH EORY AND TECHNIQUES. VOL. 43. NO. 9. SEPTE~ I BER 1995
2030
are sufficient to produce less than 0.5% error in the Green· s functi ons [91. Thi s techn ique also allows the separation of the contribution o f the surface waves. The expressions for the (magnetic current) mi xed potentials (F1 and m) are the dual o f the co rresponding electric current mi xed potentials given in [9]. Hence, when converted, (8) gives the following matrix equation
j ~""
J:~le
J:ourc~
(9)
(a)
where J,~xt . I~~t are co lumn matrices for the edge electric and magnetic currents, respecti vely, radiating in the ex terior region, and [Ycxd i s the 3D ex teri or complex image admittance matri x defined i n [ I 0]. T he minus ign in (9) is added so that the defi nition o f [Y,.xt] i n (9) conform s wi th the standard definiti on gi ven in 1101, [ Il l.
Co . ll o
J:~.
'~
J:XI
Eo Er . llo Co .ll o (b)
Fig. 3. Splining the resonator strucwre into interior and exterior regions. (a) The interior hom*ogeneous 2D region. (b) The exterior region in a dielectric sub~t rnte of thickness 211.
where l i""' is the co lumn matrix of the (edge) electric current in the interi or region, and [Yiutubstrates:· IEEE 7i·rm .l'. Micmu·m ·e '1111'01:r Tech .. vol. 36. no. 2. pp. 32133 I. Feb. 1988. 171 A. A. Omar and Y. L. Chow. "A solution or coplanar waveguide wi th airbridges using complex image,:· IEEE 7i'tms. tllicrou·m ·e Theory Tech .. vol. 10. no. II. pp. 20702077. Nov. 1992. lSI R. F. Harrington. Time tlarmrmi al 1he Semiconduclor lnslilulc of Chinc>e Academy of Sciences. Berwcen 1980 and 1982. he worked on oproeleclronic; a; a Visiri ng Profe>>or :nrhc Tcchnicall n, lilliiC of Aache n in Ge rmany. He !hen relurncd 10 \\Ork on helcrojunclion devices in the Chine!.e Academy or Science' and Wa> promoted to Full Professor. From 1986 to 1988. he wa> a Vis iting Sciemist at Heinrich1le rtz ln, titute and th..: Institute or solidstate physics in Berlin Technical Univers ity, working on optoelectro nic devices. He j oined Lehigh Univen,ity in 1989. He has publi>hed more rhan 10 technical paper!..
~ II C ROWAVE
TII EORY AND T ECHNIQUES. VOL 43. NO. 9. SEPTEI.IBER 1995
James C. l\ 1. II wang (l\1'77Sl\1'821"91} recciv..:d the B.S. degree in physic> from the ational Taiwan University in 1970. He rec..:ivcd the M.S. and Ph.D. degrees in materials science and engi neering from Cornell Univcr,ity in 1973 and 1976. re,pec li\l!l). lie b curre ntly Profe!.>or or Electrical Engineering and Director of Compound Semiconductor Technology Laboratory. Lehigh Univcrs iry. Prior to joining Lehigh in 1988. he had 12 year> of indu, trial cxp..:rience working a1 IBM . AT&T. GE. and Gain Electronics. He has been a Cons ultant for th..: Air Force Wright Laboratory and the Aero~ pa cc Corp.. in the ar..:a of microwave device' and integrated circuil'i. He ha' also been a director of Quantum Epitaxial Designs. a GaAs epitaxial wafer ' upplicr which h..: helped found at Lehigh Univcrs ity"s . mallbusiness incubation fa cil ity.
IEEE TRA;\SACTIO:.'S O;.:
~ IICRO WAVE
THEORY AND TECIINIQUES. VOL. 13. NO. 9.
S EI'TE~IBER
1995
2f).lf
Scattering at the Junction of a Rectangular Waveguide and a Larger Circular Waveguide Robert H. MacPhic, Fel!oll'. 1£££, and Ke Li Wu, Member, 1£££
AbstractA full wave, formally exact solution is obta ined fo r the prohlem of scattering at the junction of a rectangular wa veguide a nd a la rger circula r waveguide. The genera l case of an arbitrary ofl'set of the wa veguide axes is considered. E field mode matching over the 1·ecta ngula r aperture of the smaller guide is faci litated by a transformation of th e ci rcular cylindrical BesselFourier moda l fields of the circular guide into a fi nite series of exponential pla ne wa ve fu nctions in rectangul ar coo rdina tes. T his permits an analytical fi nite series solution for each of the clements of the E ficld mode matching ma tri x [M ] from wh ich the scattering matrix [S J of the junction is easily obta in ed. Num erical evaluation of th e S parameters for the dominant T E 10 (rectangular) and TE,, (circula r) modes in the cases of junctions with no otl'set and with offse t is presented. l\ loreove1·, the practical case of a circular ca vity resonator with smaller input a nd out put recta ngula r guides is considered and excellent a greement is found between the calcula ted and measured S paramctcrs.
arc compared wi th experimental measurements wi th excellent agreement obtained in all cases.
I. I TRODUCTION
II. ELECTRIC FI ELD MODE M ATC HING AT Til E J UNCTI ON
A RECE T paper Ill. a ri gorous full wave sol ution was obtained for the problem of electromagnetic scattering at the j unction of a circular waveguide and a larger rec tangular waveguide. In the present paper we consider the complementary problem of the junction of a rectangu lar guide wi th a larger circular guide. The transverse Bessel Fourier modal £ fields of the circular guide arc expanded in terms of a series of cxponcniial plane wave functi ons 121 in cartesian coordi nates for 8 fi cld mode matching at the junction's rec tangu lar aperture. Thi s problem has been very recently treated by Keller and Arndt 131 but no specific detail s of the coupling integrals 13. (5)1 were provided by the authors. In Secti on II we deduce expressions for the clements of the £field mode matching matrix [.\!] for the junction analogous to those given in Ill. M oreover. in the Appendix i s presented the deri vation of the plane wave seri es ex pansion of .J,1(hp) cxp (.iqc/1). the Bessel Fourier modal eigenfuncti on in the circular wavegu ide. Secti on Ill indicates how the junction' s scaltcring matri x [S] i~ obtained from [.\/ ] and the modal admillancc matrices p·d and [} '2 ] of the two guides. Section I V presents numeri cal results for j uncti ons w ith no off~ct and with offset. M oreover. the practical and interesting case of a circular waveguide resonator connec ted to smaller rectangu lar guides is considered and the theoretical results
Fig. I indicates the geometry or the junction of a rectangular waveguide wi th a larger ci rcular waveguide. The rectangular guide. wi th cartesian coordinates (.r'. y' . .:') . is transversely offset from the circular guide wi th cartesian coord inates (.r. y . z) such th at .r' = .r .r 1 • y' = 11 !It ·
I
t>. Ianu,cript received June :w. 199l: revised May 25. 1995. Th is work '''" 'upponcd by the Natural Science!> and Engineering Research Counci l onator fed by WR75 rcctanguh1r waveguides: (a) no off>ct of the rectangular guide>. (b) one of the rectangu lar guide; with offset.
~~••••••••••eo\~6~... •••••••••••·••·..••o.__...,tt.a6.·5
~
·10
. · 
·,
:::: · IS
·. .
4"...
~..
·~
•
\
.
~,.,_ea06
•;;;;; • •
l
....
·20
·__ .'&••. _. _ _
APPEND IX PLA E W AVE SERIES EXPANSIO
OF .lq(hp)eJ'!';,
From Strallon 18, pp. 37 I 372] one obtains the BesselFourier series expansion of a pl ane wave !'unction
·25
"'oe ·lO +~ c g
· 35 ~
Qj
+      1     ·•5+   
.E
This paper has provided a rigorous full wave modal oluti on to the problem of scallering at a rectangulartocircul ar waveguide junction when the circu lar waveguide is larger in cross section . The new and very use ful plane wave series representation of the BesselFourier eigenfunctions which represents the modes in the ci rcular guide leads to analytical series expressi ons for the element of the £ field mode matching matri x [1\/] from which the scaucring matrix of the junction can easi ly be deduced. This totally eliminates any numerical integration which. heretofore, has been used in such problems. The usefulness and accuracy of the technique has been demonstrated by the excellent agreement between calculated and mea ured S parameter of a circular cavity resonator fed at each end by W R 75 rectangu lar waveguide. Obviously the general callering solution of this junction will be widely used in the characterizati on of various wavegu ide systems such as the slot iri coupling in circular wavegu ide fi lters, certain transformers in antenna feed networks and channel fi lters for space application
___
! .
~ 5 Ci
.··•• 
~a.~
~.................
CLUSION
= ejkpSi n O,(COS ¢  ¢ ,)ejk COSO, z
e /k, ·r
·40
· ~ ++~~~~~~ II
10
13
12
14
q=
15
(24)
Frequenc y In GHz
Fig. 5. The calculated and measured 5paramete rs of the circular cylindrical cavity shown in Fig. .J(a).
Letting z
L
·5
~
· 10
= 0 and
= h,
h· sin 0;
we simpli fy (24) to
1 qejqrp , .lq (hp)ejq
= ejltpCOS ( o ,) .
(25 )
q=
Z: ·I S
.9
c · 20
5 Ci
· 25
i•
"'oe ·lO c ~ · 35 Qj
.E
· 40
But if jqj exceeds hp the amplitudes of .lq(hp) and ·'q (hp) quickly become negligible [5, p. 3591. Therefore, to a very good approximation
Cois o f the asymmetric rectangular iris in c ircular waveguides." IEEE Micmtt·m·e and Guided IVm•e Leu.. vol. 3. pp. 185 187. June 19 93. 1..:w )41 R. E. Collin. Foundations fo r Micrott·m •t• Engineering. 2 nd Ed. York: McGrawHill. 1992. )5 1 M. Abramowitz and I. A. Stegun . NandiJOok ofMathemmical Fun ctions. New York: Dover. 1965. )6) R. SafaviNaini a nd R. H. MacPhie. " Scall..:ring at rectangulartorectangular waveguide junctions." IEEE 7i·an.,·. Micro11·m·e Theory Tech .. vol. MTr30. pp. 2060 2063 . Nov. 1982. )7 ) R. M iu ra and S. \V. Lee. Analytical Techniques in1ile Theory of Guided ll'a t·es. New York: Mac millan. 1971 . )81 J . 1\. Strauon. Electmmagnetic '111('(n :1·. New York: McGraw Hill. 1941. 191 R. N. Bracewe ll. The Fourier 1iw1.ifo rm and Its ApplicatiunL 2nd ..:d. New York: McGraw Hill. 1978.
TABL E II CONVERGEI'CE 0 1' T il E S ER IES E XPANSION OF C ARTESIA K COORDINMES FOR
J,. (II (J) exp j 11 (1
IN
= 8. II fJ = ]!).G/0 . = 38° 0
11
} (J ) I' _ j ,yr.' /~ ' rp e = N l•o J 1
N
= p'.,IR.
h
19 21 23
p
= 0.8R
=
= 19.670. =38°
+ j0.0162 155 + j 0.0161307 + j 0.0161273
0.0111659 0.0110807 0.0110 03
1,(19.670) ,J""'
J.fi{Crl • S:'l
t
: hp
2(J.15
0.0110803 + j 0.0164273
where
n,. _ J;(r':\'") .J,._ ( ,y /'·P) ej(r :Y") . N  t ' /(C' 1 \ \It = ie  J·t y7eJ p · +5IY ) . N
(29)
Equation (28) is 1he discrete Fourier transform 19, pp. 358, 3591 of the sequence {Do . D 1 ... IJ.v  1 }. Conversely .\'  1
2::
13,.
(' jl,. :df
w,.
(30)
t= O
Finally , i f q
= r N
we can use (29) and (30) to obtain ''I ,\'  I
.J,(hp)('Jt/9
=L
"" eJtd!f!eJ'•(C,.•·+SI!J) .
JV L..
(3 1)
Ro h er t II. 1\lacPh ic (S'57 M'63SI\1'79F'91) was born in \Ve,ton. ON. Canada. o n September 20. 1934. l ie received the B.A.Sc. d egree in electrical eng ineering from the University of To ron to. Toronto. ON. Canada. in 1957. and th..: M.S. and Ph.D. degrc..:s from th..: Univers ity of Illino is , Urbana. in 1959 and 1963. respectively. In 1963. he joined the Universi ty or Wate rloo. \ V:uerloo. ON . Canada. as an A's istalll Profcs,or in Electrical Engineering and where he i' now a Professor o f Electrical Engineeri ng. His c urr..:nt researc h interests focus o n waveguide scallering theory. scalleri ng fro m prolate spheroid systems and dipole antennas. From 1991 to 1992 he was on sabbatical leave a s a Professeur Associ.! at the University of Aix1\ larseille I. France. working in the Dc parle mc nt d e Radio.!lectricitc.
t= O
Table II provides some numerical results showi ng the convergence of this N term series. The case considered is for a typical higher order T E mode wi th associated Bessel Fourier eigenfunction
(32) A typica l point in the guide is chosen (p. r/)) = (. ' R. :3' 0 ) : consequemly 24.587 TI = 19.670 and '< /) = 304° . In Table I we see that when the number N 2JV + L of terms in the series exceeds the argument of the Bessel functi on ( 19.670) the series rapidly converges to the true soluti on obtained by the product of the conven tional series expansions for .78 ( 19.670) and for eJ 30·1 .
=
R EfER ENCES I ll J. D. \Vade and R. H. MacPhie. "Scau cring al c irc ularlorecta ngu lar waveguide junc1io ns." IEEE Trans. Micrott·m·e 111eory Tech .. vol. MTT34, pp. 1085 1091. Nov. 1986.
KcLi Wu (1\1'90) received the B.S. and M .S.E. degr..:es fro m the East China Institute of Technology. China. in 1982 and 1985. r..:specti vcly. and the Ph.D. degree from Laval U niversity. Canada. in 1989. all in electrical e ngineeri ng. He was a Research Assistant in the East China Institute of Techno logy. from 1985 to 1986. From 1989 to 1990 he was a Postdoctoral Fe llow at Mc Master Univers ity. Hamilton, Ont .. Canada. He j o ined the Integrated Antenna Group. Com municatio ns Research Laborato ry. Mc Master Univcr,ity. in 1990 as a R..:search Engineer. Si nce 199 1. h e has been an Ass istant Professor of Electrical a nd Computer Engineering at McMaster University. lIe joined C OM DEV Ltd. Cambridge . Canada in Marc h 1993. where he is an Advanced Me mber o f the T..:chnical Staff in the Corporate R&D De partment. His current research interests include all aspects o f numerical methods in electromag netic> with emph asis on rigorous analysis o r waveguid..: systems. integrated antennas. and microwave integrated ci rc uits. H..: has published more than 20 journal papers . l ie contributed to Finite Element and Finite Difference Methods in Electromagnetic Scau ering. vol. 2 of Progress in E/ectromagnetics Nesearch (Elsevier. 1990). and to Compwmimwl Electmmagnetics (NorthHolland . 199 1).
IEEE TRANSACTIONS ON
~ IICROWAVE
TII EORY AND TECHNIQUES. VOL. 13. NO. 9.
SEPTE~IBER
1995
A Fullwave CAD Tool for Waveguide Components Using a High Speed Direct Optimizer Ferdinanda A lessandri, Member, /£££, Marco Dionigi, and Roberto Sorrenti no, Fellow, 1£££
AbstractAn extremely effic ient optimization tool, whet·e the full wave mode matching s imulato r is driven by a quasiNewton optimizer using the adjoint network method, has been d eveloped for the C AD of a class of rectangulm· waveguide components. T his includ es filt ers, phase s hiftet·s, branch gu id e couplet·s, etc., with s tep in either the E or Hplane. With respect to the conventi onal finit e dill"erence computa ti on of the derivati ves, a s peed up factot· of more than 10 times is easily achieved.
~ :.. . ..
...
: .:::·.:.:.·: . . . .' .
H·plane diaphragms
E·plane stubs
I. INTRODUCTIO
Symm etrical Eplane stubs
Symmetrical Hplane diaphragms
W
AVEGU IDE technology is still the key technology in many applications, and particularl y in space appl ications [I[. [2 [. The availability of ri gorous and efficient CA D tools is of paramount importance in order to reduce ti me and costs assoc iated with tunin g and trimming of the components. Design techn iques based on ri gorous fullwave models must be used to obtain the required accuracy. In spi te of the recent improvements in the efficiency of full wave methods. such as the mode matching technique, these methods usually invo lve signi ficant computer expenditures. The optimi zation of a single component may easil y require hundred s or tho usands anal yses. A s a conseq uence, the CAD of components with tenths or hundred s degrees of freedom may easily become unaffordable even with the most efficient full wave analysis tools. To keep the computer effort w ithi n affordable limi ts it is therefore necessary to adopt very effi cient optimi zati on algorithms. A novel implementati on of gradientbased optimi zation methods. the adjoint network method (AN M ). has recently been presented which permi ts an extremely effi cient evaluation of the grad ient of the objecti ve functi on [ 3[. Since ANM avoids the fini te difference evaluation of the mu ltidimensional gradient of the objecti ve fu nction. i ts efficiency increases with the nu mber of the degrees of freedom. thus the complex ity of the circuit. Based on AN M . we have developed an ex tremely efficient CAD tool of a class of rectangular waveguide components such as filters. phase shifters. and branch guide couplers realized with steps in either the E or 11pl ane. as depicted in Fig. I. The tool consists of a direct optimi zationdri ven ful l wave sim ulator based on a modified mode matching techn ique [4[. A high accuracy combined w ith high efficiency arc achieved by joining the rigorous electromagnetic simulator wi th the ANM optimi zation procedure. A number of waveguide components M an u~c ript received August I. t99l: revised May 25. 1995. This work was ;upportcd by the European Space Agency. ESTEC Contract 9918/92/ UNB. The authors arc with the lstituto eli Elcttronic:t. Univcrsita eli Pcrugia. 106 100 Pcrugia. Italy. IEEE Log Number 911 3128.
E·plane bnmch guide coupler
Fig. I.
H·plane directional coupler
Rectangular wav.:guidc components.
p.e.c. or p.m.c.
p.e.c.
Fig. 2.
Schematic of the basic wavcguid.: structure.
have been designed and tested show i ng very sati sfactory results.
I I. MO DELI NG OF THE B AS IC
W AYEGUIDE STR UCT URE
Fig. I shows the schematic of different waveguide strucwres. They can all be reduced to the same basic ·tructurc (Fig. 2) consisting of a wa veg uide sec tion loaded w i th either E or 11 plane stubs; the latter are termin ated by either perfect electric conductor (PEC) or perfect magnetic conductor (PMC). The waveg uide dimension ("a'' or "b" ) can al so vary from sec tion to section. A number of different components such as fi ltcrs, fi xed phase shifters. branchguide couplers ei ther in the E or Hplane [51 171 can be reduced to the basic struclllre of Fig. 2. A rigorous and effi cient analysi s of Fig. 2 can be performed using the so ca lled ce llular segmentation technique [4 [. Thi s consists of parti ti oning the geometry i nto the cascade of rectangul ar ce lls. Each cell represents a stub or a waveguide sec tion and is modeled as a multipart network in terms of the Generalized Admittance M atri x (GAM ) . A n example of
0018 9180/95SO.J.OO © 1995 IEEE
,\LESSA\'DRI t•tal.: FULLWAVE CAD TOOL fOR WAVEGUIDE CO:.IPONENTS US ING A IIIGII SPEED
OPTI~ II ZE R
cellular segmentation of a threestub waveguide structure i s depicted in Fig. 3. The deri vation of the GA M for the uniform waveguide section and for the stub i s illustrated next. Consider first a uniform waveguide secti on of length L. The electromagnetic (EM ) fi eld is expanded in terms of an orthonormalizcd set of modes e, . h , of the waveguide .\I
E1
(a)
=L
I ·,(z)e ,(.r. y)
11=1
.\I
H,
=L
!,( z) h ,(.r. y)
( I)
u=l
E 1 . H 1 being the transverse EM field component , and ;\/ the number of modes taken into consideration. The GA M constitutes the matri x relation between the I .. s and I' s coeffi cients at both ends of the wavegui de length L. Using the transmi ssion l ine theory. by virtue of the orthogonality of the modal eigenfuncti ons. one easily obtains y
_ [Y y1 1 Y 11 21
(2)
!J
where Y
!t' J
arc diagonal submatri ccs or size i\1 x i\1 given by
Y y11
/ . [ 'J}c, = Y !t'.!:! = (/(/_(} (fJ, l ) ] t
Y y12
= Y y:! 1 = diof} [ Sill .~j/I
j
. \II
f";)]
111
= I. 2 · · · i\I
(3)
Ill
c·,,
where 13,, is the phase constant and } the characteristic admitlancc of the 111 th mode. The stub structure can be regarded as the cascade o f three waveguide secti ons A. B. C. the central B secti on being larger than the ot her ones. We assume the terminal sections A and C as having zero length. Depending on whether E or Hplane stubs arc considered. the fields within the cell arc ex panded in terms of LSE or TE modes respecti vely. To obtain the GAM or the stub. the mode matching technique i s applied to the step di scontinuities A  8 and BC. For the GA M Y., or the stub one obtains
y
·'
= [Y .,11
Y ,21
(b)
Fig. 3. circ uit
(a) A 3~tub waveguide Mruc!Urc (b) and ih gcncrali1cd cqui' a len!
cell s (i.e., stubs and waveg uide sections). T his com pletes the characteri zation of the waveguide com ponent. Observe that. for simplicity. we will assume that the reference planes of the component arc put far away from the nearest disconti nui ties. in such a way that higher order modes at those planes can be neglected. The component can thus be characterized by a 2 x 2 adm iuancc matri x relative to the dominant modes. In view of the application o f the AN M descri bed i n the next section. however. the additional informati on or the voltages at the intern al or connected pon s is required. As a consequence. a different algori thm has been developed in order to compute with minimum effort all the i nformation requi red for implementation of the A 1M. Consider the cascade of multiport networks sketched in Fig. 3(b). Let V , (I,) be the voltage (current) vector at port n (with 11 l · · · N) composed or J\1, clements corresponding to the 111, modes at the port 11. Usi ng the notation of Fig. 3{b) we can write. for the fi rst network
=
(6) For the internal k th network con nected between port k and k + 1 we have
h ] [  I k+t
=
[y~·l Y~ 1
k
= 2.
· · · .V 2. (7)
For the last (N  l)th network we have
[~t,~ ] = [~~:=: ~2=:] [V~·.~~] .
0T ]
W c/J
where I I ·:~u is the coupling matri x between the A and IJ waveguides and \l 'ac is the coupl ing matrix between the D and C waveguides
11 ·.1u,,
= {
e,. .. ,. e,,udS.
(5)
./S. l
As noted above, for optimum effi ciency of the algorit hm, di fferent modal sets are to be used depending on the type of step discominuitics. Expressions for the coupling matrices for both E and Hplanc discontinuities arc given in the A ppendi x. The total Y matri x of the c111ire two porl network can be build up by cascading the GA M matri x o f the individual
{8)
All currents at the connected (i nternal ) ports k = 2 · · · N 1 can immediately be eliminated from (6}(8). We obtain the following SCI or N 2 equat ion s in the N  2 voltages at the connec ted ports
[Y~ 2
+ Y~ 1 ] V 2 + Y~2 V:.~
=  Y~ 1 V 1 1
1
Y~~ V k1 + [Y~i + Y ~IJ V~ = 0 k = :3. 1 . ... v  2 .\' 2
.\' 2
V s2 + !Y 22 =  Y1.\'1 2 V ,y .
Y 21
+ y ~·2y ~·+t
+ Y .\'1 11 ] V st (9)
!048
IEEE TRANSACTIONS ON MICROWAVE THEORY 1\ND TECHNIQUES. VOL.
The olution of the above ystem gives the voltage distribution at the connected ports for any excitation at the external ports
I and N. The currcms at the external ports arc then computed from the remaining equations. i.e., the first of (6) and the last of (8)
( I0 )
0PTIM IZATIO
llY Til E
ANM
Ba cd on a full wavc analysis technique, a rigorous description of the waveguide component as a two port microwave ncl\vork has been described in the previous secti on. To achieve a prescribed frequency response. we need to modify its dimensions according to a suitable optimization strategy, o as to minimi ze a properly defined objecti ve functi on. Di fferent objecti ve functions can be used. depending on the type of response required. For example, the following functi on can be used for the optimization of a filter
=L
{J(ii[i811
SEPTE~IB ER
1995
V TI =0
rrv = 0
( 12)
where V and I arc branch voltage and current vectors on the two networks and T stands for transpose. Consider now a microwave network. The incremental changes ~ V. ~I produced by a perturbation of its element satisfy the Kirchofr s laws and Tellegen·s theorem. From ( 12) one therefore obtains ( 13)
Using an admittance matrix description and neglecting higher order term s. the current incremental change can be expres ed as
SF
F
NO 9.
one complete analysis of the network. Dramatic computation savings, particularly when the method is used in connection with full wave models of microwave structures, can thus be obtained. The deri vation of the basic formula of the ANM is bricny described here using an admittance matri x description of the microwave networks. Tcl lcgen's theorem states th at for a pair of networks which are topologically identical the following equations hold
The 2 x 2 admiuancc matrix, then the sca!lering matrix, of the entire network can fi nally be evaluated by straightforward algebra. Ill. Q UAS I EWTO
~ 3.
~I
(J;)I  ISfflf
= Y ~V
+ ~ YV.
Substi!Uting into ( 13) gives
i= l
( I I)
where j , is the ith frequency in the prescribed band. /(1 ;. K 2; are weighting coefficients. lsi~ 1. lsi~ I arc the required amplitudes of the scattering parameters (in dB). Effici ency of the optimization strategy is of paramount importance. incc the analysis technique, alt hough efficient, still involves a considerabl e numerical effort. Optimi zation methods based on the knowledge of the gradient of the objecti ve functi on, i.e., its first deri vati ves with respect to all geometrical parameters. appear to be the most efficient ones. In this paper. the quasi cw ton method 181with BFGS updating formu la [ 121 has been used. A considerable numerical effort i usually invol ved in the computation of the gradient of the objecti ve function. The effort increases wi th the complexity of the component and with the number of geometrical parameters that have to be optimi zed. One has, in fact, to compute the deri vati ves of the objective fu nction with respect to all free geometrical parameters of the component. Conventionally. the derivati ves are computed by numerical differentiation. i.e.. by performing one additional analysis of the device for a small increment of each free parameter. The computation of the gradien t thus requires a number of addi ti onal analyses of the device equal to the number of free parameters. In the example of Fig. 3, assuming the dimensions of the feeding waveguides to be given. we have I 0 degrees of freedom. Therefore. I 0 additional full wave analy es arc required for each gradient evaluation. Ba. ed on Tcllcgcn·s theorem. the adjoint network method is well known in circuit theory 191 f I 11 . For a reciprocal network, all partial deri vati ves of the objecti ve function of a reciprocal network can be computed by ANM using only
Suppo e now that the following relation holds
rl = Y V.
( 15)
We obtain from ( 14) ~ V TI
 ~ lT V
=  y T ~y T y
( 16)
.
Equati on ( 15) defines the adjoint network whi ch i characterized by the transposed of the) ' matrix of the origi nal network. It is remarkable that a reciprocal network is equal to its adjoint network. Consider now the case of a global network composed of N  l subnetworks. Equation (4) can be generalized as fo llows
11= 1
( 17) where the index "ext" refers to the external ports of the global network, V , and V , arc the voltages at the ports of the n th subnetwork and ~ Y , is the change in its admittance matrix. The upper bar refers to the adjoint network quantities. Observe that in practice the summation involves only those subnetworks with non zero ~ Y , . i.e .. those who arc actually affected by the change. Substituting the incremental changes wi th the derivatives with respect to a free parameter p of the network. ( 17) becomes !l yT E.rt
u
D
p
I E J·t 
! lJT
.\'  l
u E .r t V ,   '""' y T () E.rt L " p u= l
!ly T
_u _, V ( 18) () It • p
ALESSA:IDRI t•t ul.: FULLWAVE CAD TOOL FOR WAVEGUIDE
CO~ IPONENTS
L'SING A IIIGII SPEED
OPTI~IIZER
20·1'1
IS I
Assuming a constant unit voltage excitation at port i of the original network and at port j of the adjoint network. ( 18) reduces to
11 dB
1: ~~~I~ 1 j
20
w here v ;, and V f, arc the voltage vectors at the internal ports w hen the network is excited by a unit vo ltage at the extern al pon i or j . respecti vely. / ,.J rcprcscnls the current al port i due to a unit vo ltage exci tation at port j. By definition it co incides w ith the cl ement ) /) of the admittance matri x of the network. I f all subnetworks urc reci proca l, we can drop the tran spose of the Y ,. matrix and obtain i.JY c,·;J
UfJ
= .\L· r v ,r 0;)y" v j U]J ,.. II
( 19)
u=J
The above formula is the basis for the application of the At M. The purtial derivat ives of y c of the g lobal network arc expressed in terms or the partial deri vati ves of the admittance matrices Y ,. of constituelll subnetworks. The summation in ( 19) includes all subnetwork s. but in practice is restricted to the only subnetwork s uffcctcd by the parameter p. si nce all other deri vati ves arc zero . ll i~ apparent that the segmentati on method described in the previous section lends itself to the implementation or the A M. The admittance matri ces at the righthand side of ( 19) are kn own anal ytically bot h for the waveguide Secti ons Ill and the stub I V. The deri vati ves w ith respect to all geomet ri ca l dimensions can easi ly be evaluated analy ti ca lly. The vo ltage vectors at the connected port. arc computed by sol ving twice the linear system (9) for
1",
= J:
l :y
I ·,
= 0:
I
·30
t
T Stan Optim1zod
 40
  Objoctovo
5o ~~~~~~J~~~~~~~~.
11 .85 11 .9 11. 95
12
12.05 12.1
12.15
Frequency (GHz) Fig. l. Rcspon:,cs of a five irisc' 11planc Iiiier h..: fore and alkr oplimitation. Design p;munclcrs arc: Center frcquenC) = 12 Gt tt: Ripple = 0.0 I dB: Band"idth =50 1\lllt. Stopband aucnuation  JO dB ;tl f = 11.92 Gtlt and f = 12.08 Gilt. Geometrical dimcn,ion' of 1h..: lil1..:r oplimitcd arc gi,cn in Tabl..: I. TABU: l GL0\1U ll\ 01 ·ntt 0JYr1\llZE1> f'u !Til o r 1'11 \VII)!! I: . I 19.05 1\11\1
=
= ()
:y = l.
Thi s is the only additional analysis effort req uired for the computation o f the deri vatives. The co mputational saving w i th respect to numeri ca l differentiation is dramatic and is hig her the higher the complex ity of the microwave structure. I V . RESULTS A CAD tool has been set up using the A 1M in conjuncti on w ith the full wave anal ys is method. High accuracy of the theoreti cal predictions is obtai ned thanks to the fullwa vc models adopted. On the other hand. the g radi cmbascd optimization strategy w ithout numerical differentiation leads to an extremely high efficiency. ll sho uld be stressed th at the optimi zation operates directly on the microwave structure rather than on some equi valent circu it. so that the components designed arc ready for fabricati on. Several waveguide compo nents. fillers. couplers. etc .. have been designed w ith the above described tool. Some examples arc i llustratl!d and di sc u ~sc d here. The computed response of a bandpass waveguide filter using 5 symmetri ca l Ifplane diaphragms (4 cav ities). prior and after optimi zati on is shown in Fi g. 4. The fi ller was required to have an attenuation lower than 0.0 I d B in the frequency band
11.975 12.025 G H z and higher than 30 dB below 11.92 Gllz and over 12.08 G Hz. The dimensions of the fi Iter optimized arc g i ve n in Table I. T he optimization in volvl!d 5 geometrical parameters and was made on I I frequency points within the passband and 4 points outside the passband. The mode matching analysi used 7 modes for field expansion in the narrowest iris and 60 modes in the waveguide secti ons. The solutio n was achieved in 52 iterat ions. th us required 52 x 60 780 sing le analyses and 5x780 deri vati ves. One complete analysis including derivati ve computat ion required approx. 3.5 son a HP Apo llo 7 15/50 workstation. T he enti re optimi;;ation process took about +5 minutes CPU. To esti mate the efficiency of the present approach compared to the conventional numerical differentiation. we have empirica lly deri ved the following relati on for the speed up facto r
=
.'i = Tro  ,= 1' "( .\', . 7 . 1.\"
/,
111
+ I)
w here:
=
I ) TFo C PU time of the optimi zation using numerical eli fTcrcnl iation.
IEEE THANSACTIONS ON ~IIC ROWAV E TIIEORY AND TECII NIQL'ES. VOL. ~3. NO. 'J. Si1'11:.\IBER llJolropy uf 1) 1 and 1)11 al k (c·u, 1.·  I ). where /) D1 nr O n.
=
=
2"/b. ~/) (k· .ll)
=
O(k·.ll ) 
but unlike D 1ciko.''/ f' 1 ko J· and D 2 eiko J'/ C'•k•.•·. D 0r •h·''j r ·~·u i s virtually i sotropic. Since /o is a function of /,·. we would expect that thi. isotropy can be valid only at the wave number. k0 . used to compute /O · It turns out. however. that while the isotropy of D o docs degrade away from k 0 . it is still much better than that of D 1 . T his point is discussed further in A ppendix A . Il l. A
EARLY EXACT SECOND ORDER
ALGO RITtl ~ t
Let us take T to be the standard secondorder FD operator gi ven by
r 2 T f(/ )
= f(t + r) + f (t r) '2.f(/)
( II)
(6) where At first sight thi s approach docs not look very promi sing. because the con tant parameter that we seck is a function of both 1.: and B. It turn s out. however. that the 0dcpcndencc of 1 is quite weak and we can eliminate it by evaluating 1 at a fixed value of 0. Oo. Defining
Do(k . 0)
= ro( k )Dt (k. 0) + ( I 
/ o(/,·)) /J2(k. 0)
(7)
where
lo(k) =
~t( k.
(9)
D 0 (/dJ) ::::::: cos /,·  I
is thus an excellent approxi mation ror all 0. The l inear combination
+ (1 
is the time step size. Setti ng
/o( A· ))D 2
( 10)
thus constitutes a nearl y isotropic finite eli fference Laplacian expression for \ 2 . Whi le \ 2 c·ikoJ'/I'' ~·er =  k2. D 0 r•k • ·•)eikoJ· ::::::: 2( mea!.ured i n time steps. The nearl y exact isotropy of D o allows us. at fi xed wave number and frequency. to construct a nearly exact algorithm to sol ve ( I ) by modifying the form of (2). Since D = D o is nearl y isotropic. the deviati on or T and D from iJ11 and \ 2 can be compensated ror by rep lacing I' in (2) by an adjustable parameter. 11
(T  u 2 D )t{r . t ) = 0.
(8)
Oo ).
L et us examine the ani sotropy of JJ0 l scc Fig. l (b) l. We find that the choice 00 = 0. L '20:37r symmetrizes ~Do ( k.B )I(cos k  I ) about zero such that its maxima and minima have the same absolute values. Thi s symmetrization is. moreover. independent of/,; to an excellent approximation. Compari ng Fig. l (a) and (b) we sec that the an isotropy o f /J 0 is less than w·l that of D 1 on a coarse grid ().. = ') . and
D o(/,·)= /o (/>)D t
T
T r·;..,, we fin d T r·'"" 1 = '21'(..v)r··~·t. where T (w) = o (.r.l ).:(w. A·. O)
( 13)
where c(w. k . fl) = 'l'(w) 11 2 D 0 (k . fl) i!. the sol ution error. Using (9). £ can be made to vani sh at fixed k and u·. b) choosing 11 = 11 0 . w here .,
lljj=
('OSW
f
t·osk1
.
( 14)
Inso far as (9) holds. the soluti on error van ishes at fixed = D 0 . 11 = 11 0 ) in ( 12) thu s define the opt imal finite diiTerence approximation to ( I). T he replacement o f I' by 11 i!. actu ally a generali;ed application of the methodology descri bed in 161. We can now construct an FDTD algorith m to sol ve the wa\e equation or the form
I• and w . The choices (D
tj{r.l + I ) = '2tl{r. l ) v(.r. l  I)+ 11 (.r )2D t •(.r . l ). ( !Sa)
COLE: A FDTD ALGORITH,\1 TO SOLVE PROPAGATION AND SCATI"ERING
I V. STA BILITY CONDITIONS A D SI1\I ULATION SPEED
2E7
Equation ( 12) can ex pressed in the form E
( 18) :!E7
~
(h)
(a)
=
= •·)
Fig. 2. Solu1ion error ani>olropy for (a) I he ( D D 1 . 11 and (b) Ihe (D D o . 11 11o) algori1hms :11 1.2«/8. and ...:fl.= 2/3.
=
=
=
We write ~1 (.r ) to emphasize that u can vary spatial ly. For the inhom*ogeneous wave equation with sources and attenuation 121
(uu P(.r)
2
\
2
= lj;(.r . I )+
11
( 15b) If rhc refracti ve index o f the medium depends on position. then at fixed frequency k k(.r) . and we can define D o and locally as functions of position. In the case of locall y variable perm itti vity. c(.r). and conducti vity. a (.r ). we can replace ( l2n )/( l + 2n). and 1/( J + 2n) by (2c a)/( 2c +a). and 2/(2E+a ). respecti ve ly (sec for example Ref. 7). Boundaries can ab o be accommodated by a local definiti on of D as show n in Section VI. Because we sample only point adjacent to .r. on a si nglein. tnlcti on mul tipledata (SIMD) computing architecture. intcrproce. sor communication is minimi zed and thi s algorithm can be rapidly iterated. Using tim edomain display graphics output we have viewed the evolution of wave propagation and scauering processe. in compi icated en vironments 13115 1. In Fig. 2 we compare the ab olute size and ani sotropy of . elution error (c) as a fu nction o f f) for the choices (D = D 1.11 = u) and (D = D 0 . 11 = u 0 ). We sec that for the latter lei is more than four orders of magnitude smaller than for the former. The primary source of solution error at a single frequency is due to the error in the phase. rather than the amplitude. o f the computed wave fronts. When ( 15a) is iterated. a mode of wave number k and frequency w wi ll appear to propagate w ith a velocity 11' such that ( 12) is satisfied. Deno ting by ~ 11 1 the maximum amount by which 11 1 deviates from 11 wjk. the maximum distance. R. that a wave front can propagate on the grid before the pha. c error accumulates to 21r f A is
u6
=
=
1,
1? = .
( 17)
~ u'
For ex am ple at /\ = ' and w /A· = 2/:3. 1• ) . whereas for (D = D 0 • 11 =
R / A = .u for (D = 11 0 )
!? f/\ = 2.8 x LO:;.
2
< 
( 19)
lllax( D ) ·
The stability condition can be expressed in the form A/ P = < 11111 ax· which can be rewri tten as
A ) 1 >
1  ?n( r) )  (. ) ('l/;(.r. t)  if;(.r . I 1)) ( J + 2o .r
+ ( l + ~ n (.r )) u(.lf(D ~;(.r. t) + 8(.r . t )) .
D 1. 11 =
•)
~~
)v(.r. t ) = o(.1Y.~(.r.t)2n(.r)u11/{r. t ) ( 16)
where 8(.r . t ) i s a source term and n is the attenuation. it can be shown 131 that ( 15a) becomes
v(.r . I+ 1)
where. for simplicity . the spatial dependence is ~upprcsscd and discrctizcd time is denoted by a subscript. Postulating a solution o f the form ~ ·1 = u•1 and i nserti ng i t into ( 18). we obtain w 2  2bw + L = 0. where h = L+ u 2 D. For ( 18) to have an oscillatory solution we must have b2 < l. This implies the constraint 11 2 < D. Fol lowi ng Ref. II J. the upper bound on 2 11 is determined by the max imum possible value or ( IJ). whi ch yields
(20)
i11110lX
w here I'"'"" is determi ned from ( 19). Since /\ and P arc measured in term o f gri d un its and time steps. respective ly. the wave ve locity. u A/ P. represents the number of grid units that the wave front propagates per time step. The larger U111ax · the fewer the number or ite rations needed to solve a gi vcn problem. For (D = D 1. u = u) .111ax( 1J ) = 1 and ( 19) y ields the well kn own constraint
=
1' 111 ax( D
= D 1.11 =
1' )
=
.../2 O. tO. 2::::::
(21 )
For (D = D o. u = uo). 111ax ( /J) = lro and we find that I /(21'o) . Using the racts th at li111k·o 'fo(k) 2 j:~ and 'fo(h· > 0 ) < 2/ :3. we obtain 11~ < :~j I for all k > 0. ln~crting w ko into ( 1l). we obtain
u5 <
=
=
1'111 ax
(v'} .
= /;2 i\ITSIII. T
)
Slll{k/2 ) .
( 22 )
Wh at value or k should we usc to evaluate I'"'""? An arbitrary signal co ntains a mix or rrcqucnc ics. so to ensure stabil ity we mu ~ t usc the mi nimum va lue of t' 111ax wi th respect to /.: . Since l'max decreases as /,· increases. we usc /,·'""" 27r j:3. which corresponds to A :.l, the shortest wavelength that can propagate on the gri d (sec Appendix B). Thu s
=
=
ll"'ax( D = D o.
11
:3 = uo) = arcsin 7i
(3) 1
=
:::::: 0. '()
=
(23)
D 1 . 11 I'). w hich is about ll7'r larger than 1' 111 ax (D The (D = D o. 11 = u0 ) algorithm thu s requires fewer iterati ons than the (D = D 1 • u = I") algorithm. while deli vering superior accuracy. The price to be paid i~ that we must ca lcul ate D 2 v in addition to D 1 tl at each time step. but thi s is more than compcnsmccl by the high accuracy that can be achieved w ith a small number or grid points.
2056
IEEE TRANSACTIONS ON
Y. EXTENSIO
I
TO THREE 0 11\I ENSIONS
The pre vious developments can be ex tended to th ree dimensions. Here there are three di ffcrc nt secondorder FD Lap lacians and two angular degrees of freedom. In three dimensions (3a) and (3b) become
~ II CROWi\VE
TIIEORY AND TECIINIQUES. VOL. 43. :\0. 'J. SEI'rE:O.IIJER 1995
o:::::: 0.11 8 11 rr. Since Do(k .O.O) = D 0 (k.O ). the 0dcpcndcncc of Do can be characteri zed by Fig. I (b), while its
(30)
For (D = D o. ll = 11o ) using linlkon 1(k) = 7 / L) and lin 1A·on2(k ) = 2/ 15. we find th at 111ax ( D ) = IG/ L·:>. whi ch yields < J 5/2:1. Repeating previous considerations we obtain
ui\
1' 11 nx( D ..
;j = D o. II = llo) =7i
. ( HITSIII
fi.0 J:l) · 
2:3 2
::::::
O.t.3. (31)
(24c)
We obtain /.domain expressions analogous to (5) o f the fo rm
Da(k. 0. (/1) =
+ (2/o
n2 = 'io ( I  lo) (24a)
 Gf(.r . !J. ::)
=
= 'io( l  ~,o)
In three dimensions 11 111 ax( D = D o. u = uo) is thus about 28% larger than 1111 oax(D = D 1. 11 = 11). In three dimensions. using the (D = D o. 11 = 11 0 ) algori thm. we achieve the same reduction uo·') in solution error as in two dimensions. ow however we must calculate D 2 v and D :1 v in add ition to D 1t' at each time step. but th i~ is more than compensated by the high accuracy th at can be achieved with a low number of grid points and by a reduction in the numbe r of iterati ons needed . V I. BOUNDARY CO DITtON$ AT TilE S UBGRID LEVEL
To apply the algorith m to solve practical problems on a coarse grid. it becomes important to implement boundary conditions at the subgrid level. For example for a TM wave in a complicated metal st ructu re we wou ld like to be able 10 implement the co nd ition E: = 0 at a metal boundaries without using a large number or grid points to describe their locations. The bound ary cond it ions o f greatest interest arc
v(.ra )=b
(32a)
n • \ d .ra ) = b
(32b)
where b is a constan t. n is a local normal to the boundary. /J. and .raE fl . The basic problem is to approximate \ 2 t"(.r. /} in the neighborhood or a boundary point in terms of the bound ary cond itions and know n fie ld quantities inside the boundary whi le eli mi nating unknowns outside it. when .ra falls between the lattice points on the grid. To illustrate how (32a) and (32b) can be implemented in thi!> case. consider a o nedimensional vi brating string. At .r 0 = 11 +a. where 0 < o < I. and 11 is an in teger. (32a) becomes I'(II+H./) =b. Assuming that waves impinge on the boundary rrom the region .r < 11 + o. we seck a FD expression fo r iJ.,..,. tj1( 11. I) in terms
~057
COLE: A FDTD ALGORITIL\1 TO SOLVE PROPAGATION AND SCATrERI NG
wt
Fig. 3. i\ni~otropy of Do in the at.irnuthal direction. .::..Do(l..;; j l. o) Do( k .;;fl.o) (cw. k  l ) at h· Variability with respect to o i. maximal at II = ;; f 4 .
=
=
where 2;;/8.
(b)
(a)
Fig . .J. l"(.rH ) = cou'>ta ut on a curved boundary (il) that passes between the grid point,. Dolled straight line segments indicate the effective D 1 and. (b) D Do. approximation to D for the cases (a) D
=
=
h _ ~T (a)
APPENDIX A APPLICATION TO M ULTIFR EQUENCY SIGNALS
Since both D o and 11 0 are functions o f 1.:. the (D = D 0 . u = solution error ri ses for spatial frequencies that deviate from the va lue, /;0 , at which D o and u 0 arc defined. The smaller 1.:0 . the smaller the solution error away from 1.:0 . Even on a coarse grid ().. ,...., ' ). howeve r, the maximum ani sotropy of D 0 (k 0 )eikeJ)eik• .•· is still less than 10 2 that of D 1 eike.•;r ikeJ·. so Do (ko )eik• .r;ei ~····· :::::: 2(cosk l ) is still a reasonable approximation even at 1.: =/= 1.: 0 . The main source o f solution error at k =!= k 0 thu s ari ses from the difference between D 0 (k0 )eik•.•;eiku :::::: 2(cos k  1) and D 0 ( k 0 ) e i~·o •·•;ei~·o •J· :::::: 2(ms 1.:0  1). To handle broadband signals one should therefore set the (smalles t) wave length. A1. (i n terms of grid units), corresponding to the highest waven umber, k 11 • such that the difference. (cos kL  cos ku ). is .. sufficient ly .. small. w here kL is the lowest wave number. The broader the frequency range. the larger )..~., must be for a gi ven error tolerance, and hence the finer the grid relati ve to the wavelengths. ··suf!icicntly small'' depends on the details of the particular problem at hand. such as signal bandwidth. the ma ximum propagation distance and the tolerabl e phase error. 11 0 )
~
(b)
Fig. 5. 11 • \ r ·(.r H) = ~""'""" " on a curved boundary ( 0) that pas,cs bcm cen the grid poitm. where n b a local normal. Dolled straight line >egmcnt; indicate the effective approximation to a for the case; (a) D = I) 1 and. (b) D = Do.
of v( n + a.l ). ¢1( 11.1 ), and t/1(11 J. I). Expanding t/l( n +o .l ) and t/J (11  I . I ) in Tay lor series about .r = 11 . we obtain
u.,·.r 1,(n . l ) ::::::
()..j iJ = t:l) the sol ution error i s less than that o f the conve ntional FDTD algori thm using (D = D t· 11 = 11). To att ai n the same accuracy with the (D = D t · 11 = 11) algori thm one would need to operate at )../ IJ = 11 10. This superi or accuracy comes at the expense of a greater computational load at each grid poi nt. but it is more th an offset by the low ,\ f h rati o that can be used. as well as by a decrease in the number of iterations needed. In addition. bou ndaries can be more accuratel y characterized. A l though optimal performance can be achieved at only one frequency. good results can still be had wi th mul tifrequency signals. Our algori thm is based on an i otropic secondorder FD L aplacian. which can be used in FD approximations to other eli fferential equations. The same approach used to construct it can be applied to deri ve other kinds o f isotropic FD operators.
?
(  )[n¢;(11  l.l )( l +a)¢1(1/. .l.)+b]. (/ 1 +(I
(33a) This res ult can be generali zed to higher dimensions. The ex tension of (33a) to two di mensions using D = D t is equivalent to approx imating /3 by the dashed line as shown in Fig. 4(a). whil e usi ng D = D o is equi vale nt to the approximation shown in Fig. 4(b). Returning to the string. (32b) takes the form u.,.t/l( u + o. 1) b. and it i s easy to show that
=
u.,. ,.¢'(11. I) :::::: 0 + 1I / 2 [¢1(11 L. I ) 
¢1(11. I)+ /;].
(33b)
Thi s result can al so be ex tended to higher dimensions. In two dimensions. using D = D 1 • (33b) is equi va lent to approx imating 13 by tangents perpendicu lar to the coordinate axes as shown in Fig. 5(a). while using D D o is equiva lent to approx imating /] as shown in Fig. S(b).
=
VII. SUMI\Ii\RY In thi s paper we have deri ved a new FD approximation to the wave equation for usc on a coarse grid to sol ve wave propagation and scattering problems in compl icated environments. Using just eight grid units per wavelength
APPE1 DIX 8 SPACETIME Si\MPLING OF THE W AVEFIELD
Let f be a function of the form f (.r) = asi u k.r + bcos k.r on a onedimensional grid. w here k = 21r / .\. For ).. = l and 2 (grid units) the sine component of th e signal vanishes at the sample points. To unambiguously determine both a and /; on a discrete interval of length ).. . at least four sample va lues arc needed. We thu s obtain the constraint ).. ~ 3. For the (D = D t. u = o) case this constraint appl ies in both two and three dimensions. for (D D o. 11 llo) . however. the ca lculation of D 0 ¢; invol ves gri d points that arc separated by a di stance of .JJ . where rl is the dimensi onality of the gri d. thus for D = D 0 we must sati sfy
=
=
(8 I )
Rounding ofT to next highest integer val ues gives ).. ~ .). and ,\ ~ G in two. and three dimensions. respecti vely. We have
2058
I EEE TRANSACTIONS ON
=
used A 8 in our calculations in order to resol ve the peaks and troughs of the wavefi elds. ACK OWLEDGME T
The author would like to thank R. A. Krutar (N RL ) for hclpf'ul suggestions and comments. A . R. Haralampus. a teacher at Bishop o· Donne I High School. and E. Kirkland, a student at Gwynn Park High School. who spent the summer of 1993 here under SEAP Program. casting these equations into a working computer code. and verifying that the new algorithm actu ally worked. Without their ef'forts thi s paper could not have been wri tten. R EfERENCES Ill S. K. Godunov. Diffen'lll'l' Scheme.\·. Am;,terdam: North l lo lland , 19R7. 121 J. ~ t. Pear,on. A Th eory of \\'tli'I'S. BoMon: Allyn a nd Bacon. 1966. 131 J. B. Cole. R. A. Krutar. S. K. umrich. and D. B. C reamer. "A cellular automaton methodology for ;,olving the wave equation ... in Proc. 7lh AC/11/S/GARCH /111. Co11j Supercompu1ing. Tokyo. 1993. 111 R. 1\. Kruwr e/ a/.. "Computation of acou,tic fie ld behavior u;,ing a lattice ga;, model.'' in Pmc. Oceam 9/ Conference. vol. I. llo no lul u. 199 I. pp. .J16152. 151 S. K. Numrich. R. A. Kr utar. a nd R. Sq uier. "Computation of acoustic field' on a mas;,i1 ely parallel proce;,sor using lattice gas method;,.'' in Compultuimwl Acou.llic.\. Pmc. Jrd IMA CS ( /111. Asso. for Mtuhem(lfics a11d Comptuers i11 Simulmion) Symp.. R. D. Lau e1 a/.. E(b .. 1991.
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161 R. E. Micken;,. "A new finitediffcrem:e scheme for schroedinger I) pe partia l di fferentia l equations:· in CmnJmltllional Acou11il'.l. D. Lee. i\. R. Robinson. and R. Vichncvetsky. eds .. 101. 2. Am'>terdam: Ebe1ier. I993. pp. 233239. 171 0 . .'l laeshima. T . Uno. Y. lie. and S. Adachi. "fDTD analysis of twodimem.ional cavitybacked antcnna for ,ubsurface radar." IECE Tm111. h'h•clrtm.. vol. E76C. no. 10. pp. I.J6R I.J73. Oct. 1993.
James B. Cole n:ceived the B.S. degree in phy'>ics from the lllinoi., Institute of Teehnolog) in 197:!. the M.S. (h;gree in information engineering from thc University of IllinoisChicago in 19R I. and the Ph.D. degree in physics from the University of Maryland in 1987. He became interested in hi'> present field. interdisciplinary problems on the boundary between electrical engineering. information '>cience and physics. as a result of hi' po'>tdoctoral experience at the NASA Goddard Space Flight Center when: he worked on mode ling a possible production mechanism for antiproton' in the earth'., atmo;phere. a nd while working on new types of' high energy panicle detector,. He joined the Army Re;,carch Laboratory in 1981\. lie wa' an im ited Gue;,t Scientist in Japan at 1 ippon Telephone and Telegraph Corporation·, Basic Rc,earch Laboratory in 1990. where he applied group theory and differential geometry to human and computer vision problems. lie 1\a' ;1lso at the PhysicalChemical Research Instit ute in 199.J. where worked on e lectromagnetic problems. lie ha; been with the Naval Research Labnrator) in wa.,hing ton. DC s ince 1990 where he has been working on parallel algorithm' to ;,imulate wave propagation and scattering in complicated em ironment' coupled with ad vanced vbualitation.,.
IEEE TRA:'\SACTIO:'\S 0:'\
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NO. 9.
SE JYJ' E~ I BER
1995
2059
Inverted Stripline Antennas Integrated with Passive and Active SolidState Devices Jul io A. Navarro, Me111ber. 1£££. and Kai Chang. Fellmr, ! £££
Abstract Integr a ted antennas can reduce the size, weight, and cost of many microwave systems by incorpora ting compon ent functio ns directly at the an tenna terminals. T heir usc in m any comm ercial system appli ca tions can produce com pact, lowcost prod ucts. C urrently, acti ve integrated antennas arc used for distrib uted osci llators in s patial a nd quasio ptical power· combining. T he inverted striplinc a ntenna configuration was developed to easily integrate with solidstat e diodes or t ransistor devices fo r switching, tunin g, m odu lation, amplification, and osci llating functions. T his antenna configuratio n offers good performance, beam sharpening flexibility, and nondes tructi ve optimiza tion. Good switching. tuning, and osci llat ing performa nces have been demonstrated.
I. INTRODUCTI ON VER THE last few years, integrated antennas have recei ved a great deal of attention because they can reduce the size. weight and cost of many transmit and recei ve systems I ll. Passive and ac ti ve solid state devices can be used to combine several component functions at the termin als o f the antenna. For example. active devices can be used to design acti ve integrated antenna oscillators. ampli fi ers and multipliers. These active integrated antennas arc ideal for current inve tigations in spatial and quasi optical power combining. Integrated antennas pose an interesting problem i nvol ving several diffe rent areas of microwave engineering such as solidstate devices. circuits. components and an tennas. Knowled ge of solidstate device characteristics and ci rcuit integration arc criti cal as we ll as component speci fi cations and antenna perform ance. Combining guidedwave circuits wi th radiating structures often leads to several tradcoffs in performance. M ateri al properties which enhance ci rcuit performance ortcn degrade antenna radi ation. DC bia. i ng circuits and device packages also di sturb antenna characteri stics. Similarl y, an antenna' s radiati on may degrade a component" s performance. T hese difficulti es need to be overcome so that integrated antennas arc able to mee t system requi rements in military and commercial applica ti ons. Solidstate devices such as two terminal diodes and threeterminal transistors arc small. lightweight and easy to reproduce. These devices arc used to develop switch ing [21. tuning 131151. detecting 161. mi xing 171. amplifyi ng 181 and oscillating components. The choice of dev ice depends on the
~ J anu,cripr recdved Sepremb.:r 12. 199l: revised May 25. 1995. Th i' \\ork was ;upponed in pan by NASALewb Research Center in Cleveland. O JI. rhc U. S. t\rmy Research Office. and rhc Texas Higher Education Coordinar ing Board·, Advanced Technology Program. The author' :tre with the Department of Electrical Engineering. 1\:xa'> A&tlt Univer,ity. College Station. TX 778133 I 28 USA. IEEE Log Number 9l I 3120.
type or microwave component. operati ng frequency. power output and other considerations. A lthough transistors offer higher dctoRF con ve rsion effi ciencies, diodes reach higher operating frequencies and power levels. Diodes normally need higher de input level s but tran sistors require more complicated biasing schemes. The choice of solidstate device for an integrated antenna also invo lves its effec t on the radiation characteri sti cs. A lthough integrated antennas can reduce the si ze, we ight and cost of many microwave sy tcms. acti ve integrated antennas have shown a deterioration of both the antenna and component performance. I f an integrated antenna can mai ntai n component speci fications with l ittle degradation in the radiation characteri stics. the approach wo uld be very attracti ve for many commercial and military systems. For the past decade. integrated antennas have mainly focused on acti ve microstrip patches [9J[ 16j for radi ators in power combini ng applications. The patch serves as the resonator which compensates for the acti ve dcvicc·s reactance at the oscil lati on frequency. The diode position along the antenna determines operati ng frequency. output power and radiation performance. An aly tica l models and commercial software packages can ca lculate resonant frequencies and input im pedance al ong the length o f the patch w hich ca n be used to find an optimum position for a solidstate device. These model s. however. seldom account for device packages. bi asi ng l ines and integrat ion discontinuities which disrupt the fi elds and currents of the antenna. These disturbances o ften cause changes in operating frequencies. lower conversion effic iencies and higher crosspolarization levels (CPL). Variations in ci rcuit dimensions. dielectric characteristics and diode parameters also introduce errors in the final design. These errors arc typicall y tuned out after assembly i n order to consi stently meet frequency and power specifications. The abil ity to i ntegrate shunt devices easi ly is of particular importance for acti ve applications. Shunt connections in microstrip requ ires drilling i n M l C's and via hole processing in monolithic M IC' s. Some al ternati ve unipl anar geometri es have been demonstrated for acti ve and integrated antennas 11711 22 1. Unipl anar transmission lines such as copl anar waveguide. slotli nc and coplanar stri ps do not require drilling for shunt connecti ons alleviating the hybrid i ntegration disconti nu it ies encountered in microstrip. A l though not a true uniplanar li ne. inve rted microstrip docs not require dri lli ng for shunt co nnecti ons. T his allows nondestructi ve ex peri mental tes ting as well as position opti mization o f diodes and coax ial probe inputs. This trait makes in verted microstrip attracti ve for hybrid applicati on . When used for integrated antennas. inverted microstrip provides a
00 18 9.J80/95SO.J.OO © 1995 IEEE
2060
IEEE TRANSACTIONS ON
built in radome for protection. By carefu ll y matching the thermal expansion and conducti vity coe ffi cients of substrate materials and housing alloys. the integrated antenna can also be hermetically ealed for improved system durabil ity and reliability 123 1. 1241. DC biasing of devices can be achieved on the substrate or through the ground plane underneath. For increased metal volume and isolation, the an tenna is enclosed by a metallic enclosure which can then be classi fied as trapped inverted microstrip. However. thi s is a special case of general triplinetype configurations 1251. Gunnintegrated acti ve inverted stripl ine antennas have been demonstrated in 126J, 1271 for beam steering and spatia l power combi ners. These acti ve antennas ex hibit good radiati on pattern . . low crosspolarization levels. easy device integration and good heat sinking capacity. Furtherm ore. this configuration is useful in passi ve probe fed applications as well as integration with FET's and other dev ices. In thi s paper, a cavity model is used to ca lcul ate the re onant frequencies of the inverted configurati on and compared to measured results. A novel test fi xture was devi sed to nondestructi vely determine input impedance as a function of probe posi tion. The fixture also allows testin g of di fferent antenna dimensions. cavity diameters and substrate dielectri c constants and thicknesse. . Switchable and tunable probe fed antennas were demonstrated using PI ·s and varactor diodes. respecti ve ly. A n FET integrated in verted striplinc antenna was also developed with excellen t osci llati on and radiati on performance.
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(a)
(h)
Fig. I. Configuralion of" !he invcncd '>lripline alllcnna (lSi\) and le'>l fi \lure (a) lop view. (b) ~ide view.
u:.).
and the corresponding resonant frcqucnc) conve ntional cavity model the resonant frequency using 128 1
!·,.
(\Ill,('
= D..tr ,J4ff
i~
Using a calc ulated ( I)
71
II . D ES IGN AND T EST OF P ASS IVE A NTE
AS
T he inverted microstrip configuration removes the ground plane from the substrate backside and in verts the conductor over a ground plane support. The electromagnetic fields arc primarily concentrated in the air between the patch and the ground plane providing a lower effecti ve dielectric constant (c,.1r ). a longer guided wavelength and higher characteristi c im pedance over a comparable li ne width in microstri p. A lthough seri es or shunt devices arc easy to integrate. the inverted configuration is prone to exciting surface wave modes. Surface waves cause considerable crosstalk in densely packed circuits. reduce rad iation efficiency and distort an tenna pattern s. Shorting pins or metallic wa lls to either side of the conductor can el iminate these un wanted modes. The patch antenna in thi s configuration can be etched w ith an arbi trary shape while a circular enclosure i s more easily man ufac tured. In this investigation. circul ar patch an tennas were used to maintain circular symmetry. Thi s structure can be classi fi ed as a trapped inverted microstrip which i s a subset of the more general striplinc typc transmi ssion l i nes. Fig. I shows the top and side view o f the inverted stri plinc alllcnna ( ISA) configuration. The important dimensions arc shown as well as the novel test fi xture. These dimensions arc used to determine an effecti ve diameter for the patch antenna
D,.,r =
D
2
1+ (
1r ·
·IJJ ) ( lu (  /) ) D · c ('tr •I /J
where(" is the speed or light. 0/1111 is 1.8+ 11 8 for the dominalll mode and /J,. 1r is the effective patch diameter due to fringing fields over the radiating edges given by 1291 as shown in (2) at the bott om of the page. where f) is the physica l patch diameter and the height ( 13) is the patchtoground ~cpara t ion. The substrate used is RTDuroid 5870 with an e,. :::: 2.3. In thi s configuration. the majority of the fie lds arc concentrated within the regi on defined by the patch and the ground plane. Since air fill s the region below the patch. the ciTectivc dielectric constant (.o,. 11·} is very nearly one. An empirica lly determined value of 1. 1 gi ves resonant frequency va lues within 3C:f of the measured resu lts which range from 3 12 G I ll.. As shown in Fig. I, the dielectric substrates arc cut in 62 mm circular inserts. T he inserts arc press fitt ed on a 3 mm deep. ""'62 mm cavity enclosu re. The inserts arc suspended approximately 1.5 mm above the ground plane housing. The ground plane housi ng has a coaxia l probefeed topped by a 3.5 mm diameter cap wh ich ensu res good contact with the antenna and avoids soldering. The probe input was connected to an HP85 108 Network Analy;er and cal ibrated up to the anten na ground plane. The analyt.er tests input impedance match. operati ng frequency and impedance bandwidth. Thi s configurat ion and test fixture allows the enclosure and substrate insert to slide over the probe feed for posi tion
'2/J (0.'2(il:)e.,rr + + 1.77 + 1.ll c,.fr + )
I
I.G:i})
NAVARRO AND CIIANG: ANTEN AS INTEGRATED WITII SOIJ D.STATE DEVICES
20(11
.
13 12
1015
\leasurod
II
~ 10
101
.
t: 9
l
I 05
Calculated
I Oll I OJ
R
I
02~
7
I
I 02
6
I Oil
<
I 01 I 005 I
I0
15
20
25
30
35
40
45
50
55
60
Pa1 ch Dnunctcn (rnm) 62 mm l>laJncter Cal. H)
Fig. 3.
Operating frequency 'erMt\ patc h diameter,, 15
r
14 ! IJ
n i soGhz
J
p !
~~~ i ~ 10 !
Labels dcpoctlhe Opcraung Frequent) on Gllz
.g 9 I
~ aI go 7 i
=
Fig. 2. Input impedance ver'u' radial offset fJ for patc h diameter' D 30 mrn: other parameter~> arc C = 62 111111. A = 1.5 111111. and lJ = 1.5 nun. The dotted li ne de note' the 2: I VS\VR circle.
! :I 
4 ;
I
J 2 !
optimizati on. Accurate probe pos1110n is measured with a micrometer. Fig. 2 . hews the !SA input impedance of the fundamental mode versus probe position from the ce nter for a 30 mm diameter patch. The best 50 n i nput match for this antenna occurs at 5.2 1 mm where the VSW R is 1.0003 and the 2: I bandwidth is 3.77%. A s shown. the probe can be positioned over a fairl y wide range and it still maintains a 2: I VSWR. T he dominant mode appears as a short circuit at the center of the patch and increa. es in impedance at the edge of the antenna. Similar results were obtained for other patch diameters. Di fferent substrate inserts w ith patch antenna diameters which varied from I 0 60 mm were optimi zed for best 50 0. match at the operating frequency. Fig. 3 shows measured vcrsu. ca lculated operating frequencies of the !SA structure as a functi on of patch diameter. The calculati ons arc acc urate to within a few percent of the measured results. For the structu re dimensions of thi s investigat ion, the operating frequencies arc prim arily determined by the patch size and patchtoground separati on. These operating frequencies are typicall y some distance away from the nearest cavity mode which should not po c any problems. Simil ar results have been reponed in 1301. Errors may be due 10 variations in patchtoground separati on. probe discontinuities and metallic wall effects on the antenna radiating edges. A lso shown in Fig. 3 is the VSWR for each antenna diameter. A the probe moves along the patch. the rc ncction coe fficient ( sll) can be optimized fo; best 50 n match. , 11 [cvc[ s less th an 25 dB were obtained for each antenna. Fig. 4 shows 2: I impedance bandwi dths versus patch diameters. As shown. the 2: I impedance bandwidth i dependent on the patch to cavity diameter rati o. Large bandwidths arc possible using a simple probefeed w ithout complex matching circuits. The bandw idth increase for smaller patch dimensions i s ex pected si nce the separation ( D) i s larger with respect to the wave length o f operati on.
I
I
a9io ~
.
' 120
.
3 • ..,
0 : .... ~. ........ ..,. ........... ~ .... ,.····· ···· ... ·····10 15 20 25 30 3S 40 45 50 55 60 Patch Diameter (mm) 62 mm Diameter CaV\ty
Fig. 1.
2: I input impedance bandwidth vcr't" patch diameter'.
The radiati on pa11crns were tested usi ng an I IP au tomated antenna test sys tem. The radiation pallern!> for the 30 mm diameter amenna are shown in Fig. 5. The E and H p!ane halfpower beam wid ths (HPBWs) arc 57.5 and 6 1.5° . respectively. The crosspolarization level is 19.3 dB below the measured gai n of I 0.5 dBi. Radiation pa11crns for other patch diameters show that the HPB W va ries wi th the patchtocavi l) diameter rati o. Fig. 6 shows the HPBWs of the E and Hp!anc pallcrn s of several antennas tested in the 62 mm diameter cavi ty. The inverted substrate and enclosure combination increase the antenna directi vity. Noticeable beam sharpening occurs near patchtocavity di ameter ratios of ::::::: 0.5. As shown in this section, the inverted configurati on offers good circuit and antenna performance and it has flexibil ity in operating band w idth and radiating bcamwidth. Just as the coaxial probe was moved under the antenna. sol ids tate device positions can also be optimized for best impedance match. The foll owing sections describe i ntegrations w ith PIN. varactor. Gunn diodes and FErs for hybrid M !C application!>. [[[. [ 'TEGRATED ANTENNAS WITH PIN'S AND VARACTORS Twoterminal device integration for this structure is straightforward. Diodes can be connected across the patch to grm7nd. Fig. 7(a) shows in tegration wi th two PI or varactor diodes. From Fig. 7(a). vary ing the diode position under the patch changes the impedance seen by the diode. Al ternat ively. the diode posit ion determines the diodc·s effect on the antenna. Since the electri c fi eld is a maximum at the radi ating edges
2062
IEEE TRANSACTIONS ON
..,..
~ II CROWi\VE
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THEORY i\ND TECHNIQ UES. VOL. 13. "0. 9.
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modate the FET as shown in Fig. 13. FET integration requires th ree de blocks for the drai n. gate and source terminals. 1t shows pri ncipal plane and crosspolari1.ation patterns of the DC bi a~ in g can be achieved from behi nd the ground plane ac ti ve antenna. T he smooth radiation patterns and low crossor etched 7o the nonradiati ng edges of the antenna. A chi p polarization level compares favorably wi th previously reported res istor connec ted across the source to gate simpli fies biasing acti ve antennas 1911 13] . 115 1. Probefed passive antennas to the device. The operati ng frequency is determined by the wit h cavity depths of 3 mm exhi bi ted HPBWs of 51 and loads at each transistor port. T he transistor can be induced to 6 1° i n the E and Hplane patterns. respective ly. T he crossosci llate at lower frequencies to reduce effects from the device pol arization level o f the passive an tenna is also  19.3 dB with package and bias lines. Lower frequencies of operation can be a oai n of 10.2 dBi. Biasi ng modi licati ons and cavi ty depth e . I accomplished with larger patch diameters. differences i n the acti ve antenna may account lor the c umgcs. A 30 mm patch was modi fied to insert the FET (N EC model The effecti ve isotropic rad iated power of the acti ve an tenna 76 18tA). A t the center of the patch. 0.4 mm gaps isolate the is 594 mW. A pproximating the acti ve an tenna gai n with the source from the gate and drai n terminals. A 0. 1 mm gap is passive antenna gain of 10.2 dB results in an osci ll ator power etched from the center to the nonradi ating edges of the patch to o f 57 m W and a dcto RF conversi on eniciency of 57 1Jc. provide de isolati on between the gate and drain terminals. T he V I. CONCLUSION FET drain lead is soldered at the center of the patch and the gate lead is approxi mately 2 mm offcenter. T hree de li nes bias A useful passive inverted stripl ine antenna configuration ~1 c ~ou rcc. gate and drain of the transistor. A ltern ati vel y. the has been demonstrated. Integration has been demonstrated source bi as ~nc can be replaced by a resi stor. T he frequency of w ith PIN and varactor diodes wi th !>ome degradation of oscillation depends on the impedance loads at the FET port s. the antenna performance. The integration create~ switchable The loads arc a function of the positi on along the inverted and tunable rad iati ng M IC components. Gun n diodes have patch antenna. Heat generated i s di ssipated by the patch. The also been integrated for act ive antenna rad iators with good lack of a lowimpedance path from the device 10 the metal heat output power and a clean spec trum . A n FET has shown sink may cause thermal probl ems in higher power devices. very good oscil lator characteri stics as well as excepti onal T his could be allcviatccl w ith a . honing pin at the center o f radiation performance. It operates at low voltage and curre m the patch to provide a low i mpedance path 10 the housing. level s w ith high convers ion efficiency. These active antennas Si nce the antenna was modified and the device terminal can be used for doppler sensing 13 11. disposable decoys. arc sol dered. it is not possible to experimentally optimize the wireless communications. phasedarrays 126 1. 1321 and power FET posi ti on. However. a ground plane sl iding short is used combini ng appl ications. Overall. these integrated and active to change the antenna cavity depth and alter the oscillation integrated antennas perform well as compared to current frequen~y. The cavity depth is used to i mprove spectral and stat;ofthcart passive radiators. Further improvement can radiation characteristics of this acti ve antenna. move these components closer to commercial appl ications. T he FET provides 57 mW at 5.69 Gll z when biased at 3.8 Combinations of several devices wi l l allow the abili ty to V and 26 mA. T he 3 dB bi as tu ning range is approximatel y integrate various component functions. 1% for a 1 V change in lltJ .,. T he sliding ground plane allows a mechanical tuning range o f nearly 6%. A t a cavity depth o f ACKNOWLI:DGI.1ENT 4. 15 mm. the measured oscillation frequency remains stable at T he authors would like to thank the Rogers Corporation for 5.695± 0.002 G llz over the antenna test sweep. T he HPBW in the substrate material which they provided. as well as L. Fan the E and Hplane patterns are 46 and 64° . respectively. The and J. M cSpadden for thei r technica l support. crosspolarization level is  19.3 dB below the maxi mum. Fig.
NAVARRO AND CHANG: ANTENNAS INTEGRATED \VITI! SOLIDSTATE DEVICES
REFERE CES Ill D. B. Rutledge. D. P. Neiki rk. and D. P. Kasilingarn. "lntc.:gratc.:d circuit alllcnna,." in lnfm red and Millime/er \Vm·es. K. J. Bunon , cd .. chap. I. vol. I 0. New York: Academic. 1983. p. 25. 121 R. B. Waterhou;c.: and N. V. Shuky. " Dual frequ.;n~y microstrip rectangular patches." Eleclmn. Lell.. vol. 28. no. 7. pp. 60o 607. Mar. 26. 1992. 1:11 P. 13hania and I. J. 13ahl. "Frcquc.:ncy agi le microstrip anthcd over 200 technical papers and several boo k chapters in the areas of microwave and millimeterwave devices and circuits. Dr. Chang received the Special Achievement Award from TRW in 1981. the Halliburton Prof.:ssor Award in 1988, the Distinguished Teaching Award in 1989. and the Distinguished Research Award in 1992 from the Texas A&M University.
IEEE TRANSACT IONS ON ~IICROWAVI: THEORY A:\D TECII:\IQCES. VOL. 11. :\0 9. Sl1'11! \IBI:R 1995
2066
Mode Conversion and LeakyWave Excitation at OpenEnd CoupledMicrostrip Discontinuities Joseph L. Cina and Lawrence Carin . Member. IEEE
Abstract:fh e m ethod of momen ts (i\loi\1) is used to s tud v mode conversio n a nd leakywave excitatio n at a n asymm etric coupl edmicrostrip discontinuity. The results show that significant mode conversio n can occur at such discontinuities and that dominant leakywave modes can he excit ed strongly. umcrical issues with regard to the i\ loi\1 analysis of such disconti nuities arc addressed as well, a nd for some examples it is shown that inclusion of a com pletedomain bas is fun ctio n for the leaky mode improves numerica l stability dramatically.
I. INTRODUCTION
VER the last several years leaky modes supported by planar transmission lines have been the subject of significant interest I I 11 121. A transmissionline mode leaks energy when it is a fast wave wi th respect to a wa ve or mode supported by the surroundi ng medium . Since planar transmission l ines are usually fabri cated on dielectric laycr(s). energy is o ften leaked into the laycr(s) in the form of a surface wave 111 161: however. energy can al so be leaked i nto parallelplate modes 171 1101and space waves II 1151. 191. For example. Oliner 131has shown that all higherorder microstrip modes arc leaky over some frequency range, w ith energy leaked in the form of surface and space waves: sim ilarl y, Boukamp and Jansen 171 have shown that over appropriate frequencies higherorder modes supported by microstrip with a top cover leak energy in the form or parallelplate modes (the leakage over appropriate frequencies associated with these particular higherorder modes is a general property of all higherorder planartransmission l ine modes). Additi onall y. it has been demonstrated that several structures support fundamental modes (characteri zed by no cutoff frequency and often referred to as .. quasiTEM'') which arc leaky over appropriate frequencies, e.g .. slot line I l l. [81. 191. microstri p on a properly oriented uniaxial ani sotropic substrate 161. coplanarstri p wavegui de I l l. [9 J, and broadside coupled microstrip 11 01. Most of the previous studies o f leaky modes supported by planar transmission lines have been two dimensional in natu re, with an emphasis on calculating the complex wave number and the transverse field profi le: there has been l ill ie work on the effects of such leaky modes on practi cal threedimensional planar circuits 11 31. Recently the fi nite difTcrencc time domain 1\l:mu.,~ript recci,ed September 16. 1991: revi>cd 1\lay 25. 1995. J. L. Ci na i'> \\ ith the Depanrnent of Electrical Engineering. Pol) technic Univcn,ity. Brooklyn, NY 11 201 USA. L. Carin wa~ wit h the Department of Electrical Eng ineering. Polytechnic Uni\ a' it). Brook lyn. NY 11 20 I USA: he il, now with th~ Department of Electrical Engi neering. Duke Univer'>ity. Durham. C 27708 USA. IEEE Log umber 9113117.
method has been used to investi gate the leakageinduced di stortion o f pulses propagating on transmission lines which support leaky modes 1121. However. we arc not aware of any previous work on the effec ts of leakymode e:..citation at tran s mi s~ ion l in c di ~co nlinuiti cs: thi s is the topic or the present paper. In particular. we investi gate the exci tati on o f zcrocutoiT frequency leaky tran smissionli ne modes at opencircuit couplcdmicrostrip discontinuities. A spectraldomain !oM 1131l 161 anal ysis is applied to model discontinuities in vol ving coupled microstri p lines i n a structure with a top cover. Such discontinuities have been studied 11 q in the context of micros trip filters. where a spectraldomain M oM analysis was applied. llowcvcr. th at study considered boxed (shielded) microstrip and therefore did not consider the c iTccts o f leakywave excitation. Further. si nce that study focused primarily on the scallcring parameters of the fi ller. mode conversion at couplcdmicrostri p discontinuitic'> was not addressed. Mode conversion has been investigated by Jackson 1161 using the MoM for th e case of coplanar waveguide terminated in a short and open circuit: howe\ cr. th at sw dy also did not address the exci tation of lcal.. y modes. Al though discontinuity induced leaky mode exc itation has not been in vestigated direc tly in these previous studi es. it has been addressed indirectl y since. from the above discussion. all higherorder transmi ssionline modes exc ited arc leaky over some frequency range. However. these modes usually damp strongly and therefore only affect currcnb in the immediate vicinity Of the disconti nuity (they excite :.urracc \\'aves. space waves. and/or parallelplate modes which origi nate ncar the disconti nui ty and propagate power i nto the surrou nding media. away from the discontinuity). Here we cmphasit.c the excitation of leaky fundamental tran ~ mi ssion  l ine modes. for which the rate or leakage is often substantially weaker than for higherorder modes. The explicit problem considered is represented schematically i n Fig. I : a single even (longi tu dinal currents arc the same on each strip) couplcd microstrip mode is incident upon an asymmetric openci rcuit discontinuity and we investigate the scallcred currents. It is we ll known that an .\" condu~tor tran smi ssion line supports .V I t.crocutoiT frcquency modes and therefore the structure we have se lec ted supports three fundamental modes (sec Fig. 2). Under appropri ate condition'> one or more o f these modes can be leaky II 01. and \\C address the excitation of such modes at the discont inu ity. Thi s work is si milar to that o f Jackson 11 61 i n that we address di scontinu ity induced mode conversion i n ovcrmodcd planar tran smission lines: however. our worl.. differs significant)) in
!XJIS 9180/95$01 .00 © 1995 IEEE
CINA AND CARIN: CONVERSION AND EXcrli \TION AT COUPLEJ\IICROSTRI P DISCONTINUITIES
SemiInfinite B.F.
I
z=O
I I
PWS B.F. Fig. I. Schematic of coupled microstrip tcnninat~d in an asymmetric opencircuit discontinuity. The scauering problem is analyzed using a Method of Momeltls (MoM) algorithm in whic h the fundamental modes arc expanded in terms of semiinfin ite basis function s (13 F) which extend over :: E ( . 0]. Piecewise sinusoidal (PWS) suhsectional basis functions arc used in the vicinity of the di,continu ity to account for the excitation of higherorder coupledmicro, trip modes.
,
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Fig. 2. Cros~ >ect ional view of a couplcdmicrostrip transmission line with a top cover. The top cover. ground plane. and dielectric substrate arc assumed in the analysis to be of infinite transverse e xtent.
thai we consider situati ons in which mode conversion occ urs into one or more fu ndamental modes which arc leaky. In addition to thi s phenomenological issue. we also demonstrate that these modes may impact the numerical stabi lity of the M oM algorithm . The remainder of the paper is organized as foll ows. The numerica l procedure used to model the discontinui ty is reviewed in Section II : alt hough the M oM anal ys is o f such discontinuities is well known. we mention brieOy several import ant i. sues relevant in the considerat ion of leakywave cxcitmion. In Secti on Ill, results arc presented which demonstrate that significant mode conversion can take place at couplcd microstrip disconti nu i ties. and highlight the excitation of leaky fundamental modes. Conclu sions and implicati ons of th is work arc addressed in Section IV.
~067
sinusoidal (PWS) subsccti onal basis functi ons i n the vicin i ty of the discontinuity (sec Fig. I ). The PWS basis functi ons model the complicated current density around the discontinui ty. and therefore they account for higherorder ( in general. highly leaky) transmission linemode excitati on. The longi tudinal variat ion o f the semiinfinite basis fu ncti ons is represented by cxp ("y;:::)U (  .;) .where 1 1 =  j rJ1 for the incident mode (an cxp (.jw t ) ti me dependence is assumed and suppressed). "Yi = jfJ; + n ; for each of th e three rencctcd fundamental modes (o 1 = 0). U(z ) = 1 for :: > 0. and U (z) = 0 for z < 0. The wavcnumbers "'(; for the three fundamental modes supported by the co upled microstri p arc ca lculated via a standard twodi mensional spectral domain M oM analys is, wi th proper deformation of the spectral i ntegrat ion path for the leaky modes ]9]. In that anal ysi s a single basis function f (.r  .rc ) = {J  [2(.r  .r c}/ w]Z } I/ :.! is used to represent the transverse (.r) vari ation of the current. w here 111 is the strip width and .rc locates the strip ce nter: thi s transverse vari ation of the longitudinal curren t density models the edge singularity and is i ncluded as well in all basis functions applied to anal yze the discontinuity. The unknown basisfuncti on coeffi cients for the rcnected modes and the PWS arc determined using a Galcrkin testi ng procedure 1131 11 6 ]. In our MoM analy sis. the expansion and testing functi ons arc expressed i n the spectral domain. Because the exci tation o f leaky fundamental transmi ssion  line modes is an i mportant new aspect of the present paper. we demonstrate how such modes arc modeled by semii nfi nite basis functions. A ssuming the wavenumber 1 = j (1 + n (fJ > 0 and n > 0). the basis function' s longitudi nal varia tion for propagat ion in the  z di rection is expressed as f' XJl
[(j(f
For 1he discontinuity in Fig. I , we apply a spec tral domain MoM analysis similar to that developed by Jackson and Pozar [13]. Only the longillld inal co mponent o f current is considered since the tran sverse component i s negl igible for the l ine w idths and frequencies of interest here. The longitudinal current density is expanded in two types of basi functi ons [1 3 H 16]: I ) semii nfini te completedomain basis function s representing the single incident mode and the multiple rencctcd fundamental modes. and 2) piecewise
scclion in bou nd a nd leaky n:girnes:· IEEE hans. Micm11m·e "JIII!ory 'fl•clr.. vol. 37. pp. 200520t0. Dec. 1989. 161 ,\11. Tsuj i. H. Shigc>awa. a nd A. A. Olincr. ··Prinledcirc uil waveguide wil h an isolropic substr:Hes: A new leakage e ffect:· in IEEE MITS Int. S.n111J. Dig .. Long Beach. CA. June 13 15. 198. pp. 7837869. 171 J . Boukamp and R. II . Jam.en. ··spcclral d omain inves1iga1ion o f >urfacc wave e xcilalion a nd radialion by rnicrostriplines and microslrip d i>k re,on:Hors:· in Pmc. l:·umpean l'dicmwm•e Conference. Nurnbe rg. Germany. Sept. 58. 1983. 181 H. S higesawa. 1\1. Tsuji. and A. A. Oliner. ··conductorbacke d s lolline and coplanar waveguide: Dangers a nd full wave analyses:· in IEEE M7TS Int. Symp. Dig .. New York. ' Y. May 2527. 1988. pp. 199202. 19 1 N. K. Das and D. M . Pozar. ··Fullwave spec1raldornain compui:Hio n of malcrial. radi:Hion. and g uided wave losses in infinile muhilaycrcd pri nlcd 1ransmissions lines:· IEEE Trans. Microll"lll"e Tlreory Teclr .. vol. 39. pp. 54 65. Ja n. 1991. 1101 L. Cari n and N. K. Das. ··Leaky waves o n broadsidecoupled microwip:· IEEE Tran.1·. Micr01wn·e Tlreory Teclr.. vol. 40. pp. 5866. Jan. 1992. [Il l L. Carin. A. A. Oliner. G. \V. Slade. and K. J . Webb. ··Packaged primed lransmissionline modal phenomena and re i:Hio n 10 leakage:· in 1993 IEEE /I /7TS In t. 5)mp. Dig .. pp. 1 195 l t 99. 11 21 M . T>uji. M. Tanig uchi. a nd H. Shigesawa. "The fc:Hure of !he narrowpub.:: lr:msmissio n o n conven1 ional copla nar wavegu ides when power leakage is prcscm:· IEEE 7iwrs. Micro11·m·e Tlr eory Tech.. vol. 41. pp. 1017 1023. June/Jul y 1993. 11 :11 R. \V. Jackson and D. M. Pot.ar. ··Fullwave analysis of microslrip openend a nd gap disconlinuilies:· lt:EE Trans. Micm11m•e Tlr eory Tech.. vol. :n. pp. 1036 1042. Ocl. 1985. 1141 P. 13. Kalchi and N. G. A lexopoulos. ··Frcquc ncyd epend enl characte rislic> of m icros!rip discomin uilic; in m illimclcr wave in1egra1ed c irc uils ."' IEEE hans. Micm11m ·e J1rem:r Teclr. vol. 33. pp. 10291035. Oc1. t985. 11 51 R. \V. Jack >on. ··Fullwave li ni1c e le mcn l analysis of irregu lar rnicroslrip discominuil ic:.:· 1/;EI:" "liwr s. Microllm·e Tlreory Tech.. vol. 37. pp. 81  89. Jan . 1989. 11 61 _ _ . ··Mode conversion al discomin ui1ies in linile w idlh conduclorbacked coplanar waveguide:· vol. 37. pp. 15821590. Oc1. 1989. 117 1 D. M. Po tar. " In put impedance and m u!Ual coupling o f rccwn gutar microslrip a nlcnnaS:' IEEE Tran.1. ,\ ntemws Propagal. vol. A P30. pp. 1191  1196. Nov. 1982. 11 8 1 N. K. Ut.unogl u. N. G. Ai a member of the.: Tau B.:ta Pi and Eta Kappa Nu honor soci.:tie>.
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TIIEORY AND TECHNIQU ES. VOL. 43. NO.9. SEir n: ,\ 113ER 1995
207J
Development of SelfPackaged High Frequency Circuits Using Micromachining Techniques Rh onda Franklin Dray ton, Swdenl Me111ber. IEEE. and Lind a P. B . Katehi, Fellow, IEEE
AbslraciA new concept for packaging high frequenc y monolithic ci rcuits is presented. It consists of developing miniaturized housings to shi eld indi vidual passive components (e.g., CPW based), acti ve clements, or combinati ons of them by employing silicon micromachining technology. At high frequencies, selfpackaged conligurations that arc fabricated in this manner provide reduction in the overall size and weight of a circuit and provide increased isolation between neighboring circuits. T herefore, the resulting characteristics make these mi cropackagcd components appropriate for high density, mult ilevel inte1·conncct circuits. T his papct· will describe the fabrication procedures used to develop selfpackaged components. Pe rformance curves for typical high frequency circuit geometri es that arc implemented in this configun1tion arc shown fo1· measured a nd theoretical t·esults.
I. I1 TRODUCTION ICROWAVE and millimeter wave ci rcuit desi gn relics heavily on the usc of planar transmission li nes such a microstrip. stri plinc. slotl inc. and cop lanar waveguide to achieve design ncx ibility as we ll as case in mounti ng active components. Whi le inherit ad vantages arc offered wi thin each of the ex isting technologies. limitations arc still present and are related primari ly to issues regardi ng parasiti c mechanisms such as electromagnetic co upling and parasi tic rad iat ion. Si nce circuit performance is often time compromised and successful designs arc conditional on circuit perform ance requi rements. the development of a novel ci rcuit gcomc1ry th at oilers the capabil ity to mini mize these effects i s warranted. One sol uti on. therefore. is to provide shielding 10 individual ci rcuit components whi le preserving the structural characlcristics o f the conventional uniplanar technology. Thi s approach can provide a new dimension to high freq uency circui t design and can al low for more compact circui t configurations which conseq uently address the issue of high density packagi ng. These individually shielded circuit components arc achieved by developing monol i thic cavities. rc!'crrcd to as " micropackages ... around existing planar circuits through novel usc of we ll establ ished Si or GaAs micromachining technologies. Thi s results in improved circuit performance due to 1hc reducti on of parasi tic electromagneti c coupl ing in the substrate (substrate modes) and air (space waves). In typical applications. where metall ic housings arc used to protect circuit componen ts from the envi ronment. unwanted
M
IVIanuscripl received September 26. 1 99~: revised May 25.1995. Th is work was ~ upportcd by the Office of Naval Research Contract NOOO 1192J I 070 and 1hc Army Research Office. T he authors arc with T he Radiatio n Laboratory. Electrical Engineering and Cornptucr Science Dcpartmenl. The Univcrsily of Michigan. Ann Arbor. M l ~81092122 USA. IEEE Log Number 9~13~39.
Upper Cn II) (:urhllcd)
Fig. I. T hreedimensional cross scclion of a micropackaged circuit where 1hc shield and line arc i1uegratcd monolilhically.
paras1!1CS. and multiple resonances arc introduced due to the presence of a l arge metall ic shielding package. Micropackaging, on the other hand. eliminates those resonances by deve loping shielded structures that arc small i n size. can follow individual circuit paths. and ca n be fabricated monolithica ll y w ith the circuit. A dditionally. micropackagcs arc also easily compatible and integratablc wi th existing conve ntional circuit arrangements and can be designed to electri cally and/or hermetically encapsulate planar circui t componen ts or subsys tems. These selfpackaged ci rcuits can therefore be ciTccti vcly used w ithin larger conventional housing struclllrcs either hybridly using a fl i pchi p technology or monolithica ll y in a minialllrizcd package config uration where the circuit response is now electrically dccouplcd from interaction s with the l arger metallic package. RF micromachining. while still in its infancy. has been recentl y used i n the development of monoli thic waveguides Ill and high perform ance membranesupported circu i ts 121. Even though micromachining of silicon is a wellestablished technology for sensor and biomedica l applicat ions. many other applicati ons exist in high frequency circui t design that can benefit from its usc. In an ef fort to ou tline the potential or thi s technology . thi s article addresses packaging issues and presents the development of the first generation or micromachinccl, selfpackaged circuits. The ci rcuit components presented arc primarily of coplanar waveguide (CPW) type and arc surrounded by an airfilled cavity i n the upper region and a substratcfi lied cavity in the lower region as shown in Fig. I. Since comprehensi ve understanding of fabrication capabil i ties is critical for the development of these ci rcuits. an ex tensive study of vari ous fabrication techniques has been performed and a brief description of the fi ndings is presen ted. Wh ile the structures presented in this paper represent only one approach for implementing micromachining techniques to high freq uency applications. several basic clements have been chosen to i llustrate the appli cabi lity of thi s technology i n thi s area. These simple componen ts con ist of tun ing stubs and li ltcrs that arc developed. measured and compared to the
00 1 89·Hl0/95$0~.00
© 1995 IEEE
IEEI: TIV\NSACTIONS ON
~IIC'ROWAVE
T II EORY AND TI:CIINIQUI:S. VOL.
~ .1.
NO. '1.
SI: I Yri:~IBER
1'!95
performance of theoretical results. In Section 11 A, a general
to design tradeoffs. Ideally. a given circuit has an optimum
overview o f the design approach is give n followed by an cxtcnsi\ C discussion o f the fabrication procedures developed and im plemented. In Section 118. measurement considerati ons arc di scussed. fundamental line characteri sti cs arc presented. and experimental results for various circuit components such as tunin g stub clements and filters arc shown.
package si ze that is small enough to elimi nate resonances within the range of operating frequencies and th at is physically far enough away from the circuitry so that it docs not i nterfere w ith the circui t' s electrical performance. Although circuits in open environments do not face the previously described problems. they arc prone to parasitic radiation which is mostly associated w ith the exci tation of substrate modes. Since the exc itation o f these modes is mos tly dependent on the operating frequency and the physical thickness o f the substrate. careful layout configurations in less dense circuit environments can sometimes reduce such parasi ti c radiation. provided there i s fl exibilit y in circuit placement. In practical applications. however. circuit requirements greatly l imit the flexibility in rearranging the locat ion o f the va ri ous circ uit components such that any layout modifications. at least. ca n only weak ly reduce such parasitic loss 1151. During the characteri zat ion efforts performed . prelim inary findings indicate that substrate modifications alone may have a substantial impact on substrate mode exci tation. While researchers have been aware of the issues discussed above in planar designs si nce the late seventies. the ex isting technology at that time could not provi de alternati ve solutions to reduce the substrate mode excitation. Recent advances in silicon micromachining techniques. however. allow for unique. yet simple practical solutions to the above probl ems. In the development of micropackagcs. di mensions can be chosen simpl y by using waveguide and cav ity models to predict geometrical dimensions that avo id un wanted resonances. In addition. the shape of the cav ity can be designed so that it follows the circuit and docs not physical ly affect its performance. For open circuits. substrate mod ifications using micromachining can be implemented to eliminate these unwanted substrate modes entirel y. resu lti ng in improved ci rcui t performance.
II. D EVELOPI\I ENT OF MICROI\ IACHI 'ED CIRCU ITS To i llustrat e the versatility o f micromachin i ng. thi s effort concentrates on developing ci rcuits that reside in partiall y or compl etely shielded environments. Poor electrical perform ance or circuits operating in open environments is attributed to freespace radiation and substrate mode excitation. w hi lc in shielded environments it is main ly due to package resonances. The open environment described herein refers to circuits which arc printed on a dielectric substrate and radiate into free space. On the other hand. the term " shielded geometries" implies circuits which arc parti al ly or completely shielded by cavities in the upper and/or lower regions. Since substrate modes occur in the dielectric substrate. the introd uct ion or phys ical alterat ions to the substrate itself can result in elimination of these parasitic waves and in improvement or circuit performance . The following sec ti ons outline the steps needed to design the circuits mentioned above and ex tensive ly describe the silicon (S i ) fabrication processes required for their development.
A. Desig11 Approach The first step in the des ign of high frequency circuits is the spcc i fication of the geometrical parameters needed to provide the desired electrical response. Whil e there ex ists a wide variety o f commercially available computeraided design (CA D ) toob for low frequency appli cati ons. software for high frequency design i s currently unavailabl e. Despite the lack or CAD so ft ware at these frequencies. there arc a number of circuit simul ations tool s that can be used indirectl y in design to produce sati sfactory analysis results. even though. very long design cycles. and extensi ve computation tim es arc typi ca lly required. The micromachincd selfpackaged circuits presented here have been designed through such an iterati ve approach. In this scheme. first low frequency software models that usc quasistatic approximati ons 131. 141 provide the initial design. then the geometrical parameters arc modified in an iterati ve manner and reali zed. Lastly the circui t performance is predicted using the high frequency anal ys is soft ware which is based on fullwave models in frequency or time domains 1511 1 When the predicted electri cal response closely matches the desired one. thi s iterative cycle is terminated. Concurrently the issue o f package resonances and substrate mode excitation arc al so addressed. As it has been extensi vely described in the literature 11 31. I I + 1. electri cal packages can greatly affect circuit performance either through package resonances or through proxi mity coupling. The first effect is mostly related to the dimensions of the package while the Iauer is due to crosscoupling of neighboring circuits. Unfortunately these two mechani sms require contradicting approaches which lead
:n
13. Fahricmion Procedures The micromachi ncd circui ts described herein arc comprised or a twos i licon (S i) wafer system havi ng a ( LOO) orientation and primaril y util ize sil icon micromachin ing processes. which arc fairl y standard in sensor applicati ons but arc relativel y new to high frequency ci rcuit design. T his sec tion presents an ex tensive discussion on the fabrication steps required for the development or selfpackaged ci rcuits and provides a detailed description of the empl oyed processes. A ll circuits presen ted arc of coplanar type. printed on Si. and shielded by miniature cavities in both or ei ther or the upper and lower surrounding regions. As seen in Fig. I. the upper reg ion consists of a mctallit.cd airfilled cav ity whi le the lower region is a substratefilled cav ity that is metallized on the lower side. Since the circ uits arc printed on the lower wa fer. high resisti v ity sil icon wi th r, ll.l is required. In thi s configuration the upper and lower cav ity regions provide ground plane equalization through direct con tact with the ground planes of the coplanar waveguide l ines. Important issues to consi der during design include the fabrication or acc urate alignment marks betwee n the various wafers. the design of appropriate feed ing l i nes for individual circu it cxc i
=
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DRAYTO:'I/ AND KATEIII: DEVELOPW!NT OF SELFPACKAGED CIRCU ITS
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Fig. :2. Lower wafer dcvdopmcnl. (a) Trans mission lines arc printed on the top >urfacc . (b) Lower ca vity is fo rmed by etch ing vgroovcs. (c) Lower cavity groove, arc metallized below the line forming d irect contact to the upper ground planes.
tat ion. and the development of enclosing environments that can be integrated monolithica lly with the speci fic circui ts wi thout degrading the electri ca l performance. T he following sections address these concern s comprehensive ly w hile descri bi ng the technique used for individual wa fer deve lopment. The lower wafer scheme shown in Fig. 2 i s a high resi sti vi ty. singleside polished. 350 micron thi ck silicon substrate having a silicon diox ide dielectric layer that has been thermall y grown to 1.2 micron thick ness. Si nce this wafer contains the planar lines and lower cavi ty region of the circui t. ox ide regions are removed on the upper surface using buffered hydrofluori c ac id (BHF) to allow ground pl ane equalizati on of the lower shield to cop lanar waveg uide ground plane. The planar circuits arc then defined using standard photol ithographic techniques and electroplated to achieve a three micron metal thi ckness after evaporating a seed layer o f titanium/gold/ti tani um (Ti/Alfri ). Once the circuits and al ignment marks have been printed. the backside of the wafer is patterned ph otol ithographi call y to defi ne the lower cavity regions using an infrared ( IR) alignment procedure. Prior to anisotropically etching the si licon. the ox ide is removed fro m these regions that define the cavity sidewalls usi ng bu ffered hydro fluoric acid (BHF). The ani sotropic etchant. ethylene diaminc pyrocatechol (EDP). uses the ( lll) crystal plane as an etch stop for (JOO) si licon wafers and results in an etch angle pro fil e of 54.75° 1161. The cavity formation has pyramidal sidewalls as seen in Fig. 2(b). where the lower cavity is the substratefi lled one shown wi th its w idest dimension at the upper surface an d narrowest dimension at the bottom of the cavi ty I Fig. 2(c) I. simulating an i nverted pyramid. The lower cavity shielding formation is complete after a seed layer of T il A u is evaporated and electroplated on the back side of the wafer to three microns. The upper wafer scheme. shown in Fig. 3, contai ns both upper cavities and alignment marks that arc formed by etching from both sides of the wafer. Since these cavities provide ground plane equal i zati on and shielding w ithou t interfering w ith the si gnal path. it is not necessary to use high resisti vity Si. The upper shielding in thi s work is developed using a 500 micron thick low resisti vity Si with 7500 Aor thermally grown ox ide. After defining the probe w indow and align ment marks using photolithogrpahy. a metal l iftoff procedure is employed to open the areas to be etched and to provide an addit ional masking layer o f Ti/A u metal on the back side of the wafer. Thi s layer serves two purposes: it offers protection on the back
Fig. J. Upper wafer development. (a) Probe windows and alignment ntarks (A I ) arc etc hed fro m both sid e' while the uppe r cavity (A2J is etched from one side o nly. (b ) T he upper cav ity is the n rnctalli;cd. (c) Finally. the u pper wafer 'ectional view after proces>ing with the alig nment mark' (C 1). u pper cavi ty (C 2) a nd the probe w indow (C3).
of the cavi ty regions and it acts as a mask during backside IR alignment. On the lower side. the cavities arc dcflncd and the oxide is removed to expose the silicon surrace as shown in Fig. 3. The patterns arc then etched i n EDP to a desired dimension th at is monitored using ··etch rulers:· which consist or rectangu lar widths correspondi ng to specific etch depths. Since th is wa fer must be handled frequently after etchi ng the mult iple cavi ties and wi ndows, addi ti onal mechanica l strength can be provided by incl udi ng a structural beam. located in the middle of the probe window. as seen in Fig. k A fter fabrication. the upper cavities arc aligned to the planar circuits wi th lower cavi ti es that arc located on the upper and lower wafers. respectively. The two wa rcrs arc then bonded together to complete the formati on of the mi cropackagc (Fig. 5). W hi le the above procedures conce ntrate on the deve lopment of a completely shielded or selfpackaged con figurati on, partially shielded structures can also be obtained by implementing procedures presented in either the upper or lower wa fer development scheme.
Ill.
EX PER IM ENTAL C HARACT ERIZATI ON OF
P ARTI ALLY A1 D COMPLETELY SH IELDED CIRCU ITS
A. Measure111ent Considerations To measure ci rcuit performance up to lO G H z. conventional onwa fer characteri zation is utilized in conjunction with the thru rcfl cctl inc (TRL) calibration technique 117 1 11 9 1. The measuremen t setup consists of an HP 85 I OB ctwork A nalyzer that operates up to 40 G Hz. an A lessi probe station. and Cascade M icrotcch groundsignalground (GSG) probes that have a probe pitch 1 of ISO 11m. This ca libration i s achieved using standards that include upper and lower shielded regions th at arc identical to those or the circu i ts or interest and that arc developed using the rabrication procedures described previousl y. A one tier deembedding technique is used for onwa fer probing which cali brates the system reference plane to a point w ithi n th e shielded transmission line. Thi s results in characteri zation o f all transi tion s located between the i nput and output ports of the ANA and the newly defined reference pl ane. 1 Pitch is dclined as the 'cparatio n between the signal line and the ground plane or the coplanar GSG probes.
IEEE TR ANSACTIONS ON W CROW;\VE THEORY AND TECHNIQUES. VOL. JJ. i\!0. 9.
2076
SEPTE ~113 ER
1995
TAB LE I
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SI' Crto:. \L Dt~u::.sto:.s OF SHIELDED CiRCl:l rs
C:l\il.\'
height
\\id thma\
"idth min
Upper l.tmcr
lSO '50
l liKl 950
Xh.:ppcd impedance l(l\\ pa'>s lilta having lo w impedance section. of :!0 olum and high impedance .ection'> of 100 ohms.
TAB LE II
~
Act U,\1 CIRCUl i Dt \ IISSIO:>S
5
1
z:t;
<
::<
 S , m cu .. u rcd
l.inL' l.cn ~t lh (mkrun' )
S: 1• mras urrd ·······S • FDTI) 11 ..~·· S FDTI)
11 1JHH
II
•
20
=
1:' 7111
,..
I,='IJ() l.a: 722
. Jo +~r+ I0 20 30 J 0 FREQUF.:>CY (G IIz)
t, .• •N,
Fig. 12. Compari,on between FDTD model and mca, urcd rc.ult' for rcncction CS II ) and tmn> mi ,~i o n (52 1) coeffi cient> of a completely ; hicldcd '> hortend tuning \tub.
5 ~
was taken to incorporate the specific metall ization thi ck ness and the appropriate surface resisti vity which correspond to the vari ou s secti ons of microstrip line widths. To realize I 00 and 20 ohm impedance steps, 20 and 380 micron wide co nductor lines arc used w ith slot w idths of 2 10 and 30 microns. In the low impedance secti on. the line exci tes a coplanar waveguide mode due to the narrow slot wi dth while the high impedance section excites a microstrip mode. T hi s mi xed mode operati on produces parasitic inductances and capacitances that cannot be ca il y accounted for in the quasistati c model. Despite this lim itation however. at rel atively low operating frequencies the effects of such parasitics arc reduced and. as seen in Figs. 14 and 15. the mcasurcmems agree very we ll w ith the 2 2 theoretical data. The total system loss, (1  l$ 1 1 1  l5':z 1 1) . shown in Fig. 16 shows good agreement between theory and measured results. Thi s indicates that the circuit has negligible radiati on loss and thereby confirms the effecti veness o f the micropackagc. The level of loss can be attributed to the aspect ratios of the low and high impedance sections which arc know n to cause higher loss in both the coplanar waveg uide mode [231and the m icrostri p mode 1241. A dd itional validation of the response of a sel fpackaged filter, shown in Fig. 17, is provided through a comparison between a theoret ical model ba. cd on fin ite difference time domain (FDTD) and experimental measurements. Even though loss effects have been neglected in the model. excellent agreement is observed between the rc ults.
£. Open End Tuning Stube T he openended tuni ng stub [22] shown in Fig. 18 wi ll be u cd to illustrate the elec tromagnetic effects from a variety of micropackaging configurations. Performance curves wi ll be shown for the upper (US ). lower (L S), and completely shielded (CS) configurations. These results wi ll be compared to open CPW and a di scussion will be presented on the nature of the various effec ts on the circuit response.
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F R EQ UENC Y ((; Jill
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In the case of open CPW, loosely bound fields tend to leak power strongly and destructi vely into !>Ubstratc modes. When shielding occurs in either or both regions. the propagati ng modes become tightly bound to the l i ne and radiation i~ reduced signi ficantly compared to the case of the open CPW. The ci rcuit performance shown in Fig. 19 compares the open CPW to the CS geometry. The open CPW :,hows performance degradation above 25 G l lz due to the excitati on o f a strong substrate mode. while completely shielded circ uits exhibit a very smooth response even at higher frequencies due to reduction of parasiti c radiation and surface waves. When comparing the US to the L S configuration (Fig. 20) for the
DRAYTON A ND KATEHI: DEVELOPMENT OF SELFPACKAGED CIRCUITS
2079
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11::1:1: TR..\~SAC110i\S Oi\ .\IICROWt\\'E THEORY AND TECHNIQUES. VOL. H NO.9. SEPIUIBER 1995
20SI
Generalized TLM Algorithms with Controlled Stability Margin and Their Equivalence with FiniteDifference Formulations for Modified Grids Ma lgorzata CeluchM arcys iak and Wojc iech K. Gwarek. Senior Me111ber. IEEE
AbslractGc ncra lized TLl\1 formulations based on modified grids of 2D shunt nodes or· 3D expanded nodes arc proposed. Generalizatio n consists of permitting flex ible control of the numerica l stability margin (and thus a limestep for a par·ticular discretization ). and of introducin g enhanced models for cu rved bo undari es. Formal equivalence between generalized T Ll\ 1 and FDTD algol"ithms based on the same gl"ids is proved. Simple rules for tr·:msforming cir·cuil m odels (from TLl\1 to F DTD and vice versa ) and for their equivalent excitati on arc given. It is demonstrated that the application of the generalized algorithm red uces computer resources requi red for the T U\1 a nalysis of a circul ar wa\'Cguidc by a n o rder· of magnitud e.
I. INTRODUCTION HE T LM method is widely used for the modeling or microwave strucwrcs. In the classical T LM approach 10 hom*ogeneous problems. 1he propagation ve locily along each tran sm ission line ( 111) i s chosen to exceed the phys ical wave vcloci1y in 1hc modeled medi um (1•) by a fac1or o f r. with r = .,fl in a 2D mesh of shun1 node!. and r = :2 in a 3D me),h of expanded nodes. Thus. for a panicular discrctizarion of space a. 1hc simu lation timestep is stifn y sci 10 ~ I= oj 11• = o/(rl'). In th is regard. as noted by several authors 111. 121. more ncxi bi l iry exists in 1hc FDTD me1hod where the rime),tcp can be controlled by the user. The first ve rsions of TLM were based on a regular Cancsian grid wirh a crude staircase approximation of boundaries. Subsequc111ly. other gri ds have also been ulilizcd. and 1hesc can be divided into 1wo categories:
T
I) rcc1ilinear onhogonal gri(b w ith cell dimensions va riable as ~.r = ~.r(.r) . ~ ll = ~ y ( y) . ~ ::; = ~ ,:(.::) : 10 1his category we assign 1hc early technique by Johns 131. 1hc variou s variable mesh techniques Ill. 141 and the recursive algorithm 151: 2) regu lar grid), bul in nonCancsian coordinate system),: thi s category includes curvi l inear onhogonal (e.g .. radial) grid), Ill. 171 and triangular grid s 181. In comparison w ith the regular Cancsian gridding. the grids of Caregories I and 2 improve the TLM modeling of many microwave circui l s. bur they abo main1ain some limitations. In panicular. 1he grids of Category I do not provide the tangen 1ial ~ l anu;,cript r~cciv~d Ocwb~r S. ~~~1: r~vi>cd M:oy 2~. 19~~ Thi> wor~ i> 'upponcd in pan hy th~ Commi>•ion of the Europcan ConuiHmi ti ~' Contract Copcrnicu,: 7565. Th~ author' arc with the ln .,titur~ of Radioclcctronic>. \\'ar,aw Univcr,ity of Techno log). 101\0\\ i~j.,ka 15119. 00665 War\a\\. Poland. IEEE Log No. 91 131 I 0.
and normal field components for the rigorous speci fica1ion of boundary condi ri on s a1 oblique boundaries. The grids of Category 2 arc perfectl y suited for a narrow class of circuib with all boundari es described by a constant function in one coord i na1e system. Thus in prac1ical microwave problem), or compl icated geometry. it o f"ten becomes necessary 10 rel ine the overall discreti zation which leads to excessive requirements of co mpu1cr resources. Our work addresses 1hcse limilati ons of cla),sical TU\1. and proposes ori gi nal techniques for: I ) controlling factor r introduced abovewhich has a mathematical i nterpretation o f controlling a ),lability margin and a phys ical interprclation of controlling 1he simulati on time),lcp. 2) improving l he representation of arbi1rary curved boundaries through the introduction or locally modified grids. In the MTTS paper 19 1. we approached the probk:m of the conl rolled stabi lity margin by developing the complcle dispersion relations for the stubloaded TL~I. wh ich required cumbersome ari1hmc1ic. The new boundary model wa!'. esrablished by incorporaring local in1cgral approximation), in10 TLM 191. A unif·ied approach to both problems presented herein is substan tially more general and ef'f'ecti\'e. We shall first postulate 1hc exi),tencc o f the generali1.ed T LM algorithms based on modified grids. We shall conduct a formal proof of their equivalence wit h FDTD al gori thm s (based on 1hc same grids). and deri ve simple rule!. for tran sformarion be tween the FDTD and TLM modeb. These rules enable immediate formul ation of the TUvl algorithm), wilh controlled ),labil ity marg in and w ith flexible representat ion of curved boundaries. original ly proposed for FDTD 1101.1111. In deri vations we shal l concentrate on 2D problems. Conclusions wi ll be directly extended 10 the 3 D expanded node (Exp ) which is a combination of the 2D shunt and series nodes. In the fi nal example o f a circular waveguide. we relate our generalized TLM formulations based on modified grids to classical FDTD and TLM. and to nonorthogonal FDTD of 1121. To our knowledge. thi s is the first in the lirerature comparison of accuracy o f the locally and globally irregular grids for the timedomain modeling.
II. GENERALIZED T LM A I.GOR ITII I\ IS Consider an arbitrarily shaped 2D circu i1 of Fig. I (a). Its boundaries can be shorl. open. or re),i),tiYe (which provide'>
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I
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Jr.l
i l ·~·+l 1' 1'"' I.\'  I .\1
(3)
,.1k
(f )
(a)
i ~ ~·+1
I
.\/

I
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11
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·'

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for open (p). short (.~) . matched (tn) and res istively terminated .Ar. respect ively. In the classical T LM modeli ng. li nk branches arc parall el to the coordinate axes. Their different characteri stic admittances resul t from inhom*ogeneous permeabil ity or variable dimensions of cells Ill. Slllbs have the following mean ing:
(r) slllbs at node
I (c)
Fig. I. (a) An arbitrarily '> haped 2D microwave circuit: (b). it' TLt\ 1 modd propo,ed herein. (c) and :111 equivalent FDTD model.
in panicular a narrow band model of a matched lerminati on or a lossy metal surface 161). Upon thi s circuit we impose a fundamen tally ort hogonal grid wilh possible usc of va ri able mesh Jechniqucs. and we match lhe curved boundary by mean s of modi fi cd nonrectangular cells 1Fig. I (a) 1. General izing 1he classical TLM approach and our rccen1 comriblllion 191. we claim that the circuil of Fig. l (a) can be modeled by a matrix of tran smi ssion l i nes shown in Fig. I (b). In the matrix we distinguish two types of lines (branches): link l i nes and stubs. Consider node JV w ith total number .\"b of incident branches of any types and any characleristic admittances b } ·.\'. A pul se i;I ·.t scattered into branch /J at time instant J.. ~t is calculated as
,.
~<
_
~·
,, 1 . \ '  "·"
"; I
·k .\'
b = 1.·· · .NtJ
( I)
• open stubs 11 model inhom*ogeneous permi tti vity or magnetic boundary. • short slllbs s model electric boundaries (i n 30. short stu bs at series nodes also model i nhom*ogeneous permeabili ty). • matched stubs 111 model dielec tri c losses. • res isti vely terminated stubs r model resisti ve boundaries or an actual lumped element incorporated illlo the circuit. Our general izati on consists in allowi ng that: I ) l ink branches can be placed obliquely in the matri x. in order to model the now of the current tan gential to the open boundary 1 1}' in Fig. l(b) l. 2) doublenode cells can be used to represen t the curved electric boundary (T in Fig. I ). 3) additional open stu bs (in 3 0. also short stubs) can be used to control a stabili ty margin and th us a timestep o f the algorith m. These ex tensions to classica l TLM w ill be explained in the following sections. via the equi valence w ith the FOTO formulations.
CELUCII  ~ I ARCYS I AK
AND GWAREK: GE ERALIZED TL~ I ALGORITmiS WITH CONTROLLED STABILITY
and conductance Gs such that
Ill. EQU IVALENCE OF TLM AND FDTD FORMULATIONS FOR M ODIFIED GRIDS
.Y,.,
Equations ( I )(8) applied to a model of Fig. I (b) wi th chosen initial conditions descri be the TLM simulation of fi elds in the circuit of Fig. I (a). We will prove that a formally equivalent simulation can be conducted in the FDTD method. By formal equivalence. we mean th at with tt.c usc of infinitely precise computer ari thmetic. both methods would provide identical field values at any time instant and at any location. In thi s work. we shall not compare the numerical robu stness of FDT D and TLM with respect to the computer roundoff errors. Thi s topic has been bricny add ressed in 1131. and it is worth further systematic research. However. our experience shows that in typical microwave problems the two methods arc imm une to computer roundoff CJTOrs, even if single precision ari thmeti c is u. cd 11 4 1. Therefore. their formal equi valence is a concept of practical as well as cognitive importance. Criteri a for formal equi valence of the TLM and FDTD methods concern two aspects: equi valent transformation of the model and equivalent tran sformation of the ini tial conditions. We shall discuss the two aspects separately. In the discussion we ~ha ll refer to the FDTD notation introduced in 1101. Thi s notation uses auxiliary variables: voltage and currents. Voltage is defi ned as II
= ./ /:;= r/;; =
~ l i\RG I N
f~'=fl
(9)
where /1 is the height of a planar ci rcuit. Surface curren t density is related to the transverse flfield by ( I 0)
7
Current !lowing between two nodes is the integral of .L along the common side of respecti ve two cells. for example lc.L , Fig. I (c)l
:z= ,, ) "_y.
cs =
( 13)
111 = 1
2) Each l ink branch of characteri stic adm i ttance replaced by inductance ,L such that ~I
.
 = , ).
1) •
is
(I~)
/ f.Al
3) Open branches arc neglec ted (thei r innucncc is incl uded in ( 12)) . 4) Each branch ,.} · ended with resistance I? or shortended (U = 0) is replaced by i nductance ,.L ~I
,. La
2,. ) ·
= 1 + ,.)· R
( 15)
whi le at the resistive end we pl ace an auxi l iary node,\ '' where at the nodal voltage us• is explici tl y defi ned by k
ll_yo
·k0;:; = j 1>1./_y '
( 16)
and ,.is denotes cu rrent nowi ng ou t of H. t oward~.\ ·. Formal proof of the equi valence of th is transformat ion is gi ven in the Appendi x. Let us now ex tend the discussion to the 3D ExpN modeli ng. At shunt nodes criteria ( 12). ( 13). ( 15), and ( 16) (and the associated steps I. 3 of the proof) remain valid. A t series nodes, dual reasoning can be applied. ormally. incident at series node ..VI arc i\1 1 l ink branches and .\ /, short stubs. In the FDTD model, at node "VI we locate nodal inductance L .\1 defined as ~I
L,\1 a
=
( II )
2
( 17)
~~~
"''\{' ".\I z L.../= 1' z 'j.\1 + L....=1 ·' .\1
and we neglect short branches. T he FDTD model i s a network of nodal capacitances and branch inductances 11 01. The effects of inhom*ogeneous filling and variable dimensions of cells arc described by modi fied values of C's and 1,L. The value of a in the FDTD equations denotes a fundamental cell size in the model (which is not necessari ly the smallest si ze).
A. £quil'(f/el1/ Tramformation of the Model Equivalent transformati on of the T L M model of Fig. I (b) into the FDTD model of Fig. I (c) is governed by the follow ing criteria I ) A t each node ;V with incident Nt link branches l, NI' open branches fl, N,, short branches 8, N ,. resi sti vely loaded branches rand N,, matched branches 111 we place capaci tance c.\' such that
2
 "'"'' L..1==11 }'.Y + "'"''' Lp== II' )'1\' +"'.\" L... .=1·' ,1 .S + "'.\', Lo= l'' )' 1\'
( I 2)
B. Equil'tlle/11 Modeling of Initial Conditions In the FDTD method. initial conditions must be given for nodal voltages at k 0 and branch cu rrents at k O.:'i. In TLM. initial conditions arc completely defi ned by the values of all incident pul ses in the model at a single ti me i nstant k = 0. Ph ysically . this corresponds to enforcing the values of both voltage and current in the middle of each branch at k = O ..J. T hus at fi rst sight i t seems that FDTD and T L M cannot be excited in the same way. and th is poin t has been raised in previous comparisons of the two methods 1151. We will now resol ve such doubts by showi ng that any excitation i n T L M has an equivalent in FDTD. By '"equivalen t exc itat ion'" we understand such a defi ni tion of i nitial voltages and currents in FDTD. and of i nitial incident pulses in TLM which resu lt in identical field val ues obtai ned wit h the two methods at any location and at any time instant k > 0. We shall refer to the initial TLM excitation from one branch by a single delta pulse. remembering that any other excitation waveform consists of a train or pulses. and our concl usions w ill apply by vi rtue of superposition.
=
=
2084
IEEE TRANSACTIONS ON
TIIEORY AND TECHNIQUES. VOL. 4J. :iO. 9.
SEPTE~ IBER
1995
L et us consider what happens if both TLM and FDTD arc 0. We denote by CTu.l (I) excited by a single pul se at k and UroTo(/) signals detected at a particular location in the circuit ( for example voltage at node.\'). In view or the above discussion. UTL:.1{1) L'Foro (/) + L'ro,D (/  ~/ ) and the foll owing relation between their effect i ve spec tra holds
/(
1'=0
1\r
~ II CROWAVE
=
=
(a)
IF[U,ur(t)]l = IF[Cro,o(/)
+ L'rnro (/ 
~/ )]1
= IF[Cro,o(/)]1 o I L + exp ( j2 ll f ~ t )l = IF[CrnrD(t)]l o lcos ( ll f ~ t )l .
(b)
(c)
Fig. 2. Exci1a1ion schemes for TLM and FDTD applied ro rhc cir cuil of Fig. I. (a) TLM cxci rari on of node . \ · from currem source: J.r(t) 2h{l O..:i..:J1), (b) FDTD excitation equi valent to (a): lr(l) 211>(1 Q. ;)..:J.t) b(t  O..)..:J1)]. (c) FDTD excitation of node • \ • from branch ,, L. equival ent to TLM excitation by ~. I :~ = I :
=
+
=
E(t) = t.( t
+
+
+ ..:J.t).lt (I)=
b(t
+ O.::i..:J.t).
TABL E I 'ODAI VoiTAGI s ,:;D BRA:;cll CtJRRE:o.Ts OnT,\1;IEO FOR TilE C1Rcurr 01 F1G. I DY DOTII TLI\1 '"D FDTD. AS,, RESULT OF ExcrrA'no;o~s 0 1· FIG. 2(a) A;ID FIG. 2(b}, RI:SI'I ("l'JVI:U. CO~SIDERED 1\KE 1\I ODELS OF FIG. I (b) A::D FIG. I (c) WI rH
b)"= I.
d!..l
~
=I. r · ) \· =r·)·\1 =r·)Q =0. ",. ' =
""
. •
k
0.
"•
I
I. 00
0 .00
0.00
0.00
o.so
o.so
zs
0. zs
s
I
2
u
s
0 0.
u
tl
s I.
so
0.
5
J
•
·"
ft
.•
1 ,, .
~
= t1
1
·' '
=~ ..
I
0.00
0 .00
I. 00
z.oo
0. so
z. oo
From thi s foll ows an interesting conclusion that a single pulse excitati on in TLM produces signals or cosinusoidally modulated spectra. wi th respect to signal s produced in FDTD by the same single pulse source. In some circuits strong resonances can be grouped above the frequency band of interest. at f requencies approaching the yquist point fs = 1 /( 2~/ ) . With all excited signals having modulated spectra of the form ( 18). the effect of numerical "ringing" at these resonances is diminished. and consequently the convergence of calcul ati ons improves [1 4]. Thi s may explai n bcllcr convergence of TL~I reported in examples usi ng a si ngle pulse exci tati on in both TLM and FDTD ] 15 ]. ] 16 ]. Naturally. equivalen t excitation models as specified abo ve provide identical convergence of FDTD and TLM. A n alternative TLM excitation is by a pu lse incident from one of the link lines, for example from line;:;}~\' at one node N of Fig. I. An FDTD equ ivalen t agai n comprises a twopul se exci tati on. applied to branch cu rrent at /, 0.5 and to nodal voltage at 1.: = 0 [Fig. 2(c) ].
=
C. Consequences of Formal Equi\'{/lence In view of formal equ ivalence. the TLM and FDTD methods exhibit iden ti cal properties in terms of stability. energy conservation. and ncx ibi lity in the model ing of irregular geometri es. T herefore by direct transformation from FDTD. techniques for controlling the stability margin and improving the curved boundary representat ion will be incorporated into TLM .
D. A Spurious TLM Mode and Irs Effect on Formal Equivalence 1rirh FDTD
First, we consider the excitati on of node ;V of Fig. I from the ideal current source connected by an additional stub •·xc y· = 0 !Fig. 2(a) ]. A ll olher incident pul cs at k = 0 arc ct to zero. Clearl y. the excitation stub does not innucncc further analysis. The values of selected voltages and currents in the circuit of Fig. I. as produced by TLM at the beginning of the simulation. arc gi ven in Table I. The same sequence of nodal voltages and branch currents (Table I) is produced by FDTD if the exc itation scheme of Fig. 2(b) is incorporated. Thi s means that for the equivalence w ith TLM, in the FDTD model i ng the same delta pu lse excitation must be applied to node N at k 0 and repeated at k, J.
=
(18)
=
A s a con equence of using more variables. the TLM method has more eigenvalues than FDTD. and it emulates spu ri ous modes which do not ex ist i n FDTD. In a previous paper 117] we have shown that the T LM spurious modes arc described by the disper ion term sin (21 If ~I ) 0. thus they arc supported at f = 1 /(2 ~ /). However. the TLM spurious modes do not violate formal equivalence with FDTD. They arc characterized by zero nodal voltages and zero branch currents which arc our variables of interest. Furthermore. they arc nonpropagating modes and cause the osci llations of branch voltages only where speci fi cally excited. We have established these properties by examining the complete T LM characteristic equation gi ven in 191.
=
CELl:Cif ~IARCYSIAK
AND GWAREK: GENERALIZED
TL~ I ALGOR ITII ~ IS
IV . CONTROLLED STABILITY M i\RGI
\VITI! CO, TROLLED STABILITY ,\ IARGIN
IN TLM
Ti\BLE II
Consider a hom*ogeneous region o f space covered by a Cartesian grid of cell size o. We construct its 2D TLM model with l i nk branches by and open stubs c ) ' . In a 3D TLM model we additionally adm it short stubs 1. Z at series nodes. All parameters/, } ·. c ) ' . 1. Z as well as L. (' in FDT D arc constant throughout the model. For the FDTD solutions. the numerical di:.pcrsion relati ons have been investi gated by several authors 121. l l·l!. 11 8 1. A fundamental parameter of these rel ati ons is a stability factor r defined as ( 19) The FDTD calcu lations arc stable 121, l l ..fl, 11 8 1. and 11 91 and numeri cal energy is conserved 1191. 1201 in 2D if r 2: ..fi. and in 3D if r 2: vf:J. By a stability margin of the algori thm we shall understand th e interva l ( ../2. r) in 2 D and ( vf:J. r ) in 3D. ote that ( 19) is consistent w ith the definition of factor r proposed for TLM in the introduction since (I
r
=
~~ II
= !.!!. .
(20)
II
Applying equi valence criteria ( 12). ( 14) we can express a stability factor in term s of characteri stic admi ttan ces o f the TLM branches in 2D r2
= ·lb) . + c ). _I_ = ~ [·I + ~]. 2
t, }·
2
(2 1a)
tY
In 3D. we apply cri teria ( 12). ( 17)
I
_2 _ 
·lb}.+ c }· ·l +t, )· t.Z _1[  ,, ' ' 2 2/J ) •
c)·][ + )' Z]
+ /i ) •
·I
II
I' .
=
I ) We confirm that classical TLM wi thout stubs (c ) · 0. Lz 0) i s operated in 2D at r .Ji and in 3D at r 2. 2) In 2D and 3D TLM. it is possi ble to increase a stability margin through the usc of positive stubs. Thereby we obtai n a family r+ of TLM algorithm s. discussed in the following secti on. 3) Using negati ve stubs. we can formulate a family rof stabl e 3D TLM algorithms w ith decreased stabilit y margin ( J3 ::=; r < 2).
=
=
ERRORS 1:'\ CALCULA1l:'\G ti Gt:\I·REQl:E:\CIES 01
IIX ll x20
~~~~
mode
CWIIY 111111 3D TLM (11 =2 f
ph
IC!!zl
r • yJl
r • 2
or IY1
cr IY1
10 1
13.078
0. 2 1
 0.31
110
15 152
o
24
o. 38
Ill
16.907
0. 04
 0.21
\ ~~~I)
• numerical dispersion errors decrease as in the FDTD meth od 12 1, 11 41. and 11 81. • simulati on can be conducted wi th the timestep prolonged by a factor of 2/ J3. thus the total number o f iterations is reduced by up to 13){ . A li mitation o f the r  algori thms i s that they permi t the modcling of electri c wall s only through the planes where tangential electri c fields arc defi ned. and o f magnetic walls through the planes w here tangential magneti c fie lds arc defined. T hi s limitation follows from the range o f clements admitted in constructing the TLM model. Thus the r  algorithm s are suitable for circuits of fa irly regu lar geometry. In Table II we show eigcn frequencies o f a l..f x l..f x 20 mm cavity ca lcu lated by standard TLM (r 2) and ,. TLM ( r = J3). For both algorithms we have used the same cell size o 2 mm and the same scheme for init ial exci tati on and parameter extracti on, as introduced in 12 11. T hi s scheme i s based on incorporating an auxi liary lossy vol tage source and detec ting the cigenfrcquencics as minima o f the discrete Fourier tran sform of the input current 12 11. otc that the version w ith r = J3 remains stable despite the usc of negative stu bs ( 1.)" = I. ~.z = 0. c ) ·=  1). and i t provides more accurate results (Table II ).
=
=
V. IMPROVED TLM M ODELING OF CURVED BOUND1\RIES
•
(2 1b) Expressions for r identical to (2 1) have been originally obtained in our MTTS paper 191 by cumbersome trigonometric transformat ions of the T LM dispersion relati ons for the 3 D model wi th stubs. In previous papers. only the 2D stubloaded T LM model has been considered 121. and conclusions concern ing the possibility to control the timestep have not been drawn. Based on (2 1). we draw the foll owing conclusions:
=
20R5
Let LI S first consider the TLM algorithm s or .family r  . Compared w ith classical 3D Exp TLM , they ex hibit two advantages:
A lgorithms of .family r+ enforce an ex tra stability margin into the TLM method. T hi s margin can serve to accommodate improved models of arbitraril y shaped boundaries. In the 2D FDTD method. a very ncx iblc model a. in Fig. I (c) has been developed for open I I 01 and short 1111 bou ndaries. Parameters L. C of this model arc obtained by assuming local quasi static field di stribut ion over each ce ll. Consequently. L. C have the ph ysical meaning of quasistati c inductances between nodes. and quasistati c capaci tances of cells I I 0 1. App ly ing the equivalence criteria. we can immediately transform thi s FDTD model into the TLM model of Fig. I (b). Let us assume that in regions covered by a regu lar square gri d. standard values of 1, L and C,\1 in FDTD arc chosen so that ~I
~I
(22)
(23)
where r is the desired stability factor: re lations (22). (23) satisfy ( 19).
IEEE T RANSACT IONS ON
2086
~ II CROWAVE
Applying the branch equi valence criteri on ( 14). we obtai n characteristic admitt ance of standard link lines as (24) Between di storted cel ls. quasistatic inductance i s modi fied which results in modi fi ed branch admillance. for example (25)
~I a2L
(26)
2
} '.  a .
d
A unique feature of our model is th at i t contains obl ique branches which allow for the fl ow o f a current tangenti al to the open boundary, such as ~ L . .1V
61
('
o.1L
a J2 ·
. =~ Y = 
phys i cal
mode :
H
11
f
values
..
STAIRCASE BOUNDARY, r
V21
c
NEll BOUNDARY HODEL, r • 2
ar
or lXI R/a ,. 5
R/a • 10
0. 0879
 2. 39

0. 1458
1 3. 45
IGI!zl :
I 02

(X)
R/1.1 • 5
R/a • 10
0.00
0.11
o. 00
0.00
I. 73
 I. 0 4
 0. 29
 0. 08
R/a
==
20
R/a • 20
H
0 .1829
.
 4 . 33 1.11
0. 5 7
 0. 7 4
 0. 20
 0 . 09
H
o. 2005
12 . 27
 2 88
 o. 59
 I. 24
 0. 39
 0. 14
01
J1
T hus. the slllb accounts i n a multiplicative way for two physical factors: I) increased permi tti vity o f the modeled medium. 2) decreased simulation ti mestep ~ I (increased r for the fi xed medium and spatial di scretizati on a).
B. Pmcrical Advanrages of rhe Nell' Bo1111dary Model (27)
In thi s way. all l ink branches and stubs perpendicular to the boundaries i n Fig. I (b) are defined. Finally, the nodal equivalence cri terion ( 12) is applied to define permilli vity/stabil ity stu bs. For regular cell ) vt we have ~I
1995
TABLE Il l
21
b   = ] } =(IJL a .
SEPTE~ IB ER
E RRORS 1:: CAI.CULATI::G C L'TOFF FREQLE::CJES OF TE· M ODES 1:: ,\ C!RCLLAR \V,WEGUIDE OF R = I ~I BY Til E 2D TLi\ 1 M ETHOD
H
~I
T IIEORY AND TECIINIQUES. VOL. 43. :10. 9.
We calculate cutoff frequencies o f modes i n a circular wa veguide of radius R , usi ng succes i vc ly refined discreti zation a. For the TM modes, this is a 2D problem wi th short boundary condit ions: for the T E modes a dual 2D problem with open boundary cond itions. We defi ne an error of the analysis as (33)
2
aC.\1
f""
Taking i nto account (23). (24) (29) A t other nodes we obtain for example (30) where S r denotes ce ll area norm al izcd to standard cell area a2 . The va lue of r = 2 is always sufficient to avoid negati ve stubs and to mainta in stability. A. Re111arks on l nho111ogeneo1ts Proble111s
For clarity we have so far concentrated on hom*ogeneous ci rcuits. but our study also appl ies to i nhom*ogeneous problems. As explained in 110 ]. the FDTD val ues of nodal capac itances and branch inductances increase proport ionally to c ,. and ft ,.. respecti ve ly. T hen the equiva lence cri teri a ( 12)( 17) defi ne modi fied va lues o f branch admillanccs in TLM . For example. if rel ati ve permeabi lity of the region between nodes P and S in Fig. I (a) is increased by a factor o f 11 1 • then 1 } · is gi ven by (3 1) instead of (25)
~I=~ } · =_!:__~ . ad, 1'1 a
(3 1)
Si milarl y. i f relat ive perm i11i vi ty o f the medi um fi ll ing ce ll P is increased by a factor of cJ>, then stub c ) ·P i s defined by (32) instead of (30) c ) ·p
= 2r 2 8f'Cf' [2
+ .1 } ' + 1 }'] .
(32)
where f. arc the calculated and physical val ues of frequency. As i n Sec tion I V, the exci tation scheme after 12 11 combined with the discrete Fourier transform is used. First. we consider the T E modes by: I ) standard 2 D TLM with 1' = J2 and a stai rcase boundary model. 2 and the new boundary model. 2) TLM with r
=
Results arc shown in Table II I. For comparable accuracy. sta ndard TLM requi res rd incmcnt of the discreti zation by a factor of 24. and thi s leads to the i ncrease o f computer memory and time requirements by factors of +16 and 86L respecti ve ly . On the other hand. our curved boundary model requires only slight modi fication of the standard T L M code. and i mposes prac tically no additional requirements on compu ter resources . The effect of i ncreased numerical dispersion due to increased r becomes neg ligible. i n compari son wi th the improved boundary representation. Further. for the TM 01 mode (fph = 0. 11+9 GHz for !? = I m) we conduct a systematic convergence study comparing the fo llowi ng approaches: I ) cl assica l T L M or FDT D based on a regul ar Cartesian gri d w ith staircase approx imat ion of the boundary defi ned only at nodes. 2) cl assica l TLM based on a regular Cartesian grid wi th sta ircase approxim ation of the boundary. wi th boundary defi ned at nodes or bisecting branches. 3) our TLM (FDTD) formu lation based on a locally modi fied grid. 4) nonorthogonal FDTD 1121 based on a globall y irregular grid.
CELL:CII,\ IARCYSIAK AND GWAREK: GENERALIZED
TL~ I i\LGOR ITII~ I S
n ,, ,,.iT~r.;.l::._~_;2
. . .. •
.~    o   : /
~ l i\RG i i\'
lOX7
results by dots in Fig. 3. A straight line repeated for da~sical FDTD after 11 21 represents average errors. Rc ~u lts become unambiguous for refined discret izat ion and i mproved boundary models.
H

. /~
 . ..   
.l
WITII CONTROLLED STAB ILITY
./;_
,· ··· .··
Y I. CONCLUSION
6/ • I
7
We have proved forma l equivalence between the gencrali~:cd TLM (for shunt or expanded node) and FDTD formulations based on modified grids. Si mple rules for transforming ci rcu it models (from TLM to FDTD and vice versa) and for thei r equ i valent excitation have been given . It must be emphasized that these rul es clari fy many previous misunderstand ings con cerning relationships between the FDTD and TLM methods. and their applicabil ity ex tends far beyond spec ific topics of thi s paper. Formal equi valence with FDTD reveals the pos~ibility to control a stabil ity margi n in the TLM algorithms. Algori thms of .{a111il_r ,.  with the decreased stability margin reduce the numerical dispersion errors and compu ti ng ti me for regularl y shaped 3D microwave circui ts. Algori thms of .fa111il_r ,.+ enable stable incorporat ion of enhanced m od e l ~ for curved boundaries. One such a model ha~ been transformed from the general ized FDTD method into TLM. According to the authors· experience. the T LM algorithms with the new boundary model (as well as the equi va lent FDTD al gorithm s based on the approach o f II 01. I I I I) arc competi ti ve tools for the analy si s of i rregu larly shaped m icrowa ve
I
8 L    '    L    '     '    '    '  l. 1.2 1.0 ·0.8 0.6 0..& 0.2 In ~
1
~ our TLM or F DT/J on modified grids.  0 nottortho~:onul FDTD afur f I 1J.
 · + ·
classica/ TL.\1, boundary at uodts or bisrrting brau ch,s.
 · .. · classical TL\1 o r FDTD, bomulary dtfintd at node'f.
Fig. 3. Error' in calcul;uing cutoff frequen cy of 1hc Tl\ 101 circular \\aveguide mode. a' a func1ion of di'Crcl*talio n.
Errors due to approaches ( I )(4 ) arc plotted i n Fig. 3. Application o f the locally modi fi ed grids discussed in thi s paper leads to the most accurate FDTD and T LM numerica l schemes. In comparison wi th nonorthogonal FDT D. these schemes require approxi mately hal f of computer memory and one third o f computi ng time (for the same number o f cells in the model). In case I which invo lves crude boundary approx imation. results strongly depend on the positi oning of the model w ith respec t to basic grid lines. and we have marked various
2
l:i~ ~~ > s + L:;~~ ~ ~, > s + I:~~ ~Xs + I: ;~ t,. > "x + I:;~;·~, til >\ .
[~ > ,i v~:t t + ~ y _; 11 ~:+' L_) .\ L._., /1 .\ /1 ·' I
.\
p= l
1=1
+ L~}"' ·' .\ ·' \ ·HI _\" + ~ L._.,r )"' .\ ,. \·Htl _\" ~=1
1.II_\"
2
= L::i~:,,1)_,.+L: ;'""'~.,/1},\"+_I: _~,..,~~·)_,.+ 2:: ;';;.~.1 ..)_,_+_I: _;'""'~;·;:,1_11_1)__\" ·
[~ ~ / )  ·'T~ + ~ L._.,l .) .\ T~: I .\ + L._., L ·' ) .\ ·'T~: ·' .\ 1
1=1
+~ L._.,
.\ I '
.\
·'='
t•= l
I"
,. , .~. + ~ L._.,
) " _\"I"
r· = l
1 // ~·+ _\"
(A I)
r·= l
_\"
) " Ill
I"
_\"Ill
,1. ] _\"
(A2)
•
rn = l
= +~~ Y,~ ~,~,.~ 7.'"} _~~s~ . }~~,~ ... )L...t= rl > .\"
·
+ Lt•= ll' ) .\" + L.... = 1·'
[ ~ ~·+0.::. + L~ ·' L._., l' _y l= l
.•= 1
·k+O.::;
I _y
.\"
+ Lr·= l'"
+~ / +0 ..)] L._., r _y · r= l
.\"
+ L..",1"' \"
(A3)
IEEE TRANSACTIONS ON W CROWAVE TIIEORY AND TECHNIQUES. VOL. 13. NO. 9.
20SK
~ tru c w rcs . For a ci rcular waveguide considered in thi s paper. accurate rcsulls have been obtained w ith only about 1O. "Equ ivalence of' propagal ion characleri;.~ i c> for 1he lransmi,sionlinc malri.\ and linilediiTcn:ncc li medomain nlclhotb in 1wo d imcn,ion>." IHI;E 7iwll. MitrOI\'fll'l' 71wory 7i•cl!.. vol. 1\IlT39. no. 2. pp. 354357. Feb. 199 1. 131 P. 13. Johns and G. F. Staler. "Tran;.icm anal)'i' of' waveguides wilh curved boundarie;,," El in ~D and 3D T LI\1 'imulalion' rcalitcd by rccur'i'c formula.,," IEEE Tram. Micmll'tll'e Theory Tech .. \OJ. 10. no. 12. pp. ~2672~71. Dec. 199~. 161 \V. J. R. Hoefer. "The lran,mi"ion line malri' mc1hod1hcor~ and applica1ions." I EEE 7iam. Micrmmn• '1111•ory Tech . . ,·ol. 1\ITI33. no. 10. pp. 88~892. Oc1. 1985. 171 II . 1\le liani. D. De Cogan. and P. B. John,, "The u'c of' onhogonal curvili near meshes in TLI\1 modcb." 1111. J. Numaiml Modl'lli11~: Electron. Nenmrks. 0ial.. and W . K. Gwarck. " cw TLi\1 algorilhms "ilh conirolled >Jabilily margin and Iheir applic:11ion 10 improve Ihe modelling of curved boundarie;:· in IEEE M7TS Dig .. San Diego. 199~. 11 01 W. K. Gwan:k. "Analy>i> of an arbilrarilyshape d planar c ircu it a timedomain approach." 1/:'EE Tr1111.1'. Micrm•·m·e '11wory Ted1 .. vol. MTf33. no. 10. pp. 10671072. Oc1. 1985. 1111 W. K. Gwarek and C. MrocLkowski. "Effccli ve analysis of IWO· dirne n, io nal waveguide di,conlinuities." in 1111. Conf Micmu·m·es and Optoe/ectmnic.~ MIOP. Wiesbaden. Germany. 1990. pp. 550 555. I I ~I P. H. Harm,, J.F. Lee. and R. i\l ima. "A >tudy of 1h.: nonorlhogonal FDTD method vcr'u' 1he corwenlional FDTD lcchnique fur computing re,onalll frcqucncie; of cylindrical cavi Ji e~:· IEEE Tm11.1. Micmu·m·e Theory Tech .. \'OI. ~0. no. ~. pp. 7~ 1 7~6. Apr. 1992. 1131 A. Feu wei> and G. il>che. "Numerical imegr:11ion of pani:rl dil'f'crcmial cqu:llion> u; ing princ iple> of rnuhidimcnsional wave digi1:1l fillers:· J. 1'/..S I Signal Pmce.uing. vol. 3. pp. 72~. 1991. 11~1 W. K. Gwan:k. "Analy;is of arbitrarilyshaped two dimensional microwa\e circuits by finilc diffe re nce limedomain mclhod." IEEE Tmns. MiclrJu·m ·e Th eory Tech .. vol. 36. no. ~. pp. 7387~~ Apr. 1988. 1151 i\1. Celuchi\larcy;iak. W . K. Gwarck. Z. Chen. i\1. i\1. Ney. and W. J. R. Hoefer. "Cornmcrm on 'A new finitedifference limedomain formulalion and ils equivalence wilh lhe TLM ;ymrnelrical conde nsed node' and reply to comments:· IEIC:E Tmns. Micmu·m·e '111em:r Tech.. \'O I. ~I. no. I. pp. 168 17 1. Jan. 1993. 1161 Z. Chen. i\1. M. Ncy. and W. J. R. Hoefer. "A new finilcditfercncc 1im.:domain fo nnul:llion and its equivalence wit h 1he TLM symme1rical condensed node." IEEE 7i·ans. Micrm•·m·e Them:\' Tech .. vol. 39. no. 12. pp. 21602 169. Dec. 1991. 1171 i\ 1. Celuc hi\larcy>ial.. and W. K. Gwarck. "On 1111: a'pccl> of sclccling the 'ymmelrical conden,ed node or 1he expanded node modelling for d eclromagnclic »irnul:llion,," E11r. Micmu·m ·e Conf Dig .. lle ls inki.
J9n. 1181 I. S. Kim and W. J. R. lioefcr. "Numerical dispcr;ion c haraclerislics :md ' labilily factor for 1he TDFD mclhod:· Electron. /..ell .. vol. 26. no. 7. pp. ~85487. Mar. 1990. 1191 D. Puller. Comp111111ionall'hyxics. ch. 6. New Yo rk: Wiley. 1973. 1 ~01 I. S. Kim and W. J. R. llocfer. "The numerical e nergy conservalion of lhc TDfD method." IEEE Tmns. Magnetics. vol. 27. no. 5. pp. ~0564059 . Sept. 1991. I ~ II \V. G" arek and i\1. Cl!luchi\htrcy;iak. "limedomain analysi> o f di~pcr 'in: lran. a texlbook on cleclromagrH!Iic theory and a sot'twar..: package named QuickWave (dis1ribu1ed by Argurvlcns Gmbl l. Duisburg. Germany). II i> current research interests arc in Ihe area; of e leclromagnclic field lh.:ory and compuh:raided analysis and design of m icrowave circuits. Dr. Gward. is a Me mber of Edilorial Board of IEEE T R\:>S \Clto.:s 0'> i\ IICRO\\'AVE TIIEORY A:"D TI:CII:"IQL'l:.S and a i\ !ember o f Ihe Technical Programme Cnrnmillee of lhc European i\ t icrowa,·c Conl\:renc..:.
IEEE TRANSACTIONS ON ~IICROWAVE TIIEORY AND TECHNIQUES. VOL. 43. ~0. 9. SFPTE:\IBER 1995
Examination, Clarification, and Simplification of Modal Decoupling Method for Multiconductor Transmission Lines GuangTsai Lei. GuangWen (George) Pan. Senior Member, IEEE, and Barry K. Gilbert, Senior Member. IEEE
AbstractIn the application of the modal dccoupl ing method , questions a rise as to why the nonnorma l matrices LC and CL arc diagona lizabl e. Is the definiti on of the characteristic impedance matrix z, uniq ue? Is it possible to normalize current a nd voltage eigenvectors s imultan eously, yet assure the correct constr ucti on of the Z , matrix'! Under wha t conditions do M :M ,. I a nd Z , M ,. M ; 1 ? In th is paper , th ese questions arc thoroughly addressed. We will prove the diagonalizability of matrices LC a nd C L fo r lossless transmission lin es (though the diagonalizabi lity of their complex a nalogues, ZY a nd YZ matrices, is not guaranteed for lossy lines), and will demonstrate th e properti es of their eigenvalu es. We have developed a n a lgorit hm to decouple one type of matrix dill'erential equation, a nd to construct t he cha racter istic impeda nce ma trix z,. explicitly and effi ciently. Based on this work , the congruence and similarity transformat ions, which have caused considerable confu ion a nd not a few errors in the deco upling a nd solution of the mat rix telegrapher 's equa tions, will be analyzed and summarized. In additi on, we will a lso demonstrate that under certa in co nditions, the diago na lization of two or more ma trices by means of the congruence or s imila rity transformations may lead to coordinate system " mismatch" a nd intJ·oduce erroneous results.
=
=
I. I TRODUCT'ION HE even and oddmode decomposition method and the c and 7!'modc decomposition method, can correctl y sol ve electromagneti c coupling problems involving two symmetrical and two asym metrical lines. but not more complex structures. The modal dccoupling technique is a powerful ex tension of the. c two methods. in that it handles an arbitrary number of coupled li nes at arbitrary locations. This technique has been appl ied to the analysis of multiconductor transmi ssion line (MTL) problem s for more th an two decades II HI 01. In 1973, Marx [21 applied modal anal ysis to second order matrix differential equations and computed the characteri sti c impedance of the MTL' u. ing voltage and curren t eigenvectors of the LC and CL matricc!>. where L is the i nductance matrix and C is the capaci tance matrix of the interconnect structure. In his work. Marx proved the biorthogonality between the voltage
T
~ l anuscripl received October 12. 1991: revised ~ l ay 25. 1995. This work was supported in part by ARPA/ESTO under '6600189C0101 a nd N66001 9IC0051 from NCCOSC/ ' RaD. NOOO I191J1030 from the Office of Naval Rc ..carch. F3361592C1023 from the Air Force Wright Laboratories. and 133P771 from Bodng Acropace .\'·. The inner product i' defined '" the canonical product of vector>. in the \pace\.\' and.\'·. i.e .. (iy).j.r)) = (yj.r).
(S1 BS  .A; S1 AS )S 1 Ie:) = 0 which. prcmultipl ying by (S' )
1
•
( 13)
yields
(B  .A, A )Ic;)
= 0.
( 1l)
Thus. we have proven that { lr·;)} arc the eigenvectors or the generalized cigcncquation (7). Because S is nonsingular. ( 12) indicates that ll':) for i l. 2 ..... 11 . in ( 1l) arc linearly independent and form a basis. II' they were no t. there would
=
IEEE TRt\.''IS,\(TIONS ON ,\ IICROWAVE T IIEORY ,\,\ 11) T I:C'IINIQt lES. VOL.
ex ist a set of constants { o 1
}
not all zero such that
L OjleJ} = () j=l
which by ( 12) would imply II
L
n1 lvj} = 0 j=l not all zero. a contradicti on to the independency o f
{l~ lj) }.
Si nce l\11 is Hermitian. there ex ists a unitary matri x U such that U ' MU =
D ,11
where D .\I is a diagonal matrix with the eigenvalues o f l\11 as its diagonal clemen ts. 1 otc that matri x U i s orth onormal. whi le matri x S is orthogonal. but not normalized. Here and throughout the paper. an orthogonal mat ri x is di ffcrcnt from the conve ntional definition by which its column vec tors arc normali1.cd. and a normali zed vector means a vector with magnitude of unity. SU. For simpl icity. these We now construct a matrix Q two successi ve changes of the bases arc merged into one. providing the following relation s
=
Q ' AQ
= U 1S 1ASU =
UU' = I
( 15)
and Q ' BQ
= U 1 S 1BSU = U 1 l\IIU = D .11 .
( 16)
From ( 15) and ( 16). it may be observed that the first congruence tran sformati on. S' AS. has tran sformed A into an identity matri x. whi le the consecutive transformation. U 1I U. keep!> the identity matri x unchanged. On the other hand. the fi rst tran sformati on. S' BS. preserves the symmetric properties of B . while the consecuti ve unitary transformation converts B into a diagonal form. Note th at when A and B arc transformed into the ba!>is { lc:} } by Q. in general. the diagonal c lements of matrices I and D .11 arc not the eigenvalues of the matrices A and B . and the co lumn vectors o f Q arc neither eigenvec tors of A nor B. but arc eigenvec tors o f matri x A  In or LK. Thi s transformati on that diagonali zcs the matrix A  I D is not a unitary one. Furtherm ore. the ijth clements of A in the!i~. namely. that all eigenvectors in the ~et { lc;) } arc linearly independent. Since (I .f) and (26) arc equi valent. we have thus proven that matrices A  I B or LK arc diagonalizablc. We may obtain the eigenvalues and the corresponding eigenvectors of matrix LK in (26) by solving eq uations Thi ~ i~
d5 (  (l.O l 30
 0.2295 0.1380  O.OOG8
 0.0130 )  O.OOG pF/cm 2.14 '0
=(
J 1.1600 ' .03GO
I ·1.HOO
0.1307
0. 1302
8.0:JGO
0.1:307)
0.1302 G.l020
L" :2.:200:3mission lin~> in a non ho mog~n~ous ... //:'/:'/:' 1i'a11.1. Minmrm·e J'lu•IJI:r 7l·ch .. vol. i\ ITf 2 1. pp. 55655S. i\ ug. 197:!. HI i\ 1. i\lchalic and R. M ittra. " lnl'est igationof tape red multipk mic rostrip lines for VLS I c irc uits." 1/:'EE Tm11.1. Micmll'are Theory Tech.. vol. 3H. no. I I. pp. 1559 1568. 1ov. 1990. 151 \ '. K. T ripat hi and J. 13 . Re tti g. "/\ SPICE model fo r m ult iple coupled micro,trip and other tran'>mi,,ion line,." tEE/:' han.\. Mit'l'oll·m ·e 71u•ory 7l·ch.. vol. i\ln ·:n. pp. 1513 1518. Dec. 1985. 161 A. R. Djordj~' ic. T. K. Sarf..ar and R. F. Harringto n. 'Time domain re.pon'c of multiconductor tr:u1smi" io n line,." Pmc. IEEE. vol. 75. pp. 7l:l761. June 1987. 171 A. R. Djordjcvic and T. K. Sarf.. ar. "Analy,is o f time response or lo"y multiconductor transmission line network!. ... I EEE h ans. tl!icro11·m·e Theory Tech .. vol. Mn ':l5. pp. ll98907. Oc t. 19 87. 181 V. K. Tripathi and H. Lee. "Spectraldomain com putatio n o f' charactcri'tic impedance. and multi port parameters of m ult iple cou pled mic rmtrip lines ... 11:'/:r : hall.\. Micrmrm·e '11w111y 7l·ch. vol. 37. no. I. pp. 2 1522 1. Jan. 1989.
~09\1
191 G. \V . Pan. K. S. Olson . and B. K. G ilbert. " Improved algorithmic methods fo r the predic tion of wavefrolll propagat io n behavior in mult icond uctor trans mission lines fo r high freque ncy digi tal signal processors ... 11;1~·E Tmn,,·. ComputerAided Design. vol. 8. pp. 60862 1. June 19 89. 110 1 G. W . Pan. J . i\. Prc.:ntice. S. K. Zah n. A. J. Stanist~wski. and B. K. Gilbert. " T he s imulmion of' highspeed . h ig hdensity digital interconnects in s ingle c hi p packages and multic hip module,:· I EEE hans. Compolletlls. Hybrids. and Manufacturing Techno/.. vol. 15. pp. 165t assign ment in this group was the developme nt of' the thcnuomcchanical modeling tech niques for integrated c irc uits a nd mult ic hip module>. Sub>cqucntly. >he was assigned to ex plo it new approaches for the modeling of the elcc tromagn~tic environ ment or Gaf\ s integrated circuits. printed c ircuit boards and i\ IC M's operating at high sy"cm clock rat~s and wide s ig nal bandwidt hs. Dr. Lei is a member o f Tau Beta Pi.
IEEE TRi\ NS,\CTIONS ON 1\IICROWAVE THEORY MJD TECII NIQUES. VOL . .JJ. :\0. 9.
G uangWcn (Gcori,!C) Pan (S'8 183M '84SM'94) received the B.E. degree in mechanical e ngineering from Peking Institute of Petroleum Technology in 1967. He auendcd the Graduate School. Uni,•crs ily of Science and Technology of China from 1978 10 1980. majoring in electrical engineering. He received the t. I.S. degree in 1982. and the Ph.D. degree in 1984 both in electrical engi neering from the Univers ity of Kansas. Lawrence. KS. lie worked at the lm,titute of Developme nt and Re,earch in Northwe~t of C hina in machine design a' an '' "ociate Engineer. and then as an Electrical Engineer respom,ible for de.,ign of pul.,c\\ idth modulation electronic' and digital remote lire control ')'term "'cd in petroleum 'ei!.ntic exploration. lie came to the United States in Augu,t 1980 a~ a R e~earc h A~s i stan t in the Remote Sensing Laboratory, Uni ver.,it) of Kansas. From 1984 to 1985. he wa~ a Post Doctoral Fellow at the Uni' er.,ity of Te.xas. engaged in a project on computer aided design of airborne antcnna/radorne '>Y'> tcm;. He joined the t.l ayo Foundation in 1985. engaged 111 the theoretical modeling of the e lectromagnetic behavior of high;pced integr:tted c ircuit ~ . electronic c irc uit board;. and high de n;ity ; ub;trates. placement and routing. from 1986 to 1988 he wa; an Assoc iate Professor in the Department of Electrical Engineering. South Dakota State University. In 198S he joined the Department of Electrical Engineering and Computer Science at the Univen.ity of Wi;cons int. lilwaukee a> an Associate Professor. lie ha., been the Director of the Signal Propagation Re>earch Laboratory >ince 1990 and became a Profe.,,or in 1993. His re;earch interes ts cont inue to be in the mathematical modeling of the electromagnetic eJI\•ironmcnt of high clock ralc ..,ignal procl:~~or~. Dr. Pan b cited in Who·, Who in the Midwest. a me mber of Eta Kappa Nu. and i' on the Editorial Board of the IEEE!Mn .
SH'TE~IBER
19..
IV. DISPmS IO ANALYS IS OF THE LiNEA R VA ET YPE WAVEGUIDE US ! 1G THE GSM
The dispersion propert ies of the linear vanetype wavegu ide have been previously explored using fi eld matching al gorithms 121. ]+]. 17]. ]8 ]. The linear vanetype waveguide geometry in thi s study i s the closed end space geometry depicted in Fig. I. The field matching algorithm developed by Gunderson 111 found resonances for the surface waves 161. 171 forming the first passband of the slow wave circuit. The ability to resolve the system normal modes and accurately solve for the dispersion of a peri odic circuit using the GSM algorithm i:, determined by the number of mode. kept in the truncati on of the infinite series ex pansion of the waveguide junction norm al modes. The truncation of the infinite series expansion of the waveguide j unction norm al modes invo lves the ability or the modes used in the GSM simulati on to satisfy the RC criteri on. It has been demonstrated that if an insufficient normal mode cxpan ion is used for a wa veg uide junction. then the ca lculated GSM wi ll be in error clue to the violation of the problem boundary and edge con ditions ]7] . ]II]. ]21] . T herefore. the ability of the GSM algorithm to adequately predict the dispersion charac teri stic:, of a peri odic circuit i s determined by the proper . election o f region norm al modes i n the correct rati os between reg ion. to sati sfy the RC criterion. T he RC cri terion problem i s demonstrated with the mode match ing simulation of a single peri od o f the closed end space linear vanetype waveguide shown in Fig. 3. This geometry was also analyzed using the fi eld matching al gorithm developed by Gunderson 141 for the first passband. Thi s circuit geometry has been cxtcnsi vc ly analyzed ]7 ] using vari ous experi mental and analy tical techniques. The intent o f the present study is to prov ide a method to compare the field matching and GSM di pcrsion al gorithm s with experimentally determined di spersion informati on for the analysis of the structure shown i n Fig. 3. Thi. problem was analyzed with the mode matching algorithm using three different mode ets to determine the di per ion propcrti c. as a function o f the RC criteri on. The mode. u ed for these simulati ons consi. ts o f the rectangular waveguide T E : and T.\f: norm al modes. The mode set elected for thi s mode matchi ng si mulation was based on the concept of the continuity of the transverse electromagnetic fields defined by an infinite seri es expansion of the waveguide region normal modes at the plan e of a waveguide discontinuity. The infin ite seri es expansion or the region normal modes req uires truncation for implementation on a computer. in
TABLE I CO~ti',\RISOS OF CALCL:LA1 ED ASD M EASL' IU:D DISP ERSI OS P,\Rti~IETERS lOll TilE LtSEAR VASETYPE \VWt:Gl' lD E SitOIIS IS FtG
rnode 1dc:ntrf1cr
nLO Tf:u/l:lfu Ellu
L/1'·' 0mode n/9mode 2'f"J9rnodc l"'JJmodc
.Sr./9mt)I.Jc: 5rt.fC)mO!Jc 2i:/3modc 7'tt/tJrnodc S:t/9rnudc i'Hll()liC
E..:pcnmental frequency. Gill
66600 9 JS00 8 6;EAR
v \:;tTYI'I:

I HJR
\V,\VEGl:IJ)t SIIOW:1 );>; FIG.
3
,.....__
\t I

.. :
"""'•
1dcn11hcr T/.1.11
Nt,\
Tluii llu
N/A ;>;JA ;>;lA
lllu
1//o: O·mcJJc l"'.l')mode ~r.Nmodc
rJ.lmodc .Sr.J9rnudc ~r.Nmudc
1:J.lrncldc 7r.I'Jmodc \ ;J'I)·mtldc
::r01.".!c
m:J
Gundcr\on (.SJ GS~I error. 't 2.S mode\. error. r
52 or 97 modes. error. r;
0.0 00 I0
1.2.
0.0 05 10 00 02 0.0 02
1.1
0.2
J2
0.1 0.1 0I 0.1 0.1
7.3
00 06 0.5
09 09 10 0')
12 I2
12 I~
lit
=
71:J
....
~
=
=
""II
~u
•)n
ltl(,·~~n
Fig. .J. Eigenvalues dclcrmincd using lhc PEC boundary condition with ahc GSl\ 1 for lhc linear vanetype waveguide \hown in Fig. 3.
..
=
=
=
=
111 1 :
I requ~n~) . ( ill/
=
=
.•.
1111
= 0. I. 2. :3 modes. Sim ilarl y,
=
= =
::
::!
flll
the T.\1 : mode indices for Regions I and 3 included the tilt = lli:J = 1. 2. :3 and lit = ll:J = I. 2. 3 modes. Region 2 for the TE: modes was l imited to mode indices 111 2 = 0. 1. 2. 3 and 11'2 0. 1. 2. and for the T i\ J= modes to index li mits 1112 = I. 2. :3 and 11'2 L. 2. Examination of the peri odic waveguide system normal mode cutoff and slow wave mode dispersion information shown in Tabl es I and II indicates th at the error using the GSM eli persion algorithm is greater than that obtained using the field matching dispersion algorithm. The error in determin ing the dispersion information wi th the GSM algorithm can be reduced by increasing the number of modes u~cd in the simulation to improve the RC criteri on. The second GSM imulation involves 52 modes in Regions I and 3. and 38 modes in Region 2. The Region I and 3 mode indices for the TE: modes included the tnt '":J 0. I. 2. :1 and lit n :1 0. 1. 2. :J. •1. 5. G. 7 modes. Similarl y, the TM : mode indicc~ for R eg i on~ I and 3 included the 7/lt = II!:J = l. 2. :3 and lit = II:J = I. 2. 3. 4. 5. 6. 7 modes. Region 2 for the TI::: modes was limited to mode indices m 2 = 0. J. 2. 3 and 11'2 0. L. 2. :3 . •J. 5 and for the Ti\J: modes to index l imits 111 '2 I. 2. 3 and 11'2 1. 2. :3. •I. 5. Examinati on of the di s pcr~ion information in Tables I and II indicate that the slow wave mode dispersion error ha been reduced substanti ally by improving the RC criterion in this simulati on. Finall y. the GSM dispersion error may potentially be f urther reduced by increasing the number of modes used for the simulation. The third GSM simulation invol ves 97 modes in R egio n ~ I and 3. and 7 1 modes in Region 2. The Region I and 3 mode indices for the TE: modes included the tilt = '":J 0. I . 2. J . I . 5. G and nt na 0. l. 2. J. '1. 5. G. 7 modes. Similarly. the T !\1 : mode indices for Regions I and 3 incl uded the lilt lll :J I. 2. 3. •I. 5G and n 1 li;J = I. 2. :3. 1. 5. G. 7 modes. Region 2 for the TE= mode was limited to mode indices 111 2 0. l. 2. J. ... IEEE Trans. MinrJI\'{/1'1' Theory Tech.. vol. I\1TT·34. no. 12. pp. 14901498. Dec. 1986. 1161 A. S. Omar and K. Schlinemann. "Transmi;;ion matrix represenwtion of fi nlinc di'>COnlinuitics,'' IEEE Trans. Alic:rall·m·e Theory Tech .. vol. I\ 1TT·33. no. 9. pp. 765770. Sept 1985. 1171 F. Dai. "Scaucring and transmission matrix rcprcserllalions of muhiguidc junction.,:· IEEE Trans. Microwm ·e Theory Tech .. vol. 40. no. 7. pp. 1538 1544. July 1992. 1181 F. Dai and A. S. Omar. "Fieldanalysis model for predicting dbpersion property of coupled cavity circuits:· in Proc. /993 IEEE Int. Alicl'llll'lll't' 71rl'ory 7i•ch. Symp .. Atlama. GA, June 14 18. 1993. pp. 901 904. 1191 B. T. Smith. J. M. Boyle. J. J. Dongarra. 13 . S. Garbow. Y. l ~ehc. V. C. Klema. and C. B. Moler, Mmrix Eigensystem Noutines EISPA CK Guidi'. New York: Spri ngerVerlag, 1976. 1201 S. S. Omar. Private communication. June 30. 1993. 1211 J. van 131;rdcl. Singular Electromagnetic Fields am/ Sources. Oxford. Encland: Clarendon. 1991 . 1221 Y. ~C. Shih and K. G. Gray, "Converge nce of numerical solutions of 'tctHypc waveguide discontinuity problem; by modal analy>is." in Pmc. /983 /EEE Int. Micro11·m ·e Theory Tech. Symp.. Bmwn . MA. I\ lay 3 1 Junc 3. 1983. pp. 233234.
W . Scott Best (S'79M'80) n:ccived the B.S. in electrical engi neering from the Univer;ity of Louisvi lle in 1979. the M.E.E. from the University of Louisville in 1980. the D.E.E. from the Univers ity of Utah in 1985. and the Ph.D. in electrical engineering from the Univers ity of Utah in 1995. From 1980 10 1987. he was employed at the a val Weapons Cemer. China Lake. CA as an Electronics Engineer working on the research. des ign. and developme nt of radar transmiucrs for air·to·air weaponry. He was awarded a Navy Fellowship to aue nd the Af'TER Program at the University of Utah fro m 1984 to 1986. From 1987 to 1993. he was employed at Varian Cros;cdField and Receiver Protector Products. Beverly. MA as Manager. Applied Technology in the Advanced Technology Group participating in the research. de~ ign . de\ elopmcnl. and product ion of crossedfield amplifiers and magnetrorl',. from 1993 to the pre'>ent he has been at DuPont Central Re~earch and Development. Wilmington. DE a ~ a Senior Research Engineer in the Wave Energy Group working on the research. design. and development of microwave applicator' for product proces;ing and plasma generation.
phy~ ic'
r;~diation.
Lc\\is C . Goudrich n:cei,cd the B.S. degrcc in electrical encinecrinc from the Uni,ersit\ of Utah in 1963. and tl;;; Ph.D. degree in electrical ~ngineering from the Uni,er,ity of Ut. From 1978 10 1985. he s upervised a nun1ber of di ffe rent groups in the Engineering Research and Development Division responsible for developing radio fre· que ncy/rnicrowave technology. and instrurnenh for color me;~; urcmcnl. tr;~ce monitoring. and DNA scque ncing. From 1985 to 1992. he managcd the Applied Phys ics Section in Engi neering Rcscarch ;~nd Dcvclopmcnt Di' i'ion wi th respons ibility for R&D in the area' of o,eno,or' and analy;er~. radio frcq ue ncy/rnic row;~ve techno log). and color mea,urc mcnl. From 1992 to the pre>cnt. he is responsible for managing Cent ral Research and Devclopmenl Division's R&D in the application of radio frequenC). microwa,e. pla; ma. and ultrasonic technology to proce'' impro' ernent and enhanccd chemical reactions.
£"'
IEEE TRA;>.:SACTJONS ON
~ IICROWAVE
TIIEORY AND TECIIN IQUES. VOL. 43. '0. 9.
SEJYJ 'E ~IIlER
1995
~109
ModeMatching Analysis of TEo 11 Mode Waveguide Bandpass Filters Andrea Melloni. Marco Politi. and Gian Guido Gentili
AbstractA mode matching technique for the a na lysis of wa veguide handpass filt ers employing circula.· T E011 resonators, coupl ed hy rectangular apertures, is presented. S uch a technique takes rigorously int o accoun t thickness and angulat· ofl'set of the two coupling irises, a nd hig he r modes interactions between resonat ors, whil e overcoming th e typical limita tions of the ava ila bl e approximat e models. We show, through a n o pt imization procedure. that it is possible to desig n filt ers wi th a desired frequency response, witho ut needing a ny furth er empiric adjustments . purious responses can be controlled too.
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I
I I
I.
INTRODUCTION
<
HE HIGH unloaded Qfactor of TE0 w modc in cy lindri ca l cavity is very atlractive for the reali zati on of lowloss narrowband filters at millimeter wavelengths. Direct co upled filters employing th is kind of resonators can be designed to have a very selective passband with steep skirt s. together with a very low midband in sert ion loss. One further ad vantage is the higher midband pu lse power capacity, when compared to other waveguide fil ters. The main building block of the TE011 mode bandpass filter is the cy li ndrical resonator shown in Fig. I . coupled both with the external rectangular wavegu ides and with the adjacent cavi ti c~ through two rectangular apertures (coupling irises) which arc placed on the cavity sidewall with an angular offset equal to 20. The analysi o f such fi lters is usually performed with reference to approximate models based on ·'Bcthc·s small coupl ing theory'· Ill. later modified by Cohn 121. MacDonald 131. 111 and. recently. by Levy 151. Such mode ls cannot take into account the effects o f the coupling apertures on the TE011 resonant frequency and the higherorder mode coupl ing between adjacent resonators and the irises angular offset '20. Moreover. the design of such fi lters 16.1171is compl icated by the large number of spurious modes that resonate at frequencies that arc close to the TE011 mode and its degeneracy w ith TM 111 mode. As a result. several empirical adjustments may be required when bandpass filters are designed according to the approximate model s. and particular care must be taken to guarantee satisfactory cavity tuning control and suppression of adjacent modes 181. 191. Thi s paper proposes a modematching technique for analyzing all buiding blocks o f the above filter in order to accurately evaluate their frequency response. The resulting
I II
T
Manu,cript recci,cd No,cmbcr 10. 199l: rc,·i>cd 1\l ay 25. 1995. A. ~lclloni and 1\1. Polili arc with the Politccnico di 1\ lilano. Dipanirncnto di Elcuronica c ln formatione. 120 133 Milano. llaly. G. G. Gcmili i., with the CSTSCN R. Politccnico di Milano. 1\lilano. llaly. IEEE Log Number 9l 13130.
,,
v
>
Sl·t llcul J\ A
Fig. l. The cylindrical rc.,onator coupled to the C\tcrnal wa\ eguidc' and to the adjacent resonators by rectangular coupling iri'c'.
procedure all ows the designer to take into account the effect of all coupling apertures, the angle '!.0 between them (sec Fig. I ). 1hc spuriou s responses and the higherordermode interaction between adjacent resonator!'.. The on ly restrict ion on the irises dimensions i s th at 1hc aperture \\'i dth b, must be small compared to the diamc1er U or the cavil). Small va lues or the rati o IJ, I I? results in a considerable simplification of the modema1ching procedure. Thi s. however. is not a severe restricti on as the coupling mainly depends on o,. I f. for any reason. one has to usc larger va lues or b,. the more general modematching procedure reported in 1101. 1111 can be employed. The technique we propose i~ quite accurate up to b, I I? ~ o.:.l. The method is explained in Sec tion IIA for cavi1ics coup led through irises w ilh identica l dimensions. The more general case o f asymmetrical cavities having two diffe ren t apertures is treated separately in Sect ion 11B. An exlensivc convergence anal ys is has been carried out in Scc1ion II I and 1hc efficiency of the method and other numerical resu lts arc discussed in Section I V. Some prototype fi lters with one and four cavi ties have been built as well in order to validate the method described above. In these cases. a very good agreement between simulati ons and measurements has been observed. as discussed in Section V.
I I. THE S O LUTIO
O F TilE FI ELD PROB LEI\1
The anal ysi s of the T E011 bandpass filter can be t:OI1\'Cnicntly carried out by splitting the whole structure in simpler
00l8 9480/95SO.J.OO © 1995 IEEE
2110
IEEE TRANSACTIONS ON
building bl ocks, as shown in Fig. I . Onl y two disconti nuities are present: the symmetrical doublestep formed by the junction between the rectangular external waveguide and the first (last) rectangular coupling iris and the discontinuity at the junction between the irises and the cavity itself. Each discontinuity is considered separately and its generalized scattering matri x is evaluated. The overall scaucring matrix of the total filter is then calcul ated by a suitable direct combination of all single cauering matrices 112]. In thi s work. the modematching technique is used to calculate the generalized scauering matrix of each discontinui ty since it is a method particularl y suitable when the fi elds in each region can be expressed analitically as a mode superposi ti on. The first step of the modematchi ng technique consists in choosing the most suitable modal expansion of the fields in the regions forming the junction. T he analysis of doublestep discontinuities in rectangular waveguide has been carried out by several authors. In the past 11 31. 11 41 the rigorou s T ETM field ex pan sion was success fully used. umcrical analysi s, however, has veri fi cd that the electric E.!. component of the scaucrcd field is much smaller than others in the band with of the dominant TE 10 mode. suggesting a T F field expansion I I 51 1I 7J. The TE.,. expansion has the advantage of working with smaller matrices w ith respect to the T ETM case. reducing the CPU time and storage requirements and it was therefore used in thi s work. Let z be the directi on of propagation i n the rectangular region. that is the external waveguide and the coupling iri ses, as shown in Fig. 2. T he modal expansions of the fields component are deri ved by an .rdirectcd magnetic scalar potential ll.,.x. T he expressions of potential and fi elds, assuming a dependence of the type ,,jwr . are the following ll r =
L 'I
l:o y
I
(ate 1 ••"

a;;ei••" )rp~(;r.y)
( I)
l 'f
. unJ. =  •JWfL   = L (a'I+ e ~·•  +a  e•·•)e OZ 'I '
Y'l
TIIEORY AND TECIINIQUES. VOL. 43. NO. 9.
~ II CROWAVE
SEPTE~ IBER
1995
!J
T
Elcrlrk / Magnclic Plane:
Fig. 2.
Top view of the iriscavity junct ion.
ot
n;;
Finally and arc the incident and rcncctcd modal coefficients. respectively. L et's now concentrate on the fie lds modal expansi on in the cylindrical cavity. The ex ternal waveguide operate in T E 10 dominan t mode and the coupling irises arc below the cut off. Under these conditi ons. and due to the orientation of the waveguide itself (sec Fig. I). no E,. component exists in the coupling i rises and hence only T F mode (simply T E in the following) arc excited i n the cavi ty. In practice TM modes could be weakl y excited because of unwanted asymmetries and losses, but in general their effects arc negligible and they arc not considered in the followi ng. The TM 111 modc is the only one that can seri ously affec t the bandpass filter performance becau se it is degenerate w ith the TEo1 1 mode. T his degeneracy can be split simpl y by using a noncontacting tuning plu nger or by modifing the resonator as in 18. 91, but these arc empirical adj ustments and arc not considered in this analysis. The cavity can be considered as a rad ial waveguide of height h. The termination of such a waveguide with an ideal electric surface with two rectangular apert ures at a distance 17 from the center of the cavi ty will be enforced later by the modematchi ng technique. After that. the modal field components of interest in the cavity arc [ 10 1. I 181
(2)
E~ =
'I
j w fl
L
CpP~:.. (cp . .r)
(7)
p= l
IIC: = LcP } ~~¢~:,(cp .. r )
where q1~ is the rectangul ar mode functi on
(8)
p= l
where
(4) c, is the Neumann factor w hich equals I if 11 = 0 and 2 if n =f. 0 and a normali zi ng coe ffi cient has been i ntroduced so that J~ q1~J dS = 1. 'I is some combination of m and 11 and k = w..jli"i. The mark 0 is used to di stinguish quantities related to the rectangular region from quantities related to the circular one, indicated in the following w ith 0 . In (2). (3) the modal eigenfuncti ons are C'yq
1/ being , ,1
hJ.,,
=
.1"(/
. ./0 = .JWf tf!CJ = (/·2 _ 1.2 ...
J1.;;., + J..:;, 
(5)
)A,OI...,lfJ "'.1"111 \f'q
(6)
(9)
= J/.·
rb1,,· (cp . :r) "'
J..: 2
the propagation constant.
= (2111 + I )r.I a and ky, = u:rr I b w ith
111. 11
= 0. l. ...
=
=
2  "'';. . " J·J being Hcj jr.l lt. and i 0. L. ... "'· 1 j = I . 2.... arc integer numbers dependi ng on p. In (9) the rati o between the two first kind Bessel fu nctions J of order i + 1 and i can be easily ca lculated 119. p. 3631. Moreover cp i s the modal ampli tude cocffi ciem and r/J~, (cp . .r ) 1 ' is the modal eigenfuncti on
{!!;;
cos . = 1I 1"1i  cos(HJj.l") . (1cp) S lit
( 10)
where c, = 1 if i = 0 and c; = 2 otherwise. As for the eigenfunction of the rectangul ar waveguide. a normal izing
~IELLO
'I
t' l
a/ :
~IODE~ I ATC III NG
ANI\ LYSIS OF WAV EGUIDE BANDPASS FILT ERS
J
')
factor has been added so that 5 /~~~. dS = J over the cav ity sidewall S. Note. by the way. that the field has been expanded in fir. 1 ki nd Bessel fu ncti ons. Thi s physical ly means that the cavity fie lds arc expanded in stationary modes and (7) and (8) represent the solution of the Helmhol tz equation in cy lindrica l coord inates. once the singul arity in r = 0 has been removed. M oreover modes wi th ,,_,.J = 0 do not sat isfy the boundary co ndi tion on the bottom and the top of the cavity and they can be neglec ted. Finall y. if the coupling apertures arc placed midhcight on the cavity sidewall. only symmetri ca l modes with respec t to the .r direction have to be considered. th at is /....,, = (2111 + l )11ja. ti.rj = (2j + l )11jh and 111 . j = 0. 1. 2 ... In the next two sections the conti nuity of the fi eld · in the apertures and the boundary condition on the metalli c sidewall of the cav ity permit the obt ai ning of the generalized scatterin g matri x of the reson ator.
}Ill
H~.. is the coupl ing matri x and Y 0 and Y 0 arc two diagonal square mat ri ces of norm ali zing coe ffi cient. Superscript ·T· denotes tra nspose. Note the I /2 fac tor in the first of ( 13 ). resulti ng from the i nserti on o f the elec tri c and magnetic wall in the simmctry plane. The clements of the coupling mat ri x arc defined as
expli cit form for w hich. always under the assumption of small li;. i s repon ed in Appendix. w hile ( 15)
By sol vi ng the two matri x ( 13) as usual. the two scaucring one port matri x S, and S,, result
A. Symmelrical Cm•ily
A simple bu t important case is represented by a cav ity with two identica l apertures symmetrical ly placed wi th respect to the height of the cav ity . Cavi tie s wi th two different apertures arc treated i n the next section. By taking ad vamagc of symmetry. only half o f a cav ity. obt ained by inserting either an electric or a magnetic wa ll at the symmetry plane. has to be analyzed. A top view o f hal f of the cavity so obtained is shown in Fig. 2 together with the electric and magneti c field co mponents to be matched. The two generalized one port scallcring matri ces of hal f the cavi ty arc deri vable by suitable projec ti on o f the I:: field and If fiel d cominuity equati ons over the aperture. I f I; is small compared with R. the deri vati on of the two eq uations i s strai ght forward . This assumption means that the curvature of the matching su rface. named a in Fig. 2. can be neglected. T he electri c field component in the iri s i s therefore neglected and the continuity between the cavity electric field E~ and the electri c field component E~ in the rectangular aperture. tangential to a . must be enforced, i.e.,
er:
;0 1'.,:
ll~
= { t:;~ Iaperture 0
= 11.~
( I I)
lcavi ty sidewal l ( 12)
lapcnurc.
To en force the boundary condition on the cavity sidewall that i s the vani shing of the tangential E ficld and the co mi nuity o f the electri c field through the aperture, ( I I ) is proj ec ted by using the cavi ty eigen functions (/> 0 , (
= H~.. (A l
and then the scattering matri x of the w hole sy mmetri c resonator is
S JI =
( 13)
where A += [otJ. A  = [a;;J and C = [r·1,] are the incident. rcncctcd and stationary modal ampli wdc vec tors coeffi cients.
2
S,,. S c s 21
2
( 17)
Note th at S 11 and S 21 arc real numbers because the iri ses operate below their cutoff frequency and the techniq ue can be implemen ted using only real al gebra.
B. A.IYIIIII /el r ica/ Cavily The case of a cavity wi th rectangular apertures of different dimensions i s now investigated. The most direc t approac h considers a symmetrical resonator wi th two apertures equal to the larger one fol lowed by a symmetrical doublestep to reduce the dimension s of one of the two apertures. The symmet ri cal resonator generalized scallcring matri x must be therefore cascaded 112 1to the double step one placed at a null distance from the aperture on the cavity sidewall . Thi s approach is qui te simple becau se all the needed buildingblock arc already available and one has on ly to co mbine them. However the co mputational cost is heavy. a great number of matrix operati ons arc needed and the case of two apertu res placed at different heights canno t be handled. The mos t powerful approach consists therefore in co nsidering the w hole resonator. A s for the symmetrica l case, the conti nuity o f the elec tric and magneti c field s in the two apertures must be en forced. toghcthcr w ith the vanishi ng or the tangential electric field on the metal lic sidewall of the cavity. The two equations are wrillcn as follows
+ A )
= Y 0 (A +  A  )
S,, + S,
.iWfl
L ('p/~~.. I'
_ { Lq(ot, 
+ o ~1 h !l'l + L,,1(n;i,1 + a2,1 )('2yq
(cavi ty sidewall)
( I S)
1111
IEEE TRANSACTIONS ON 1\IICROWAVE TIIEORY AND
L
cl' } ~~ (.')~:.,
TECII~IQ UES.
VOL. 13.
~0.
9.
SEPTE~IBER
1995
6.
Jl
.,..o
615 6 2
where the subscript I and 2 have been added to quantities relative to the first and ccond aperture. respecti ve ly. Equati on ( 18) must be proj ected on ¢~, and the magnetic } .,,
6.25 6.3
fieldcontinuity ( 19) on 4/?,1 and 4>~1 to sol ve the scatt ering problem. In matrix form. the three equations arc
H r(A t + A ~ )+ H 2(A t + A2) {
=C
 6·'' o~L::2':" o ~':" o :6':" 0 :8:.:!oo
Y~( A1  A ~ )= H f Y ° C
Number
(20)
Y~( A2  A2) = H r Y ° C
T)
(22) (23) (24)
(25) and (26) The two coupling matrices, whose coeffi cients arc reported in A ppendix. arc
l 
and
[~
1I,,,
and
(27)
r;J= [~f] Y [H 1 0
H 2 ].
Fig. 3. Scaltcring cocfficicnls convergence of a ') nunclrical c:l\'il) . Sec for dimc no, i on~.
h.:\1
(2 1)
 1 T
whe re
H_[H,,]
1 Ci(HJ
Numb c:r o f mod e'> Ill I he cantv
S12 = (Y 1 + D ) F (I + S22) S21 =(Y~+ G ) 1 F ( I + S i r) S22 = (Y~ + R) 1 (Y~ R ) 0
or lltOdt·~ In t he lrb
Cl
and lead. after the elimination o f the amplitude vector C to the generalized . cattering matri x
s I I = ( Y~ + T ) I ( Y~ 
~
6.35
(28)
The ~i mp l e r form for symmetrical cavity given in ( 16). ( 17) can be obta ined by simply setting Y~ = Y~ and H 1 H2 in the equations above. Also in this case only real algebra is needed.
=
Ill. CONVERGENCE PROPERTI ES In the modematching technique the field mode ex pan sions arc truncated for practi cal reasons and it is well known that the choice of the number of modes it is not always an easy matter. T he ex tensive studies on the subject, both for a general discontinuity [20]. 12 11 and for the iri scavi ty discontinu ity 110 ]. demonstrated th at the ratio between the number o f modes u~cd on each side of the junction plays a very important role in the convergence behavior of the mode matching technique. The best rate of convergence i s obtained if the same spati al resolution or the field mode expansions in the two region form ing the discontinuity is ensured 1201. Thi s means that all the modes having a cutoff frequency lower than a prescribed value must be taken into account. both in the waveguide and in the cavi ty.
In the structure of Fig. I two convergence analysis have been carried out. T he first one focuses the attention on the discontinuity between the rectangular iris and the cy lindrical cavity (for the convergence behavior or the doublestep di scontinuity sec 117]. [2 1J). Fig. 3 shows the convergence of the rcnection coeffl cient 8 11 and the transmission coe rtlcient S2 1 (both are real numbers) as a function of the total number of modes in the iri s and in the cavity. The dimensions of the considered cavity arc D = 34 rnm.h = 18 mm. '20 = 1 '0 . with two identical apert ures of dimensions a ; 9 mm and b; 6 rnm . The ana l y~i s has been carri ed out at 13.8 Gil t. and shows that about I 0 mode ~ in the iri s arc sufficient to guarantee a va lue th at di ffers less th an I Ck from the asymptotic one (obtained wi th 140 modes in the iris). The second convergence analysis is devoted to the whole structure. A onecavity filter is analyzed in order to study the i nnucncc of the various discon tinuities. The accuracy in the evaluation of the resonant frequency and the quality factor of the loaded cavi ty. the two parameters on wh ich the design of a filter i s based. is discussed. Fig. 4(a) and (b) show. respect ively . the percent error on the qual ity factor Q and on the resonant frequency for the above mentioned cavity coupled to an external R I 20 ( 19.05 x 9.525 mm) rectangular waveguide. Three di iTerent coupling iri~e~ with lengths /, arc considered and the errors refers to the asymptotic val ues. It is interes ting to note that the resonant frequency of the TE011 mode i ~ determined accurately even wi th very few modes in the iris while the evaluation of the Qfactor needs a more accurate anal ysis. Finally the convergence is faster for long irises (i.e.. I , = 4 mm ) because of the weaker i nteraction of higher order modes between the double step discontinuity and the aperture in the cavi ty sidewall. A further convergence analysis concern s the difTcrcnce bet ween the two approaches proposed for the analysis of an asymmetrical cavity. In Fig. 5 the same cavity as above but with two different i rises w ith dimen ions a 1 = 9 mm. 1! 1 = 6 m111, a 2 8 111m and b2 5 mm is considered. The scauering coefficients arc report ed as a function of the number of mode ~ used in the i rises. T he continuous li nes represent the scauering coeffic ients obtained wi th (2 1) to (24). that is considering the
=
=
=
=
~IELL0\'1 an IMPATT. The waveguide incorporates a grating surface c111illcr ( IMPATT devices at 60 GH;r were con sidered because of their high power capabili ty). The photograph in Fig. 5 identifies the critica l parts of the millimeterwave grating surface cmi ller (Mf\ IWGSE). An IMPATT i s mou nted at the bollom o f a cy lindri cal cavity in the ground plane below the waveguide (sec Fig. 6). The RF energy is coupled into the waveguide through a cylindrical post coming from the top of the IMPATT. passing through the dielectri c. and contacting the top metal. Periodic slits in the top metal form a secondorder grat ing which ac ts as the antenna.
ROS I::>. arc ex trapo lated to c . the powe r coupling coefficient for the 11 guidc (TM 1guide) is found. M ca~u rc m e nt s were made in TM Iguides with and w ithout a grating 11 9 1. On structures w ithout the grating we measured the power coupled to the guides by inserting a tapered sec tion of T M Iguidc into a standard I ·band measurement fix ture based on a rectangular waveguide. Record power was achieved: 427 m W at 62 G Hz. as measured w ith a standard I ·band metal
Jr0.
1\lca!.urcd (a) and compuacd (h ) far lidd' ( .::,. /
=
!JO
and
0 .:)< III I) .
waveguide test bed. This is a factor or four higher than power from oscillators that were fabricated from ei ther dielectric waveguide or image guide. We compared this power to the power achievable by the same IMPATT in a reduced height coaxial metal cavity and found the ratio to be 0.65. which cl osely corresponds to ou r ca lcu lation of 0.62 as shown in Fig. II. V. 2 D ACTI VE ANTE NA UTILIZING
IMPATT DEVICES 11911281 Two IM PATT diodes have successfully and repeated ly been mutually locked to one another in one TM I guide with a 939c grati ng. Fig. 12 shows a comparison of measured and calculated farfield pallerns. The ca lculated far field is based upon the theoretica l ncar fie ld pauern for a single propagating bound mode w hich va ri es as featuring three l l\I PATT diodes osci llating at the frequency o f 61.7 G Ht. was performed. In these tests, all three IMPATT devices in the linear array
ROSEi\' N a/.: 1:\\'I:S JIG \TJ0:.1 Or ACTIVE A:\TE:'\:'\A AR RAYS
111 1
10.0 dBm div
j Fig. IJ.
O nly
th ~ l~ ft
o'cillator o r
t h~
three
d~ m ~n t
linear array i' bia,etl.
Fig. 16. ano ther.
Al l three clement\ or the linear array arc bia.,cd and locked to one
Fig. 17.
:!D array.
II
!
!
ll · 61r.:
. l
PEA
,
'
!l:Hz! 3 .00 Bm j
[
'
/
I
·

==.
i

I

II
!
I


.
""""" 

i I
!
Fig. 1l. The right a nd left mc illator> o r the three clemen t li nc:u· a rray arc IO well a'> the gratings· un i formity arc very cri tica l.
VI.
A
LJ,
EA R A CTIVE ANTI: NNA U TILIZING PH E i\IT A
FIRST STEP FOR M 0 1 OUTII IC ARRAY CONFIGURATIOi'\
As a consequence o f: I) the dirticuhy in providing a single mai n beam when using an IM PATT dev ice between two sets
II: EE TR ANSACTIONS ON WCROWAVE THEORY AND TECII;./IQUES. VOL. 4J. :.:0. '1. SFPTF~IBER 1995
of grm ings (when the gratings arc radiating off the Bragg condition). 2) the increased difficulty in processing IMPATI devices utilizing the MM IC approach. and 3) the availability of potcll!ial high efficient high power PH EMT s. we have inves tigated the poss ibility of replacing the IMPATI devices by PH EMT devices. PH EMT s arc making rapid progress in millimeter wave power amplifier applications. and very encouraging results arc being reponed for as high as 94 Hz 1291. 130 1. TRW 1291 has reported a two stage amplifier with be11er than 370 mW output power. a compressed gain of 7 dB and greater than I I % poweradded effi ciency over the frequency range of 59.5 63.5 GHz. With low loss planar combiners. these modules produced an output power of 740 m W w ith a power gain of 11 .68 dB. V II. M IC ROSTRI P B ASED GRATINGS
I
COS / 111 1
=
.
z
:Jo
•
N,
= {1 + jn
I
C)
·60
z iblc for re,earch ami developmelll in the ar~a' of millimeter·\\ a\'e d~vicc;, and circuit' and microwav~ optical intcr:oction. lie hold' an appointon~nt as Adjunct Profe"or in rhe Dep:ortment of Electrical and Computer Engineering. l ie al\o hold\ the title of A;,.,ociate in Medicine at Jcf"fcr;,nn Medical College in Philadelphia. The author of more than 110 technica l paper>. lhc editor of 2 lechnical booh. and a contributor to hooks in the field' of engineering and medicine. he holds 15 U.S. patelll\ in the field, of engineering and medicine. Dr. Rthcn is the recipient of a 1972 and a 1982 RCA Laboratoric;, Oul\tanding Achic1cment Award. the 1\l icro\\:1\c Pri;e at the 16th European tl icrowavc Conference in 1986 for coauthoring the paper entitled ··optical Beam Control of tlillimetcr \Va1e Ph:"cd Array Antenna; for Communication.,:· a 1989 David Sarnoff Re,eareh C~rner Technical Achie\'cment Award for conception and implementation of highpower ortic:olly activat~d \cmiconduc tor switc hing rechnology operating in the frequency range from DC to G il t.. a 1989 IEEE Region One Award for
Paul ,1 . S tabile (S"75 "1"79 St1" 88) rccei1ed the B.E.E.E. degn:e summa cum laude from t>lanhattan College in 1979. the :'1\.S.E.E. and Ph.D. degree' from Rutger' Uni\er,it} in 1982 and 199::!. rcpcctively. In 1979. he joined the Da1 id Sarnoff Re,carch Center (formerly RCA Laboratorie,). Princeton. NJ. IIi' early accompli,lunent,. a' a 1\lcrnbcr of the Technical Staff. included re\carch and development of the fir't monoli1hic \ilicon millimeler\\:1\e ;morce'. amplifier' and control circuit'. and novel optically controlled microwa1·e de1 ice' and circuits. :'I lore recent accompli'>hmcnts included re,earch into a ne\\ dc1 icclateral It IPAn· diode' (for hi' doctoral di\Scrtation). millimeterwave surfac~ emitter'\\ ith optic all~ controlled grating.. and high power. la\crfircd 'cnoiconductor '" ilchc'. Since 1992. he ha; lead th~ development of new product\. a' llead of the Elcclronie Product\ Group at Sarnoff. Product;, include infrared and 'isihle camera systeom. comoncrcial h:o,ed video sy>tcm'. and information \ccurity technologic>. lie ha' authored and coauthored over 60 technical paper'. Dr. Stabile holds \even patents and i> a mcmb~r of Tau Beta Pi and Eta Kappa u. He wa, founder and Chair of the Princ~ton Chapter of the IEEE l.a.o.:r' and ElectroOptic'> Sodct) and is a member of th~ IEEE Joumal of Lightwa\'e Technology Steering Committee. He a l'>o ha' 'ITI Tr:ul\actioro' Editorial Board. and the IEEE 1\ ITfS International Sympo,ium Technical Program Committee.
'ignificanl contribution\ to microwa,·e lechnolog) h) the ill\ention and of microwave balloon angiopla;,ry. and a 1990 Da1 id Sarnoff Re\earch Cenler Technical Achievo.:mertl Award for concei1 ing and proposing an optically controlled. s urface emitt ing. m illimeter \\ave sto.:erable :.ourc~. H.:: is a Fdlow of the IEEE. lie ha;, bo.:~n a member of the IEEE r rrS Technical Program Committee si nce 1979: he i~ i\ ITIS Technical Committee Chairman on Biological Effects and Medical Application\: a m~mbcr of the IEEE ,'vrtTS To.:chnical Committee for LightWave Technology: 1\\\ociato.: Ediwr of th~ IEEE Jo L:K:;"t. ot LIGIII · W WL T r;cn:;ot.cxn (J 1.:n: and a memher of the Editorial Board of 1\ licrowa\e and Optical Technolog) L~tto.:rs and of :'1 I icrowave and Guid~d Wm e Lo.:ttcr;,. He ha> '>en ed on th~ To.:chnical Conunitt~e for the IEEE International Conference on 1\l icrowa'c' in "lcdicinc held in Belgrade. Yugo,l:l\ia. in April 1991. and on the Prugram Committee for clectrolllllcrnational 1985 :oml 1991. in which he chair~d the tlcdical Scs,ion\. l ie i:. a M emhcra t Larg~ of the IEEE llealth Care Engineering Policy Conm o ittc~. and has \ervcd o n the IEEE Educational Activities Board: and he hold' the tit le of Profc">ional Engineer in the Province of Bri ti'h Columbia. Canada. II i~ biography ha' been selected for inch"ion in t>larqui' \VIm", Who in t h~ World. d~vcloprncnt
ROSEN eta/.: I 'VESTIGf\TION OF ACTIVE ANTENNA ARRAYS
Aly E. Fathy. (S'82M' 84SM'92) photograph and biography not available at the time of publication.
Dean II. Gil bert. photograph and biography not available at the time of publication.
Daniel W. llcchtlc (M'90) received the Ph.D. in physics from the Universi ty of Colorado. Boulder. CO. in 1978. Since 1978 he has been a Member of the Technical Staff at the David Sarnoff Research Center, Princeton. NJ. He has designed many MM IC and discretedevice power ampli fie rs and has been involved with lowpower component development from 160 GHz. He has also designed lownoise optical receivers with stateoftheart sensiti vities. highdensity laser arrays. GHzrate fiberoptic links. He i>. currently involved with designing components and power MMIC s for wireless applications.
Walter i\ 1. .I anton served in the U.S. Army as a Communications Technician from 1939 to 1 9~5. In 1 9~8 he graduated from the Baronian School of Horology. Philadelphia. PA . From 19531959 he was a Senior Research Technician at Bell Telephone Laboratories. Murray Hill. NJ in the Model Tube Laboratory. In 1959 he transferred to SFD Research Laboratory (Varian As>.ociates). He joined RCA Laboratories in Princeton. NJ in 1972 as a Research Technician . He has worked on the electron tube injected beam transistor program and cathodeluminescent display devices. For the past ten year.. he has been responsible for the mechanical design and fabrication on the millimeterwave. pin. and varactor diode projects. as well as laser arrays and their packaging. He has coauthored 30 technical papers and six patcms.
Francis J. i\lcGinly graduated from Pennco Technical Institute in 1978 and received the Associate degree in electronics fro m Bucks County College in 1980. From 1974 to 1980 he worked in Research and Development at Milton Roy Co. whe re he assisted in development of nowcontrol instrumentation. He received a commendation for his contribution to dialysate metering technology. In 1980. he joined the Microwave Technology Di vision at David Sarnoff Research Center where he presently works as an AMTS. He has been involved in the development of microwave solid state power amplifiers and receivers and has contributed in the development of high definition TV and microwave video di>.tribution systems. He is coauthor of several publications. rvtr. McGirll) has received five tcchnical commendations including an individual Sarnoff Technical Achievement Award in 199~ .
2115
J erome K. Butler (S'59M'65Si\J'78F'89) received the B.S.E.E. degree from Loui>.iana Pol) technic Inst itute. Ruston. and the M.S.E.E. and Ph.D. degrees from thc Uni versity of Kansa>.. Lawrence. He was a Research Assistant and held a CRES Fellowshi p at the Center for Re;carch in Enginecring Sciences. Uni versity of Ktubas. In 1965. he joined the faculty of the School of Engineering and Applied Science. Southern i\ lethodist University. where he is now University Distingubhed Prof'e"or and Chair of Electrical Engineering. I lis present research is concerned with the application>. of waveguide gratings u>.cd in semiconductor lasers and millimeter wave devices. He wa>. an 1\\sistant Professor from 1965 to 1970, an Associate ProiCssor from 1970 to 197~. and in 1 97~ he was promoted to Professor. He Wl" appointed University Distinguished Professor in 1993. From 1987 1990 he was the Associate Dean and Director of the Graduate Division. Since 1990 he has held the appointment of Chair of the EE Depanrnent. 111.! has wriucn numerous hooks and journal articles in the areas of solid state injection Ja,ers. liher optic telecommun ication systems. and quantum electronics. He coauthored the book Semico~tductor Lttsl' rs a11d Hetemjrmctio11 LED's . This hoot.. was one of the first research texts pertaining to semiconductor laser>.. a major de' icc used in fiber optic communication systems and CD players. In 1977 he was given the Southern Methodist Universi ty Sigma Xi Research Award. In the sumrncn. since 1969. he has been a Staff Scicnti>t. David Sarnoff Rc>.earch Center (formerly RCA Laboratories). Princeton. NJ. He has held corNrlting appointments with the Central Research Laboratory of Texas Instruments. Inc .. the Geotechnical Corporation of Teledyne. Inc.. Earl Cullum 1\;,>.m:iate>. of Dallas. Texas and the University of California Los Alamo' Scientific Laboratory. Los Alamos. New Mexico. Dr. Butler was elected a Fellow of IEEE for contributions 10 semiconductor lasers and the theory of radiation characteristics of optical waveguides. Ik is a me mber of Sigma Xi. Tau Beta Pi. Eta Kappa Nu. and is a registered professional engineer in the State of Texas.
Gary A. Evans (S'69M'75SM'82 F'92) was born in Omak. \VA. He recei,·ed the B.S.E.E. degree from the University of \Va>.hington. Seaule. in 1970. and thc M.S.E.E. and Ph.D. in elcctrical e ngineeri ng and physics from the Ca lifornia ln;,titute of Technology. Pasadena. in 197 1 and 1975. respectively. He worked for R&D As>.ociate>.. !\Iari na Del Re). CA, and w;t> a Visiting A"istant Professor in the clectrical engi necring departmcnt at the Univcr;,ity of Washington from 1977 to 1979. I k has worked at thc Aerospace Corporation. El Segundo. CA. from 1979 to 1981. TRW. Redondo Beach, CA. from 198 1 to 198~. and RCA Laboratorics (now the David Sarnoff Research Center). Princeto n. NJ. from 198~ to 1992. In 1992 he joincd Southern Met hodist Uni versity a>. a Profe"or in the Electrical Engineering Department. Since 1979 he has worked primarily on the design. growth. and fabrication of conventional cleaved facet and grating surface emiuing semiconductor laser>.. lie has over 170 publishcd article>.. and is a co·cdi tor of the book Surfacl' Emi11i11g Semiconduc/Or Laser.\ (New York : Academic). Dr. Evans wa; elected a Fellow of the IEEE for "contrilnrtiom, to the development. fabrication. and understanding of semiconductor lascr\ ... I k is a licen>.ed professional engineer. has served on numerous IEEE commillee>.. is a past Chai rman of the Princeton La>.ers and ElectroOptic Society (LEOS). a past Chairman of the Santa Monica Bay Section of the IEEE. and is an Associated Editor of the IEEE Jm•K::At. OF QL•,\::TDt Et.Ect RO::tcs.
IEEE TRANS,\CTIO:\S ON
~ II CROWAVE
THEORY AND TECH:IJQUES. VOL. H :10. 9.
SJJ>TJ:~ I BER
1995
Improved Analysis and Design of CoupledLine Phase Shifters Charles E. Free and Col in S. A itchi son, Fellow. IEEE
AbstractAn analysis of coupledline microstrip phase shifters is presented which shows that significa nt difl'crcnccs in theoreti ca l performance arc obtained by using a n exact a nalysis in terms of odd and even mode propagat ion veloci ties rather than the approach in which the velocities arc averaged. ~ l eas urcd data arc presented and compared with theo ry over the frequency range 8 12 G Bz and the agreement wit h theory is good .
I.
INTRODUCTION
HE CO FIG URATIO o f a conventi onal coupledline phase shifter is shown in Fig. I. The circuit in troduces a transmission phase change between ports I and 2 w hich is a funct ion of the coupled leng th. L ... Thi s form or circui t was employed by Schiffma n Ill as part of a broadband 90° phase shi fter. However. Schiffman' s origi nal work was based on stripline transmission structures. where the odd and C\·cn modes propagati ng along the coupled lines have equal phase veloci ti es. Thus Schiffman was able to make usc of the we ll known expressions ror coupledline fi lters developed by Jones and Bolljahn 121. W hen this type of circuit has been designed in microstrip the same transmission equations are usual ly quoted, and the unequal odd and even mode ve loc ities averaged to provide. theoretica lly. a wel lbehaved characteristic with zero i nsertion loss at all frequencies. In thi paper a more exact analy i is performed. in term s of the independent odd and even mode phase changes along the coupled section and shows that for certain values or the electrical length, L,.. the inserti on phase departs significantly from the ideal (average mode velocity) characteri sti c. and the device presents a significant mismatch at the input port. Some authors. notably Schick and Koh ler 131. have recognized the problem and suggested modifications to the basic design to compensate for the difference in the odd and even mode velocit ies, bu t there docs not appear to have been any ex tended theoreti cal consi deration of the simple coupledline section to show the ex tent of the problem. There is a further problem which doc~ not appear to have been addressed in the literature. namely that of establi shing the actual coupled lcng lh th at should be in silllations where the circuit designer is inhibi ted from using the familiar chamfered entry by other circuit considerati ons.
s
or microwip couplc.:dli nc
Fig. I.
Con fi guration
phase 'ohifter.
Fig. 2.
Coupled micro,trip lines showing pon configuratillll,,
T
1\lanuscript
r~c~iv.:d
Novemh.:r 14. 1994; rcv is.:d May 25. 1995.
C. E. Free is with the School of Electronic Engineering. M iddl
v,
Fig. 6. Difference between exnct and approximate theoretical insertio n phase at 12 G Hz.
It is shown i n the Appe nd ix that if the exact analy sis i s employed the input impedance at port I is frequency dependent and given by
from which the transmission phase change i s obtained as ,;.. _ 'I' 
1f 
2
. I [Zo( Zoo + L,lu ,.,
t an flo Zor
('OL
2Z 0 , Zoo Lau ()" cot. (),
0,.)].
(T he detailed deri vations of (4) and (5) arc given i n the A ppendix.) I f it i s assumed that the odd and even modes have equal ph ase velocities. i .e., 00 = 0, = 0. then (5) reduces to
r/>
= ('OS  1
(6)
which is the form quoted by Schi ffma n, and usually employed as an approx i mation in microstrip designs. The approx imati ons usuall y made in coupledl ine ph ase shifters al so extend to the match of the ci rcu it. w herein it i s assumed th at the i nput impedance is given by
Zo
= VZocZoo ·
(7)
I ll. COMPAR ISON OF EX ACT A D APPROXII\1 1\TE T H EORY
Zo,  t a u•J (} ] 
';~;fo:!!o_ _ __ Z.,,. [ +ta u., f.,'' Zoo
2Zm..Zoo
Fig. 5 shows the theoretical inserti on phase. compu ted as a function of the coupl er length at 12 G Hz. which results from usi ng the approx imate and exact methods o f analysi s. It can be seen that the ex act respon se depart s signi ficantly from the approximate characteristic for cert ai n lengths and. as wou ld be expected, the di fference tends to i ncrease with the length of the cou pled section. T his is demonstrated more clearly in Fig. 6. where the magn itude of the of the diiTercnce between the responses has been plo11ed as a function of the length. and shows that the el i ffcrcnce can be as large as 60° .
IEEE TRANSACTIONS 0
"'.... ii'i ::? Ill Ill
~ IICROWAVE
T II EORY AND TFCIINJQUES. VOL.
Tt:.ST CiRCUIT Dt~ll: :IStO:IS
"' ..;
NO 9.
SI : PTE~IR ER
TABLE I Till! SY.\lllOt s \ Rt Dt tt'l.D " Ftc.
1111111
L,"·"'
'"
s
1
759
26
4 65
100
CIRCUIT
"' .;
~ 3.
L
2
1257
47
4 76
95
"'
3
178 4
30
47 0
103
4
4650
52
477
:o9
'
.,;
5
6672
5~
469
:09
J99S
I
c
~
. .. ..
:;:; Ill
c
N
.. ..;
~
6.0
7.0
..
8.0
Ill
Length (mm)
{
Fig. 7. Theoretical in,cnion lo" for pha'e 'hiftcr at I:! G il t u;ing exact
c
..
~
anal~ '>i'>.
:;:;
. '
Ill
c
The approximate theory assumes that the match o f the ci rcuit is perfect. and frequency invarian t. It can be seen from Fig. 7 th at the exact ana lys is predicts significant inserti on loss, due to circui t mismatch. As would be expected the highest loss occurs in the same region as the highest phase di ffe rence.
"'
~~8~.~o~ s .~o~~~~.~~~171.~o~12~.·o
Frequenc!l ymmetricaJ shielded micrOstrip (/t 1 lt2 It) . In thi s case a finite anal ytica l e.xpre..,sion for the line capaci tance is a\'ai lable 1161
=
=
The parameter b i~ again g iven by relation (27) and hence the capac itance o f the line is g iven by relation (5).
C. Srmcr11re C
_
To co mpute the integrals entering in formu las (7) (9) we have wrillcn
• (I
B. Srmcr11re B
COS II
I + sin 11
Hence the capacitance of the line is g iven again by formula (5) .
(29)
The capac ity of the line. and hence the c harac teri ~ ti c impedance. can be determined by usi ng rormulas gi ven in Section II.
J._ ( Ill
('. nu l
. = 2(1
/,· (k)
1
+ 12) /\"( k ')
(36)
where
1.
= tanh ( ;.iJ) /t . 2
k
1
= v~ I  k.
(37)
Here /,·(/.) i s the complete ellipt ical in teg ral of the lirst kind.
IEEE TRANSACTIONS ON .\IICROWAVE TIIEORY AND TECIINIQlJES. VOL.
TABLE I
c
C(exactly) .969823E+00 .134626E+01 .187554E+Ol .288224E+01 .488254£+01 .888254E+0 1 .108825E+02 .142160E+02
NO.9.
C 1(app1'x) .969826 E+ 00 .134627E+ O1 .187555E+ O1 .288224E+01 .488248£+0 1 .888211E+01 .108805E +02 .142072E+02
Rel.en·. .26E05 .37E05 .SOE05 .21E05 .12E04 .49E04 .19E03 .61E03
]
,,
eps.rel.=l. · ·  eps.re1. ~12.
l l
l I
I
DO\! AI" l\ IE'IIIOD (CE). \:'\D St:llSTRII'l\ IEl 1101> (55) 1:'\ Till: Cll \R\ClHUSTIC bii'EI>,\:\CE CAI.Cl: L.HI():'\S
CE
134.72 134.78 112.50 112.58 90.385 90.482 72.789 72.892 61.885 61.987 42.293 42.376 26..t54 26.503 12.726 12.745
ss
PM
CE
ss
134.63 112.43 90.325 72.74 1 61.845 42.267 26.438 12.717
109.01 90.952 72.975 58.676 49.821 33.934 21.143 10.125
109.06 91.020 73.054 58.761 49.904 34.001 21.183 10.140
108.94 90.891 72.927 58.638 49.789 33.913 21.131 10.118

I
I /
I I
.l I
20 10
TAB L E II CO~II'ARI SO:'\ 01' II IE P ROPOSED l\ IETIIOD ( P l\1). SPEC IRAL
PM
1995
I
·~
w/ h 0.1 0.2 0.4 0.7 1.0 2.0 4.0 10.
SEPTE~ I BER
ECr~~~~~~
c
CO~IPARISO:'\ Ol· liiE (EXACT) II'ITII (APPR) 1:\ THE CASE o1· Snt~IETRIC,\L CovERED STRIPu:.:~:
h" = h/( 2b ) 5.000 2.000 1.000 .500 .250 .125 .100 .075
~ 3.
~
%~==~,;0:::;2;0==~~0~===40~==5~0====E~0===:7:0==~e~o_jro (teta·alfa)/2
r:ig. 7.
Change of the capacitance with strip width.
It is easy to sec that the proposed formul as (5){ I 0) yields calculated impedance val ues which lies between the values obtained by CE and SS methods: the maximum relative error wi th respect to the values gi ven by substrip method is less than 0.07o/c and the comparison wi th values computed by spectral domain approach gives a maximum difference less than 0.2o/c .
TABLE Il l CO~II'ARISO:'\ Ol· liiE PIWI'OSED M ETIIOI> (PM ). SI'I:CII(I\L
DO.\tM:> l\I ETIIOI> (C E ). A:'\1> 5UilSTtW' M ETHOD (55) I:> TilL CIIAR,\CTI· RISll(' bti'EDA:'\CE C\1 Cl' I.,\110:'\S
w/ h 0.1 0.2 0.4 0.7 1.0 2.0 4.0 10.
PM
CE
ss
PM
CE
ss
94.670 78.955 63.312 50.870 43.166 29.357 18.258 8.7260
94.718 79.015 63.381 50.943 43.238 29.415 18.282 8.7392
94.605 78.902 63.270 50.837 43.139 29.338 18.247 8.7 197
65.578 54.658 43.787 35.143 29.792 20.212 12.536 5.9720
65.612 5.t.699 43.835 35. 194 29.843 20.249 12.555 5.9808
65.534 54.622 13.759 35.120 29.773 20.200 12.528 5.9678
We compared the values for the capacitance given by proposed formulas (5)(9) wi th the fi nite exact capacitance given by relation (36). The results arc gi ven i n Table I. II i s to be noticed that the results given by the new formul as are better than those obtained in 1131. In fact the maximum re lative error h/(2b) 0.075) instead of 2 2D 2 I A. where f) is the largest linear di mension or the
11: 1:1: T RA:\SACIIO:IS 0:<
~IICROWAVE
T IIEORY AND TECHNIQUES. VOL. IJ. ,'10. 'l.
S l : l'fE~IBER
l'l=
5 ·25
20
w
oc ~0 +~~~~T~ 0
30
60
90
120
150
30
180
9 .0
ANGLE (degre es)
9.5
10.0
10 .5
11 .0
FREQUENCY (GHz)
Fig. 'J. Thcon:tical and c'pcrinu:mal Eplanc pallcrrh for a :1 x 3 broadband amplifier.
l"'ig. I I. Gain vcr't" frcqucnc) plot for a 11\'o>tagc 3 x 3 naml\1 band :unplifkr with ncarficld 'cparalion between the two '!age,.
the maxi mum. However. for some system appl icati ons the multi stage spatia l ampli fier should be viewed as a " black box" without correcting for the I / r2 loss due 10 the separation of indiv idual stages from each other. In this case. the ncl gai n should be cal cu lated usi ng the procedure descri bed in Section 1A for a single stage ampl ifier by using (I) or (2). The path loss between 1hc different ampli fier stages is cons idered 10 be inherent to the spatial ampli fier and the terms com pensating it in (3 ) and (+) should be neglected. Following thi s approach the ca lcu lated ncl gai n at I 0.0l G Hz is only 0.34 dB and 10.05 dB using H )l. One possible way 10 improve these gains is 10 usc lcnsc. 10 focus the beam from one stage onto the nex t stage. Another possibl e soluti on is 10 place the spatial amplifier stages w ithin their ncarfield regions. 10
'2 ......... .,_..
(a)
(b)
Fig. 10. 1\l ulli,lagc >p:llial an1plificr mca>urcmclll 'clllp. The >lagc> arc placed in the farficld of one another. (a) Gain mca,urcmcnl (mca,un:mcnl of Pr). (b) Calibr:llion (nH.::Nircmcnl of P. ).
circuit ) as shown in Fig. 10. In order lo determine the gain of a cascaded spatial amplifi er ( I) and (2) were ex tended for the general case or 11 cascaded stages
c:1c:2 ... (: II _ 1I..,,.' ,.
rfr§
( I . 1 + 12 . )''
11  l
c ....'t ( ,' I (,'2 ... ( ,, II

,_,
II
II
('"):! c,·1 A1
iul
)
( l7itlj)._
II ~? c c·
;= I
' 
1
0111)
rT r§ P,. ..,...'=....,...,.,. ~ ~. 2
tage amplifier utiliting ncarficld coupling.
IV.
CONCLUS ION
Narrowband and broadband quasioptical amplifiers on double laye r microstrip circuits have been demonstrated. The common ground plane between the input and output ports provide effec1i ve signal isolation and dev ice heat sinking. Also the ncarfield coupling of the transmilting and recei ving horn s to the spatial amplifier was demonstrated. The results obtained from a twostage spatial amplifier by employing the ncarfield coupling of two si ngle stage amplifiers were al so reponed. This paper demonstrated the feasibi lity of obtaining net gain through the ncar field coupling of spatial ampl ifiers. A detai led study of the ncarfield coupling between the stages is underway.
IIBT grid amplifier: · in lEE/;" M7T S Int. Micmu·m·e SymJ>. l)i~ .. J une 1993. pp. 6 15618. C. Y. Chi and G . M. Rebcit. ",\ qua>ioptil::li amplifier:· 1/:"J:"/:" Micmum·t• 11111/ Guided \\'t11·e Leu.. , ·ol. 3. no. 6. pp. 161 166. June 1993. T. !\ Iader. J. Schoenberg. L. Hannon. :uul Z. B. Popm ic. "Planar 1\ t ESf"ET transmis>ion w:l\'e amplifier... l:"lt•ctrmr. Leu.. \ol. 2?. no. Ih include analog IC de;ign. Ml\IIC de>ign and communication theor).
ACK OWLEDGM E T
The au thors wo uld like to th ank Fuji tsu Corp. for donating the H EMT" ~ and Rogers Corp. for donating the substrate material. R EFERENCES Ill 1\1. Kim. J. Ro>cnberg. R . P. Smi th. R. M. We ikle. J. B. Hacke r. M. P. Dcli'>io. and D. B. Rutledge. "A grid amplifier." JED:" Micrm•·r,.·e {Ill(/ Guided \\'ttt'l' i..t'/1.. ,•ol. I. no. II. pp. 322321. ov. 1991. 121 ~1. Kim. E. A. Sovcro. J. B. llachcr. 1\1. P. De Li.,io. J. C. C hiao. S. J. Li. D. R. Cagnon. J. J. Ro.cnbcrg. and D. B. Rutledge. "A IOO·ch:menl
Amir lu r tazawi (S"87l\t"t)()) wa' born in Iran in 1962. l ie received the B.S. degree in electrical engineering from the Stale Uni' er,ity of Ne\\ Yorl. at Stony Brook in 1?87 and the ~I.S. and Ph.D. tkgrec' in electrical engineering fromlhc UniH:I'\il~ of Tcxa\ al Au>lin. in 1988 and 1990. re,pccli' cl) . In 1990. he joined the Uni\er,il) of Central Florida. where he i> currcnll) an i\"ociale Profc"or. ll is research inlere.1> include rnillimelcrwave power combining o'cill:nor' and amplifier,. qua 0 increases. consistent wi th the fact that wavefronts that contribute at later porti ons of the scat tered waveform must tra ve l longer distances from the input plane th an contribu tions arri ving at earl ier Limes. As demonstrated by Fig. 4. in which for simpl ici ty we consider the special case or TE,,0 modes (for which the modal rays travel in the .r;; plane). (I 0} further implies that as time proceeds the wavefront seen at the observati on point origi nates from different localized portions of the input fi eld profi Ic.
IEEE TRANSACTIONS ON
~ II C R OW;\VE
TIIEORY AN D TECI INIQL'ES. VOL. 43. :
0.50
"
:9 0.00
'0
.~
l
ti:
L)
0.2S
·c:;
H
UJ
\v
0.00
0.25
0.50
~
~
I ~
\
0.05
900
1000
1100
1200
1300
1400
1500
1600
1700
Time ( ps )
100
100
l"ig. 6. Calculated timedomai n fields observed 30 cm from the op.:n end of a rectangular due In pul sed planewave exc itation as in l"ig. 3 (11, = l:i 0 ). The ineident pulse shape is described in l"ig. 5 and th.: wav.:guidc dimensions arc " 1.55 em and /, 1.55 cm. The solid curve rcpr.:sents 1h.: r.:suli s of an FFT of ( II ) while the points were calcu lated by the asymptolic expression in ( 14 ).
Time ( ps) (a)
=
Spectrum IS
=
TE 10 mode 

0.0.1 .
E
>
900
1000
I""'
1100
1300
1400
Time ( ps)
""'
1000
1700
(a) n ::,. mode
O.Ol r~~~~~~.;..=!!!!..::;::.:, 20
40
60
80
100
Frequency ( GHz)
(b)
E
>
~
~ .o.oo i..vw.M!IIIItltl
i.i: L)
Fig. 5. Pul se shape and speel rum for 1h.: Raleigh wavclel 11g1 used as lhe incident waveform in our calculat ions. (a) Timedomain pulse, (b) spectrum.
~ .O.Ol""''S;;;;;;;~::v."""7;..~;;;;~;;.,,;;;;,;j=:,; 1000
1100
1:00
1300
1400
I :>Oil
1600
1700
Time ( ps)
Il l. EXPERI M ENTAL R ESULT S i\ND
(b)
T IME FREQU ENCY PROCESS ING
To demonstrate thi s phenomenology experimental ly we have performed optoelectronic measurements in which freely propagating shortpulse radiation is generated by switching planar antennas photoconduclivcly. We usc coplanarstrip horn antennas fabricated on oxygenbombarded si licononsapphire wa fers. The opti ca l pulses arc generated by a mode locked CW Nd YLF laser which is pul se compressed and frequency doubled to produce approximately 4 ps duration green pulses (A = 527 nm ) at a 76 M IIz repetition rate and 150 mW average power. The optical pulses arc split into pump and probe beams, with each beam subsequen tl y coupled into si nglemode opti ca l fiber. The opticalfiber fed pump beam i s used to switch a debiased antenna, generating a freely propagating burst of radiation with instan taneous bandw idth from 15 75 G IIz; the probe beam is time delayed w ith respect to the pump beam and is used to time gate the waveform on the receive an tenna,
T E,. mode 0.01
8
>
l
~
'0
u
o.oo
i.i: L) '5
4
•t . ••
~
Ul
.0.01 900
1000
1100
1300
I""'
1400
I :>Oil
1600
1700
Time ( ps)
(c)
l"ig. 7. Timcdomuin fi elds for 1hc T E 10 , T E:w and T Er.o modes for th.: geometrical paramct.:rs and excitalion puis.: considered in Fig. 6. The solid curve represents the results of an I"I"T of ( II ) while the points were calcu laled by I he asymplolic ex press ion in ( 14 ). l ndicalcd by arrows arc I he approxi male modal lurnon t imes as gi ven by Fig. 8. (a) TE to. (b) TE:w and (c) TE;,o.
its optical path length being adjusted by mirrors placed on a precision co mputercontrolled translational stage. The time
~148
IEEE TR A!\S ACTIO:.IS 01\ ~IICROWAVE TII EORY A:: D TECII :.IIQUES. VOL. 4}. ::0. 9. SEPTE~ I B ER 1995
TE 10 TE,. TE,
2E4 Q)
u
~
C.t E4
E
~
2 500
.;

.
.:: 5E3 u
Q)
a.
'§'
..
en
·c:
TE
::::l
25
.ri
~
u
Qj
TE,.
u::
50
75
tOO
Frequen cy (GHz) 0
c
Q)
u
·;:;
.s
TE,o
1000
1200
1400
1600
1800
2000
Time ( ps)
Fig. 8. Ti m~depe n dcn1 modal di; pers io n curve s described in ( 13a ) and ( 13b) for timedomain TE..,o modes in a r~c l an g u lar waveguid e: the physical parameter; arc as considered in Figs . 6 and 7. Shown dashed is the pulse spectru m of the incide nt wa veform. with amplitude uni ts as d escri bed in Fig. 5. T he arrows indicate the ap prox imate time; at which the respective timcdc pc ndcm mo dal dispersio n cu rves first reside within the exc itatio npulse spectrum. idc mify ing the approximate moda l wrnon time : these arrows arc also d isplayed in Fig. 7.
gated wa veform on the recei ve anten na is averaged by a lockin amplifier. Detail s about the experimental facil ity arc gi ven in [9]  I IIJ.
A. PulsedBeam Excitation In our fi rst set of measurements the pu lsed radiation was collimated using a fusedsilica hemi spherica l lens, generatin g a pulsed beam wi th approx imately linear polari zati on. These field s were made incident at B; = 15° on the open end of a rectangular waveguide wi th o = 1.55 em and (J = 1.55 em. and the electri c field was polari zed paral lel to the .11 axis (sec Fig. 3). We have demonstrated previou sly th at the beam is approxi mately uniform over a 3 em diameter and therefore for thi s incident polari zation we anticipate that on ly TE 111 o modes will be exci ted (no variati on in the y directi on). The field s were measured by placing a second copl anarstri p horn antenna inside the wavegu ide (Fig. 3) and drilling a small hole i n the side of the waveg uide through which th e probe beam wa directed via optical fi ber. Due to the smal l diameter of the optica l fiber (radius of 0.25 mm). the hole in the side of the waveguide and the fiber i tsel f introduced negligible perturbation to the waveguide fields. T he coplanartrip transmission li ne anten na feed consisted of 20 11 m wide lines with 10 t tm interstrip spacing: i n the horn region the maximum separation between the outermost edges of the two lines was less than 4 mm . Thus the anten na itself also introduced only a minor pertu rbation of the fie lds inside the wavegui de. T he waveguide ex tended in back of the probe antenna for a sufficient di stance such that no reflecti ons from the end of the waveguide were present in the measuremen t time window. T he incident pulse and spectrum are shown i n Fig. 9 (measured in the absence of the waveguide) and the results of the waveguide measurement are demonstrated in Fig. I 0. A s discu ssed above. the coplanarstrip horn antenna has been
·2500 0
50
100
I SO
Time (psec)
Fig. 9. Ty pica l n:fcrc nce pube a nd ib spectrum. measured in the absence o r the wa veguide.
shown previously to be approxim ately linearl y polarized. and for the measurements in Fig. I 0 the recei ving antenna was oriented such that Ey was measured. The results in Fig. I 0 arc presented as fol lows : the bottom plo t represents the measured fields E y wi th the probe antenna placed i n the center of the waveg uide z = L7 em from the open end. the left plot represents a numeri cal Fourier tran sform o f the entire measured waveform. and the center plot represents the results of a shorttim e Fourier transform (STFf) using a Gaussian w indow wi th a 90 ps 3 dB width. In the tim e frequency phase space (center) are al so plotted the timedependent dispersion curves gi ven by ( 13a) and ( 13b) for the TE 10 . TE:JO · and T E.>o modes. The resul ts of the ST Ffwhich demonstrate the timefrequency behavior o f the timedependent waveguide modesarc in close agreement w ith the predicti ons i n ( 13a) and ( 13b); however. due to the pul se bandwidth only the T E 10 and TE:.~o modes arc excited strongly. while the TE;;o mode is exci ted much more weakl y. Note that as ti me proceeds the frequency content o f the TE 10 mode approaches 10 G Hz. an d from Fig. 9 it i s seen that the incident pulse has little energy around thi s frequency. Thus. at earl y time s mode TE 10 i s exci ted strongly but as time proceeds its excited strength di mini shes as its instantaneous frequency lowers and mo ves below the spectrum of the incident pu lse. At late times the fields are represented al most enti rely by the T E:.~o mode. A comment should be made with regard to the results in Fig. I 0. When performing a pumpprobe optoelectronic measurement. one actu ally measures a convo lution of the actual fields (Ey in th i s example) with the impulse response of the photoconducti ve detection system. However. by exa mining the timefrequency resul ts of Fig. 10. w hich arc i n good agreement w ith the expectations of ( 13a) and ( 13b). we sec that the impul se response of the detector docs no t appear to corru pt the time frequency distribution of Eu signi fi cantly . To furt her substantiate thi s point. we deembedded the response of the detector by dividing the Fourier transform of the waveform at the bottom of Fig. I 0 by the Fourier tran sform of a reference pul se (such as the one in Fig. 9 ). y ielding the impulse response of the sys tem JI(w) over the system bandwidth. A new ti me
KRAU ,., a/.: SIIORTPULSE PROPAGATION IN A HOLLOW WAVEGUIDE
2119
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50 100 150 200 250 300 350 400 450 Tim e (psec)
Time (psec)
Fig. 10.
Measured timedo main waveform (bottom), numerical Fo urier trans
fo~m (lei'!). and ~horttime Fourier trans fonn (center) for the s tructure cons idered in Fig. 5. The fie ld Ey was meas ured in the center of the wa veguide :: = 4.7 cm from the open e nd, wi th 0, = ·15° and o = /1 = 1.55 e m. Also shown arc the predic ted timedependent dis persio n c urves for the TE to · TE:10 and TE~ o mode~ usi ng ( 13a) (solid) and (1 3 b) (dashed).
domain signal for the waveguide problem was calculated by multiplying JJ(w) by the spectrum of a synthetic pulse wi th band width consistent with our measurement. A fter repeating the STFf processing in Fig. I 0 on the synthesized waveform , we observed no appreciable change in the timefrequency phase space.
B.
Loca/i~edSource
Excilation
Our second series o f measurements were conducted wi th both the transmitting and recei ving antennas pl aced inside a rectangular waveguide w ith a = I em and b = 2.2 em. For these measurements no lens was used on the transmitting antenna, and therefore it represented a local ized source of pu lsed radiation. The transmitting antenna was placed in the center of the waveguide cross section, wi th its metall ization parallel to the (y, z) plane. The recei vi ng antenna was positioned at .r = a/5 and y= b/ 2, a longitudinal distance 10.7 em from the tran smitter; the receiving antenna was so posi tioned such that we could measure modes for which Ey has odd symmetry in :~: . The results for thi s exa mple are presented in Fig. II using the same format as in Fig. I 0. In Fig. I I we have also plotted the instantaneous dispersion curves from ( 13a) [( 13b) results from the assumption of planewave exc itati on, which is not considered here!; note that the T E,. 11, and TM,.,,. modes have the same dispersion relations for like (m, n) and therefore the dispersion curves alone cannot di stinguish the TE and TM modes. For both TE,.11, and TM,11 ,. modes Ev has a sin (m7r.r / a) cos(1myjb) transverse variation; because the receiving antenna is centered in the y direction and ofT center in the .r direction 'Ill E [1. 2, 3 .... ) and n E [0. 2. 4, ... ). In STFf timefrequency processing one uses a fi xedsize window functi on which is slid across the timedomain data. A t each window posit ion a Fourier transform is taken, prov iding information about the waveform's timedependent frequency con tent. The frequency resolution is dictated by the si ze of the timedomain wi ndow (in Fig. II we have used a Gaussian window with a 90 pscc wide 3 dB point). T herefore one must
Fig. I I. Meas ured timedomain waveform (bottom). numerical Fourier transfo rm (left), a nd s horttime Fourier trans form (center) for the receive antenna placed as in Fig. 5 but with the transmitt ing antenna placed ins ide the I 0.7 e rn from the de tector. Waveguide dime ns ions: a J waveguide :: e rn and IJ = 2.2 cm, location of receive antenna: (.r· = a(). !J = IJ/2). and location of tra ns mitti ng antenna: (.r· = af2. y = IJ/2). Also plotted arc the timedepende nt dispersion c urves predicted from ( 13a) forTE..," and TM .,.,, modes (11 1 and 11 arc labe led on the right s ide of the cente r plot).
=
=
compromise between temporal localizati on and frequency resolution. In Fig. II the frequency resolution is inadequate to distinguish between the (m, n 0) and (m, n = 2) modes, w here m = I , 2, and 3 (in the vicinity of 2 10 290 ps, however, the ( I , 0) and ( I , 2) modes appear to be disccrnabl e). To obtain better frequency resolution one must increase the w indow size, w ith the inevitable loss o f temporal resolution. The w indow size used for Fig. I I y ielded the best compromise bet ween temporal localization and frequency resolution and provided the best agreement w ith the timedomain dispersion curves predicted in ( 13a) .
=
IV.
CONCLUS ION
A n asymptotic analysis has been per formed for shortpul se propagation in a hollow waveguide. T he asymptotic resu lts demonstrate that timedomain waveguide modes arc characterized by timedependent instantaneous frequencies which are large at early times and decrease to the modal cutoff frequency wi th increasing time. T he asymptotic parametrization of the timedomain fields in term s of chirped waveg uide modes appl ies to waveguides o f arbi trary cross sec ti on, but has been examined here numeri cally and experimentally for the case of rectangular waveguide. In our numerical example of a pulsed plane wave incident on the open end of a recta ngular waveguide, the frequency domain waveguide modes were converted to the time domain asymptotically as we ll as via a fas t Fourier transform (FFf). Except for very early times. the asymptotic data were in nearly exact agreement with the results o f the FFf. Moreover, the asymptotic analysi s was also capable of predicting the approximate modal turnon times. The measurements were performed optoclcctronically usi ng a shortpulse laser to switch pl anar antennas photoconductivcly, generatin g freely propagating bursts of radiation wi th instantaneous bandwidth from 15 75 GHz. The first experiment considered pulsed radiation incident obliquely upon the open end o f a rectangular wavegu ide, w hile the second considered
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IEEE TIV\NSACTIONS ON ,\ I ICROWAVE T III' ORY AND TECIINIQUES. VOl ..
a pul sed antenna placed in side the waveguide: in both cases the fi elds we re measured optoelcctro nicall y using an antenna si lllated inside the guide. The measured timedomain fields were proces sed using a short lime Fourier transfor m. and the time frequency resul ts we re found to be in good agreement wi th the predictio ns or the asymptotic analysi s.
~.1. ;>;(), 11.
SI'PTDIIIER 19'15
I 161 L. 13. Fcl,cn and N. 1\ larcuvil;. Radiation a111l Scal/ering of Englcwood Cliffs. NJ: Prcnticc llall . 1973.
11 71 H. Ling. S.W.
L~ c.
and R.C. Chou.
" 1i ighf'rcqu~nc y
RCS
\I~II'C' I.
or op~n
cavilics with rcclangular and c ircular cross 't:clion:· 1/·J :'/:' Tmn1. Antennas Propagat.. vol. 37. pp. 648654. 1\ lay I9ll9. !l XI P. ll ubral and M. Tygcl. "Anal ysi; of the Ralcigh pul,c." Gt'Of>h.l'licl. vol. 54. pp. 654 65ll. I 9X9.
ACKNOWLEDGMENT David Kralj was horn 1\lay 14. 196X in ,\ Jar del Plata. Argcnl ina. l lc becantc a United Stall:' ci1i1en in 1992. l ie rcccivcd lh~ B.S. and i\ I.S. d ~~ rcc' in dt:clrical cngineering from Polytechnic U nf, cr,il~. Brooklyn. NY in 1993 and I 995. rcspcuivdy. He is currently puNting lhc Ph.D. dcgrcc al the sam~
The last author w ishes to than k Pro f. L. B. Fc lscn o f Boston U niver sit y for several stimu lating di scuss ions on highfrequency asymptotics and timedomain structural disper sion. REFERENCES I I I L. Brillouin. 1\im·c l'mfJagation and GmllfJ \lt·locitl'. New York: Actdcmic. 1%0. 121 J. D. Jacl,son. C/aniC'II! t:lcctmdynamic. t:h. 7. 2ml .:d.. t cw York: \\'ilt:v. 1'.175. 1.11 K. If Oughslun. J. E. K. Lattn.:ns. and C. M. Balit:lsis. "Asyntplolic dcscriplion of ck clromagncliL· pubc propagation in a Jincar dispt:r· si1
400 300 200
Fig. 4. Fourport microstrip example circuit. The physical distance between ports I and 2 is approximately 14 cm.
100 0 0
3
2
4
measured  
5
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v1 _slm  
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v1 _meas 
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I
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2l
400
>
>
20
.§..
300
"' .s"'
200
20
>
100
12pole  
80
(a)
soo
33pole   
100
time (ns)
s
10
1S
time (ns)
Fig. 5. Comparison o f measured Stt (I) step response with those generated by direct convolution from 12pole and 33pole impulse response approximations.
3
2
s
4
tim e(ns)
(b) vln_slm
soo
v2_slm  
Z0 :son
v2_meas 
,  
2
400
>
4
.§..
300
"' .s"'
200
>
100
son
0 0.0
o.s
1.0
1.S
2.0
2.S
Fig. 6. C ircuit used to generate simulation data for comparison with meas ured data from the fourport circuit in Fig. 4.
tlme(ns) v1 _slm v1 _m eas
(c)
Fig. 3. Comparison of s imulated and measured data for circ uit in Fig. I. (a) Opencircuit reflect ion at port I. (b) Shortcircuit reflection at port I. (c) Matchedload transmission at port 2.
was steep compared to the response waveform feat ures be ing modeled, the input step was taken to be an ideal step, and discrete differen ti ation was used to approximate the impulse respo nse. As shown in Fig. 5, 33 poles were require d to capture the detai l of the re flection wave form at port I, al though as fe w as 12 were suffici ent to model the maj or e ffects. A lth ough the 12pole response i seen to di verge sli g htly from the 33pole response and the measured response, both generate d responses converge to zero at de. The measured re fl ecti on converges to a small negative value at de due to a slightly mismatched load at port 2. In order to evaluate the acc uracy of the extracted mode l, it was inserted into the c ircuit shown in Fig. 6. Po rt I was driven by a step inpu t th rough a 50 input impedance, port 2 was termin ated by RL = 50 kr2, and ports 3 and 4 were terminated by 50 res istors. Fig. 7 compares simulation resul ts for reflection at port 1 and crosstalk at port 4 with measured results for which port 2 was unte rminated. The simulated and measured curves are nearly overlapping at the scale shown.
n
n
v4_sim 
v4_ meas  
600 500
> .§.. "' "' 2l 0 >
400 300 200 100
1;·,.,_.,.
0 100 0
s
10
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20
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Fig. 7. Ope ncircuit reflect ion at port I and crosstalk at port 4 for the circuit in Fig. 4.
V. CONCLUS ION
This paper presented a general approac h fo r characteri zing inte rconnect c ircuitry at the board, package, and MCM substrate levels using measured timedo ma in data. The technique uses the dominant scatte ring poles and residues extracted from TOR data as paramete rs for a circuit model, and can be aut omated. The model may be implemented in a SPICEbased simul ator, and is evaluated in linear time. The approach models delay and reflection introduced by interconnects, as well as crosstalk between multiple conductors of varying geometries,
IEEE TRANSACTIONS ON ~ IICROWAVE THEORY i\ND TECHN IQUES. VOL . 43. NO. 9. SEPTE~IB ER 1995
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as accurately as these effects can be measured. A s a result, large lumpedcl ement models arc not necessary to represent di stributed interconnect networks for which ti medomain data
11 3 1 G. V. Dc varayanad urg a nd M. Soma. " An interconnect mode l for arbitrary tenninations based on scattering parameters:· Analog lmegmted Circuits and Signal Processing . vol. 5. no. I. pp. 3 115. Jan. 1994. ( 141 S. Corey and A. T. Yang. " Inte rconnect c haracteri7at ion using time·
is available. Twoport and fourport example microstrip cir
domain rcncctomctry... in Pmc. IEEE Jrd Topical Meet. E/ec. Perf.
cuits were mea ured, characterized, and simulated, and the results were compared w ith measured data to demonstrate the val idity of the approach.
Electron. Packaging. Nov. 1994. pp. 189 19 1. (l S I E. K. Miller. cd .. 1imeDomain Measureme/1/s in Electromagnetin. New York: Van NostmndRe inhold. 1986. pp. 8793. 116 1 F. B. Hilde brand. Introduction to Numerical Analysis. New York: McGraw Hill . 1974. pp. 457162. ( 17 1 D. T ufts and R. Kumarcsan, "Estimation of freque ncies of multiple s inusoids: Maki ng linear prediction perform like maxim um likel ihood ... Proc. IEEE. vol. 70. no. 9. pp. 975 989. Sept. 1982. 118 1 G. A. Bake r. J r. a nd P. G ravesMorris. Ptule Approxima/1/.L New York: AddisonWesley. 198 1. ( 191 A. Semlyen and A. Dabuleanu. " Fast a nd accurate switching tra ns ie nt calculations o n trans mission lines with gro und re turn using recursive convolutions:· IEEE Tram. P01rer Apparall/s 5)·st.. vol. 94. no. 2. pp. 56 157 1. Mar./Apr. 1975. 1201 V. Raghavan, J . E. Bracken, ;md R. A. Rohrer. "AWESpice: A geneml tool for the accurate a nd effic ie nt simulation of inte rconnect pro ble ms:· in Proc. 29th A CMIIEEE Des. Automat. Conf.. Ju ne 1992. pp. 8792. 12 11 J. T . Yao and A. T . Yang. "Cons iste nt nonlinear analysio. based on improved harmonic balance techniq ues." in Pmc. 1993 European Design Automation Conf.. Sept. 1993. pp. 430135.
ACK OWLEDGtvi ENT
The authors would like to thank the A dvanced L aboratory Instru ments Test and M easurement group at Tektroni x for measurement equipment, test circuits. and helpful suggestions.
R Ef'ERENCES Ill A. Ruehl i. "Equivalent circ uit models fo r three d imensional multiconducto r syst em~:· IEEE Tran. Micronm•e Th eory 7ech., vol. 22 . no. 3. pp. 2 16224. Mar. 1974 . 121 A. E. Ruchli and H. Hcc b. "Circuit mode ls for threedime ns ional geometries inc ludi ng die lectrics.'' IEEE 7i·ans. Micmn•m •e 71teory Tech.. \'OI. 40. no. 7. pp. 1507 15 16. July 1992. 131 V. Raghavan et al.. "AWEinspired:· in Proc. 1993 IEEE Cu.\'10111 lmegrated Circuits Conf., May 1993, pp. 18.1. 1 18. 1.8. 141 P. Feld mann and R. W. Fre und, ''Efficient linear c ircuit analys is by Padc approximation via the Lanczos process." in Pm c. Eum pean Design Automation Conf. . Sept. 1994, pp. 170175. (5 1 F. H. Branin. "Transient analysis of lossless transm ission lines:· Pmc. IEEE. vol. 55. no. II . pp. 201 2201 3, Nov. 1967. 161 F. Y. C hang. "Transient analysis of loss less coupled trans mission lines in a no nhom*ogeneo us dielectric medium.'' IEEE Tran.1·. MictrJn•m •e Themy Tech .. vol. 18. no. 9. pp. 6 16626, Sept. 1970. 171 A. Djo rdjevic and T. K. Sarkar. " Analysis of time re~pon se of lossy rnulticond uc to r transmissio n line networks:· IEEE Trans. Microwm·e Theory 1eclt.. vol. 35, no. 10. pp. 898908, Oct. 1987. (8 1 13. J. Cooke. J. L. Pri nce. a nd A. C. Cangcllaris, "Sparame te r analysis of m ulticonducto r integrated circ uit interconnect syste ms: · IEEE 1i'{//t.\'. Compw er Aided Design. vol. II. no. 3, pp. 353360. Mar. 1992. (91 R. Kipp. C. H. Chan. A. T. Yang, a nd J. T. Yao, "Simulation of highfreq uency integrated c irc uits incorpo rating fullwave a nalysis of mic rostrip discontinuit ies:· IEEE Trans. Microwave Theory Tech. , vol. 4 1. no. 5. pp. 848854, May 1993. (1 01 D. S. Gao. A. T. Yang. and S . M. Kang, " Accurate mode ling and simtllation of interconnec tion delay and crosstalks in highspeed integrated circuits:· IEEE Trans. Circuits Syst. I & II. vol. 37, no. I. pp. 1 9. Jan. 1990. ( Ill D. Winklestein. M. B. S teer. and R. Pomerleau. "Simu la tion of arbitrary transmission lines with no nlinea r te rminat ions:· IEEE Trans. Citt·uits Syst. I & II, vol. 38. no. 4. pp. 41 8122, Apr. 199 1. ( 12 1 L. P. Va kana~. A. C. Cangellaris, and 0 . A. Pa lusinski. "Scatte ring parameterbased simulat ion of trans ie nts in lossy no nlinearlyterminated packaging intercon nections," IEEE 1i'tms. Component.\', Packaging and Manufaclllring Tee/mol.. Parr 8 : Adl'tmced Packaging. vol. 17. no. 4. pp. 472179. ov. 1994.
St even D. Corey n:ceived the B.S. a nd M.S. degrees in e lectrical e ngineering from the Unive rsity of Was hington in Seattle in 1991 and 1994. respective ly. He i ~ curre ntly working toward the Ph.D. degree a t the Univers ity of Was hington. He spe nt the s ummer of 1994 worl.ing on package mode ling and measurement at Te ktronix. Hi ~ research interests include numerical methods. s imulation a nd m ode ling of linear interconnect. and linear network reduc tion.
Andrew T . Yang received the B.S. degree in electrical e ngineering and computer science from the University o f Cal ifornia at Berkeley in 1983. and the M.S. and Ph.D. degrees from the Uni vers ity of Il linois a t UrbanaChampaign in 1986 and 1989. respectively. From 1983 to 1984. he was with the Ad vanced Micro Device Corporat ion. CA. Since 1989 he has been w ith the Uni ver:.ity of Was hington at Seatt le where he is c urrently an Associate Professor of Electrical Engineering. His c urrent research interests include simulation of mixed analogdigital circuit'>. timing simulation with e m phasis on analog modeling, and mode ling of semiconductor devices. Dr. Yang has served as a membe r o f the technical program comm ittee of the IEEE Inte rnational Conference on Com puterAided De:.ign. In 1992. he received the NSF Yo ung ln\•estigator Awa rd.
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MICROWAVE TIIEORY AND TECHNIQUES. VOL. ~3. NO.9. SEPTEMBER 1995
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Short Papers_ _ _ _ _ _ __ _ _ __ _ __ _ A ConvolutionBased Approach to the SteadyState Analysis of Nonlinear Microwave Circuits Using SPICE Patrick Halloran and Thomas J. Brazi l
,\ bstractA convolutionbased, blackbox approach is proposed to incorpora te linear circuit blocks into a nonlinear timedomain a nalysis program (SPICE) for the purposes of obta ining steadysta le results. T he technique used is str a igh tfor war d and offers a n en·eclive method for incorporating linear circuits descr ibed in the frequency domain into a nonlinear simulation. T he technique a llows SPICEusers to cou p le the fl exibility a nd accuracy of generalpurpose linea r microwm•e simula tors together with SPICE nonlinear device models, a nd thereby ohlain steadysta le results which a rc comparable in accuracy to popula r alternative methods such as harmonic balance.
I. I TRODUCTION A wide range of numerical techniques has been applied to the si mulation of microwave circui ts which contain both Jumped nonlinear components and distributed linear components. A complete review is given in [ 1[. Techniques used include harmonic balance [2[. 131 and power series analysis [4]. i n addi tion to several others. e.g .. [5 [. [6[. In recent years. harmonic balance has become one of the most popular freq uencydomain methods for the steadystate analys is of microwave circui ts w ith periodic excitation. Prior to the ad vent of harmonic balance, traditional timedomai n simulation techniques (for example. as appli ed in SPICE) formed the most i mportant largesignal nonlinear analysis tool avai lable to the microwave engineer. Thus. there has been considerable in vestment in timedomain si mulation model s and tools. anti they continue to be w idely used. One of the major problems with timedomain analysis is the inability directly to utilize accurate frequencydomain descriptions of linear distributed clements which form such an important part of any highfrequency ci rcuit design. These descripti ons need to incl ude important effects such as Joss and dispersion. Direct modeling of distributed linear components wi thin a transient si mulator is limited only to simple idealized types. The description of all component s must either be in the form of an eq uivalent circui t or a dyadic i mpulse response. Much work has been carried out on buildi ng suitabl e timedomain. SPICEcompatibl e descriptions of frequencydomain transfer f unctions with the main goal of determi ning the response to an aper iodic excitati on, [7], [8], 1101. T hat is. in developing the timedoma in model. the aim is to synthesize an impulse response which approximates the desired frequency function over that region of the frequencydomain where the signal s incident on the component have nonncgl igiblc spectral energy. In general. where real distributed components arc being simulated, it w ill not be possible to obtain an impulse response which exactly represents the desired function over the required band. thus the simul ated transient w ill always offer an approx imation to the acw al transient response of the ci rcuit. Equi valent circuittype model s, on the other hand, are inherently ti ed to a particular circuit, as, for example, in the case of model ling of multiple coupled transmission lines in 171. (8 [. In this approach, an Manuscript revised April I, 1993; revised May 25. 1995. The authors arc with the Department of Electronic and Electrical Engineering, University College Dublin, Dublin l. Ireland. IEEE Log Number 9l 13l II.
equi valent ci rcuit i s proposed for a structure. and standard parameter extraction techniques arc used to ca lculate the equivalent circuit parameters. With either approach, a transient simulator solves the circuit equations progressi vely in time and so any impulse functions synthesized from a frequency function must be causal [I I[. Since the frequency functi ons are generally represented numerically and onl y over a l imited band, th is almost always impl ies that the impul se response offers an approximation to the corresponding frequency function. Thus, if a transient si mulation w ith periodic excitation i s simulated into steadystate, and the results arc compared to a frequencydomai n method such as harmonic balance, then some di screpancy will be observed, clue to the fact that the impulse response is only approximating the desired frequency function at the fundamental excitation frequency and its harmonics. Of course. there are many other factors which may contr ibute to the discrepancy but the simulations can be set up so that these innucnces are mini mi zed . In view of the fact that SPICEtype transient simulators are used in certain applications to per form steadystate analysis of microwave ci rcuits, it is apparent that it would be useful to the microwave circuit designer to have a technique which would allow the direct incorporation of frequencydomain data within such simulators in a form which would guarantee the correct steadystate result. In this paper we present such a method. The discrete Fourier transform (OFf) is used to sy nthesize a ti medomai n model from frequencydomain data. this model being exact at a certai n set of frequencies. These frequencies are chosen to coincide with the exc itat ion freq uency and its harmonics. A device has been added to the SPICE simulator to accept the result ing di screte impulse response and to perform the appropriate convolutions. A standard SPICE simul ation is then performed, until steadystate is achieved. Result s can be presented directly in either time or frequencydomain by estimating the Fourier series of the response by means of the OFf. T he usc of convolution to simulate a nonlinear c ircuit which contains components specifi ed by their frequencydomain behav ior has been reported by many authors. one of the first being Silverberg and Wing [9[. Djordcv ic eta/. [ 10[. have also utili zed convol ution to simulate distri buted lossy di spersive interconnects w ith nonlinear ter minations. In the context of the present con tribution. what is important is how the i mpulse response i s calculated and how the subsequent convolutions arc performed. Existing convoluti onbased techniques seek to offer an approxi mati on to the desired frequency response at all freq uencies, their aim being to make poss ible a fu ll transient analys is of the networks involved. In thi s paper. a technique is presented which allows the incorporati on of frequencydomain data into a generalpurpose timedomain simulator such as SPICE. for the purposes of obtai ning only steadystate results. Whi le the method is exact for the steady state. the transient part of the solmion (which must be first integrated through) i. considered mea ningless. The overhead of performing the convolutions is not significant in the si mulation, since the impul se responses are short a~1d discrete convolution is used. In Section II the timedomain model i s presented. foll owed in Section Ill by details of the incorporation of this model into SPICE. Finally. as an example. the steadystate analysis of a 10 GHz microwave ampli fier is presented in Section I V and the results are
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compared wilh an harmonic balance simula10r. Concluding remarks arc given in Sec1ion V.
II. L INEAR CIRCUIT REPRESENTATION In the steadystate regime. vohages and currenls al any point in a network are periodic and so have spectral energy onl y at discrete frequencies, de. f o. 2/ o . . .. where /o is lhe exciJation frequency. Also for mos1 practical cases a frequency can be found beyond which the slcadystate responses have negligible spectral energy. Thus, i f we arc only interested in the steadystate behavior of a network which has linear circuit blocks embedded wi thin it, we only need an exact rcpresentalion of those blocks at de, the excitalion frequency, and a finite number of harmonics of that frequency. Lei H(f) represent some frequencydomain system function associated with a linear network which is embedded into a nonlinear network . Suppose it i s required to simulate the steadystate response of 1hc network to a sinusoidal excitation at /o. We propose to sample H(f) at de. fo. 2/o .... .Y/2/o and form the sequence, H (kfo) /.: 0. 1. .. . .Y/2 (.Y assumed even for illustrati on). A ppl ying li (11/o) and it s hermitian pan to the inverse DFf y ields the realvalued sequence h(nT) given by ( I ) wilh T 1/.Y/o
=
=
h(nT)
= .~·
k=Xl
L
H(~·fo)eJ2"("kf/o)
k·=O
n=O. l. 2 ... .. .Y  l.
(I)
I f h(nT ) i s considered to be a (finite) train of impul se functions then its continuous Fourier integral becomes exactly a (fini te) summation (i .e., a discretetime Fourier transform) and this resuhs in a continuous periodic frequency func1ion Ii (f) gi ven by u= J\"t
fJ(f)
=
L
h(11T }rJ2"fn1
(2)
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The circuit equati ons are so lved progress ively in time so that all device models must be causal. This is in contrast to simulation techniques w hich solve for the steadystate only, such as harmonic balance, where exc itations and responses are peri odic and ex ist for all time. The bandlimiting which is implicit in developing a timedomai n model from a limited set of frequencydomain data. frequently causes the impulse response to become noncausal. The noncausal portion cannot be simulated, so it must be discarded and the positi vetime portion of the impulse response adjusted in some way to compensate. Normally this is done in a manner which en urcs correct de performance of the approximation. However. it docs cause an approximation error at all other frequencies of interest. In thi s paper, the sequence obtained from ( I ) is interpreted as a positi ve time sequence with no reference to it s relationship 10 the actual continuous timedomain impulse response of the original frequency function. An advantage of ( I ) is the fact that i t is discrete. This implies that the convolution opcraJions are not as timeconsuming as would be the case for a continuoustime model since only the convol ving variable with a continuous character needs to be interpolated, and calculation of the convolu1ion integral also reduces exactly to a simple sumof· products operati on. This interpolation i s the only source of error in the simulation of the linear device. We consider for illustration a linear twoport, represented in the timedomain by a set of impul se responses as described in Section II. If a scauering parameter representation of the twoport i s used then the defining equations may be written in lime as
= It 11 {nT) * 1(1) + h 12( uT) * 112(/ ) b2(t) = h 21 (nT) *a,(/)+ h 22( 11T) * a2{l).
b, (I )
11
(3)
In (3) * denotes convolution, a and b arc the incident and reflected power waves, and the h.,,., may be deri ved from ( I ) with H(f) being 1he corresponding cauer ing parameter. T he It""' are discrete and finite and so 1hc convolutions in (3) arc exactl y summations
u=O (\ '  l
This frequency function i s equi valent 10 the original function II(/) at de. f o. 2/o .... ( .Y/2  1 )/o. and it i s continuous at all points si nce it is a finite summation of trigonomctrical functi ons. The relationship between H (f) and if (f) at points other than the original samples will depend upon the behavior of the original function and on the sampling process. I f a transient SPI CEty pe analysis is performed using these impul se samples, the steadystate rcsuh will be correct and the simul ated transienl wi ll be that of a network which contains an embedded linear block described by a frequency function given by fJ (f). However. the resulting steadystate response to a periodic excitation w ith fundamental frequency at / 0 , is still valid for the original system function, because ll(f) and Ii(f) arc equi valent at de. f o. 2/o . ... (.Y/2 1)/ o. A t the same time, the transient portion of 1hc simulation is in valid. since over the continuous frequency band where the transient response has spectral energy, the original function and its approx ima1ion arc, in general, not similar. Using an approx imate timedomain model w ithin any convolution based simulator can resuh in an unstable rcprc entation of the overall network. The technique presented here can suffer from this problem because the timedomain model only guarantees an exact rcprcscnlati on of the original frequency function at the fundamental frequency and its harmonics.
Ill . INCORPORATING DISCRETE CONVOLUTION INTO TilE SPICE SllllULATOR A direct ti medomain simulator. such as SPI CE, formulates and sol ves a set of coupled nonlinear integrodiffcremial equations (associated with a nonlinear electrical circuit) directly in the timedomain.
h,,.,.(nT) *a,. (I)=
L a,.(l kT)h.,.,(J.:T ).
(4)
k=O
At each iteration at each timcstcp, (3) may be wriuen in the following form for which a modified nodal analys is 11 2] stamp is readily calcul ated
+ II, i2{ l ) + Io, i2(1) = G2 u2(1) + 1\·2 "'(I)+ Jl2i, (I)+ l o2 · i, ( I )= G,P, (I)+ /\.,1'2(/)
(5)
Equation (5) may be represented by the equivalent circui t shown in Fig. I . Thi is the resistive circui t presented to SPICE at each iteration, which may be readil y incorporated w ithin the normal timedomain numerical solution. It is a straightforward maller to extend this kind of representation to a Ppon linear network . if required. In the following, a SPICE analysi s incorporating this kind of description of linear frequencydomain blocks is described as a ·modifiedS PICE' solution.
I V . EXAMPLE OF APPLICATION OF M ETIIOD A s an exampl e of the technique descri bed above, a I 0 GHz M ESFET ampli fier on microstrip i s presented in Fig. 2. The acti ve device is represented by a standard nonl inear model. and the linear ci rcuitry w ithin w hich it is embedded is fi rst analyzed in the frequencydomain using HPMDS [ 13]. The microstrip structures which form the matching and bias networks are both lossy and di spersi ve. and eight harmonics are included in the anal ysis. The linear input and output matching and bias networks are represented in the frequency domain by two 3port scaucring matrices deri ved from HPMDS, and
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2! ,.~~~...,~~
i,(t)
o•1(t)
2 t 59
1
1.5
... ~
0.5
.2..
i, (t)
CD
g"'"'
j
1.,
1.l
.3 ·0.5
v,(t)
·1
Fig. I.
Equivalent circuit presented to SPICE at each iteratio n. ·1.5 ·2'::':::'''''''   '    
2.8
2.82
2.84
2.86
2.88
2.9
2.92
2.94
2.96
2.98
3
Time (ns) Fig. 3.
Fig. 2.
Load voltage waveform. X harmonic balance.  modifiedS PICE.
10 G Hz MESFET am plifier.
the corresponding timedomain models are calculated as outlined in Section II. A transicm simulati on using the modifiedSPICE simulator is performed until a steady state regime may be identified. A n harmonic balance anal ysis of the same network i s performed. also using HPMDS. Identical device and circuit models are used both in the harmonic balance and modi fi edSPI CE anal ysis. Si nce both simulation methods offer similar capabilities this test provides a good veri fication of the technique proposed. Fig. 3 shows the output voltage in the ti me domain at an input drive level of 12 dBm with the MESFET bi ased i n the linear region. The harmonic balance result has been transformed into the timedomain using the inverse Fourier series. The waveform obtained using harmonic balance and that obtained from the modifiedSPICE simulator proposed i n this work, arc seen to agree very closely. Defining t'H as the harmonic balance result and t •,. .• as the modifiedSPICE result. we define the relati ve difference as
~( I)= lt 'lo&(l)  P,. ,, (l)l_ lt '/,1,(1)1
(6)
The average relati ve difference over one period is less than 0.5%. Simulation run time on a HP700serics workstat ion for harmonic balance is five seconds while the mod ifiedSPICE simulator requires 22 seconds. The duration of the timedomain tran sient, which is dependent on the circui t and the way in which the linear components are modeled, dictates the runti me for the modi fiedSPICE simulator. Note that the impulse response records need contain onl y 15 samples each in this case. To further demonstrate the capabil ities of the modifiedS PICE simulator a power sweep is performed to an input dri ve level of 20 dBm. First. second and thirdharmonic powers arc plotted agai nst avail able input power in Fig. 4. Again. both harmonic balance and the modifiedS PICE simulator offer good agreement. A simulati on using idealized transmi ssion line models for the microstrip l ines is also shown to demonstrate the effect of failing to account for the lossy substrate used i n this example.
· 1 ~~:o1~5~~ 1 o~.~5~o5~1~o~ 1 5______2o Input Power {dBm) Fig. 4. Power sweep showing output power at I st ( x ). 2nd ( o ). and Jrd (*) harmonics. x/o/* modified SPICE simulator.  harmonic balance,  idealized am plifier.
Y. CONCLUSION A technique has been presented which allows the usc of frequencydomain data within a SPICE analysis for the purposes of obtaining steady state results. The modificat ions to the SPICE simulator are at a device level. The technique allows SPICE users to couple the ncx ibility of a linear microwave simul ator w ith SPICE nonlinear device models. and obtai n steadystate results. I f circuit transients arc long then the technique wi ll be inefficient si nce in order to obtain the steadystate results an art ificial transient must be integrated through. However it i s certai nly useful in situations where these transients do not persist over a long time interval. A realistic example of a 10 GHz microwave amplifier is presented, and results compared to an harmonic balance analysis showing excellent agreement. REFERENCES
Ill V. Rizzoli and A. Ncri. ''State of the art and present trends in nonlinear microwave cad techniques," IEEE Trans. Micr01mve Theon • Tech .. vol. 36, pp. 343 365, Feb. 1988. 
IEEE TRANSACTIO S ON
2160
121 M. S. Nakha and J. Vlach. ··A piecewise harmonic balance technique for determination of periodic response of nonlinear systems:· IEEE 7i"lms. Circuits Sy.l"l .. vol. CAS23. pp. 8591. Feb. 1976. 131 K. S. Kundert and A. SangiovanniVinccntclli. ··Simulation of nonlinear circuits in the frequency domain:· IEEE Trans. ComplllerAided Design, vol. CAD5. pp. 521  535. Oct. 1986. 141 G. W. Rhyne and Michael 13. Steer. ··Gencralit.cd power scric> analy>is of intermodulation distortion in a MESI"'ET amplifier: Simulation and experiment."' IEEE 1i"rms. Micrmral'e Th eory Tech.. vol. Mrr35. pp. 12481255, Dec. 1987. JS J D. R. f'rey and Orhan Nonnan. ''An integral equation approach to the periodic steadystate problem in nonlinear circuits."' IEEE Trans. Citruits Syst. 1 & II. vol. 39. pp. 741755, Sept. 1992. 161 A. Buonomo, 'Time domain analysis of nonlinear circuits with periodic excitation."' Electron. Le11 .. vol. 27. pp. 6566. 1991. 171 V. K. Tripathi and A. Hill. ··Equivalent circuit modeling of losses and dispersion in single and coupled lines for microwave and millimeterwal'e integrated circuits."' IEEE Trans. MiCITJI\m·e Theory Tech .. vol. 36. pp. 256262. Feb. 1988. 181 J. I. Alon!.o. J. Borja. and F. Perez. "A uni versal model for lo.,sy and dispersive transmission lines for time domain CAD of circuits."' IEEE Tmns. Micrmrm•e 7/wory Tech.. vol. 40. pp. 938946, May 1992. 191 M. Silverberg and 0. Wing. "Time domain computer solutions for networks containing lumped nonlinear clements:· IEEE 1i"rm.l·. Circuit Theory. vol. CT15. pp. 292 294. Sept. 1968. 110) A. R. Djordjcvic and T. K. Sarkar. "Analysi' of lossy tr:msrnis>ion lines with arbitrary nonlinear terminal networks." IEEE Trans. Minmrm·e Theory Tech .. vol. MTT34, pp. 660666. June 1986. Jill Thomas J. Brazil, "A new method for thl! transient simulation of causal linear systems described in the frequency domai n," in IEEE Mit'l'm,·m·e 111eory Tech.S Dig. 1992. pp. 14851488. 1121 C. W. Ho. A. E. Ruehli. and P. A. Brennan. "The modified nodal approach to network analysis:· IEEE Trm1s. Circuits Syst .. vol. CAS22. pp. 50+509. June 1975. 1131 HP85 150B Microwave Design System (HPM DS).
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makes the four material parameter relati ons d yadic wit h nine components in general. and the total number of independent material parameters of bianisotropic media is hence 36. The material parameter are limited by principles dictated by physical Jaws. For example. the wellknown causal ity requirement leads to the Kro nig Kramers equations. And since the resu lts of bianisotropic clcc tro magnctics arc re levant to the growing number of microwave applicat ions of complex materials. it is evermore important to find o ut and articulate physical restrict io ns for the magnctoclectric paramete rs. A recent publication Ill provides us with an argument that gives one restrictio n to the 36 parameters for linear bianisotro pic media. Lakhtakia and Wcig lhofer usc te nsor analysis and the general covariance requ irement s given by Post 121 for unifo rm med ia and structural fields. and the c laim resulting from their analysis is strong: the su m of the traces of the magnctoclcctric dyadics must vanish. Applied to the spec ial case of biisotropic mcdia 1• the concl usion of the analys is in Il l is that isotro pic materials have to be reciprocal. In other words. the claim stands that NRBI (nonreciprocal biisotropic materials) can not ex ist. It is the purpose of this article to discuss possible counterexamples to this NRB Inoncx istc ncc result. For that end. let us reformulate the constitutive re lat io ns (I )(2) into a form where reciprocity b eco m e~ visibl e
= r · E + (\ t  jlir )JJt ofo · IflJ = (\ + f';; )JJtoFo · t + ~ · lr 
(3)
D
(I )
where now the magnctoelectrie parameters arc contained in the chirality dyad ic':: and the nonrcciproc ity dyadic \ .The superscript T denotes the transpose operation. 2 The nonrcciproc ity decomposition (3) (4) is in accord wi th the reciproc ity definition for bian isotropic media 141 =r c= r .
Ar e Nonreciprocal HiIsotropic Media Forbidden Indeed?
=t
I' = I' .
= =, f. =  ( .
(for rec iproca l media). (5)
Ari H. Sih vola Biisotropic =medi a have material dyadics that arc multiples of a Abstract Doubt is cast in this article on the universality of the
conclusion that linear hiisotropic media have to be reciprocal, which cl:1im has recent ly been set fort h by Lakhtakia and Wciglhol'cr.
I.
I NTRODUCTION
unit
dy~tdi c
D=~·E+t.lr
( I)
il=C·E+~ · fr.
(2)
Here the material parameter dyadics arc permi~ivity ~ . permeability ~ . and the mag netoclcctric crosspolarisations f. and (. Non isotro py
Manuscript received July 7. 1994: revised May 25. 1995. The author is with the Helsinki University of Technology. Electromagnetics Labomtory, FIN02150 Espoo. Finland. IEEE Log Numbcr9413413.
i ~ot rop i c
chiral medium has d yad ics
of a sample of NRB I material is of the fo rm \ = \f. where \ =F 0.
I I.
Bianisotropic materia ls have a more compl icated response to electrici ty than ord inary isotropic materi als. The polarization behavior of bianisotropic med ia is contained in the constilllt ivc re lation s between the e lectric ( D) and magnetic (.ll ) d isplacements. and the electric ( E) and magnetic ( ii ) field vectors
I. The wellknown
'::=,,.[ and \= 0 131. And in particular. the nonrc; iprocity dyadic
CONSEQUENCES OF T il E M AGNETOELECTRIC T Rt\CELESSNESS
The crucial result of [II is a conditio n for the trace of the magnctoclectric dyad ics. In particular. it restricts the non reciprocal part of these dyadics with the condition
rr{\}
=0
(6)
where trace means the sum of the diagonal e lements of the dyadic. Note that the constilllti vc relations used here re late the pai r ( D. [J ) to the pai r (E. fr ) whereas Ill follows the "BoysPost" relations where D and lr arc given as material functio ns of the primary fields E and [J. However, the magnetoelectric parameters have the sa me 1 For biisotropic rnl!dia. the four dyadic'> rl!duce to scalars since there is no direction dependence in the medium. The magnetoclectric coupling remain;, through two parameter,. 2 Jn (3)(4). the additional cocflicients (the imaginary uni t j and the free!.pacc parameters l' o. r 0 ) have been included for conformity with earlier notation 131.
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meaning for non magnetic materi als (ji = fl o!. which is the case in the following analysis) in both syste ms 3 h is clear that the lracclessness condition prohibits the existence of NRB I since rr{ \ } = 0 leads to \ = 0 in case of bi isotropy. As a maucr of fact, the condition can even be used as an effecti ve refutation of the fol lowing argu menl that has been presented to construct a sample of NR BI mediu m. Take a piece of nonreciprocal bianisotropic material (e.g., chromium oxide), and cut it into pieces. Then mix these inclusions, randomly oriented, in isotropic matrix material, like epoxy resin. Hence the anisotropic nonreciprocity of the origin al inclusions wi ll be averaged within the mixlllre that certainl y becomes isotropic. However. the effecti ve nonrcciprocity of the final mixture is proponional to onethird of the trace of the inclusion nonreciprocity dyadic. Therefore. if the trace vani shes. the mixture is reciprocal and not NRBI. In other words. the tracclcssncss says that if there is nonrcciproc ity in the material, its magnitude has to be of di fferent sign for differen t field directions.
Ill.
THE CASE OF CIIROMI UM O X IDE
h seems, however. that critical objections can be raised against the traceless property of the nonrcciprocity dyadic. First, let us consider the case of chromium oxide (C r~ O:I) that is known to di splay the magnetoelectric effect 15]. i.e .. the electri c field induces magnetic polarization in the material (effect labeled as MEr: ), and the magnetic field produces electric polarization (MEu ). In measuremenl s, the magnetoelectri c effect in chromium ox ide has been shown to be uniaxiall y ani sotropic, and hence one might not expect it 10 be an example of isotropic nonreciprocal material. However, the effect is also very sensitive to the tcmperalllre. In facl. the tcmpcralllrc dependence of the magnetoelectric components along the crystallographic c ax is material "as confirmed by D. N. A;trov. "1\lagnetoclcctric effect in chromium oxide:· Sol'iet Physics JETP. vol. 13, no. 4. pp. 729733. 1961. 161 V. J. Folcn. G. T. Rado. and E. W. Stalder. "Anisotropy of the rnagnetoclcctric effect in Cr~ 0 :1 : · Phys. Rel'ie11· Leu.. vol. 6. no. II. pp. 607608, June I, 1961. 171 T. H. O'Dell , The Elecrrodynamics of MagneroEiectric Media. Amsterdam: orthHo lland, 1970. p. 282. 181 R. Raab. private communication. 191 T. H. O'Dell. "Measurements of the magnetoelectric \usccptibility of polycrystallinc chromium oxide:· Philisophical Magtdne. vol. 13. pp. 921933, 1966. II 01 A. Sihvola. "When doubting Tcllcgen material gi,•c her the benelit of the doubt,'' Chiml discu.uion forum CHIRAL L til listserv @DEARN . BITNET alias listserv@Vt·I.G!·lD . DE. on 21 Dec. 1991. (Ill B. D. F. Tcllcgen. "T he gyrator. a new electric network clement.'' Philips Res. Rep .. vol. 3. no. 2. 1918. pp. 81101. (121 Reference (7). pp. I I. 18.
Four ierTransform Analysis for Rectangular Gr oove Guide ByungTak Lee. Jac W. Lee. Hyo J. Eom. and SangYung Shin
V. CONCLUSION
The present aniclc is an attempt to defend the position that N RBI media arc not forbidden. at least not because of the covariance principles expounded in Ill. Based on the examples of chromium ox ide and phenomenological Tcllcgcn material. the traceless property of the nonrcciprocit y has been challenged. Since the mathematical derivation leading to the tracclessness in I l l seems to be sound. the arguments of the present paper lead to objections against its Maning premise. The inevitable conclusion here is that the covariance property in the form of Post 121 docs not hold universall y in nature.
;\ bstract T he r ecta ngu lar groove guide is analyzed using t he Fourier transfo r m a nd the modematching technique. The e n force ment of the boundary conditions at the groove a pert ur es yields the s imultaneous eiJUa tions fo r th e fi eld cueflicient ins ide t he grooves. The s imultaneous el)ua tinns a rc solved to re present a dis1>crs ion relation in analytic ser ies fo rm. T he nume rical computation is performed t o illus trate the behavior of t he guided wave in terms of frequency and gr oove s izes. The presented series s olut ion is exact and ra pidlyconvergent so that it is effici ent for nu merical COI11J>utation. A s imple dis pers ion relation based on the d omi nantmode analysis is obtained and is s hown to be ,·c ry a ccura te for most practical a pp lications.
ACKNOWLEDGMENT
The author wou ld like to thank the following sc ientists for usefu l discussions: J. H. Cloetc, B. Jakoby, G. Kristcnsson. A. Lakhtak ia, I. V. Li ndell. R. Raab. S. A. Trctyakov. A. Viitancn. W. S. Wciglhofcr, 7 Due to domain nucle:llion. also simpler media like permane nt magnets will eventually decay into the timesymmetric state. thw, being unstable. The time constunts involved in the process. however. may be millions of years.
Manuscript received July 20. 1991; revised 1\l ay 25. 1995. The authors arc with the Department of Electrical Engineering. Korea Advanced Instit ute of Science and Technology. 373t. Ku,ong Dong. Yusung Gu. Tacjon. Korea. IEEE Log Number 91131 11.
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l. I NTRODUCfi ON
t.Y
T he groove guide is a useful structu re due to i ts lowloss highpower capacity above I 00 GHz. Experi memal studies were performed in [ l ) to determine the dispersion relation of the groove guide. T heoretical studies were also per formed using the approx imate equi valelll network [2] and the numerical approach [3]. A lthough the di spersion relation of the groove gui de has been relati vel y well understood, i t i s of interest to obtain an exact analytic soluti on. In th is paper, we obtai n such a solution by utili zing the Four ier transform and modematching. T he solution i s in rapidlyconvergent series so that it is very efficient for numerical computation. In lowfrequency l imit, the solution simplifi es yielding a dominantmode approxi mate dispersion rel ation which is very accurate for most practical appl ications. In the next sections, we derive the di spersion rel ations for the TM and T E modes and discuss the numerical efficiency of our solution. T he notations used in the paper cl osel y follow those in [5).
II.
Regi~n (Ill)
PEG
id PEG
·····:Reg1on (II)
b
2a .
PEG
Fig. I .
X
Geometry of rectangular groove guide.
Ill. TMMODE ANALYSIS
=
We consider the T M case, E : f. 0 and H: 0. T he enforcement of the boundary conditions is necessary to determi ne unknown coefficients p, and r, . 1 T he condition E~ (.t . 0) E~(.r. 0 ) gi ves
=
F IELD R EPRESENTAT IONS
Assume a TE or T M mode propagates along the ;; direction inside the groove guide in Fig. I. W ith the ti mefactor eJ"' 1 suppressed the propagating E field E(.r , y, ;;) is gi ven as E (.t:. y )e;fh . In regions (1 ), (II), and ( Ill) E: and H : are
_.!._ {
{t.
2rr ./_ 00
c£; + £;
)r  J'"' dC
I'm , ;., o.,( ,
=
+ o) , ;., ((,.,d)
for for
00
L
=
E; (:r. y )
L
q, cos a,(.c + a )
where E~ 1 (.r. 0) and
m=O
. co s
E : (.r . y)
€... (y +d)
( 12)
"
t;; + E; are a Fourier transform pair and eJ 9 (
(2)
l oo
= ?
p,,a, sin (€,, d) a 2 F,(( a)
, ,=1
00
1
( I I)
00
£; + £; = L
( I)
· si n €, (y +d)
II
l·rl a.
Taking the Fourier transform of ( I I ), we obtai n
p, si n a, (.t: +a)
n1= l
H ;(.r,y)=
PEG
f
Regipn (I)
F,(g) T he condition H;' (.t·. 0)
oo
(3)
_.!._ 2rr
loo .
=
1)"' 
= H;(.r. 0)
·(E +  £: )
. oo
Jn
:
e;y
( 11/ 'Tr /2)2
g2 
:
for e
l.rl
;(r
( 13)
. < a gives d(
=
00
L
(4)
E;
11
(.t·. y )
L
=
r,., si n n, ( .r + n )
p,,€,, sin a,.,(.r +a) cos( €,d).
( 14)
,, = J
Similarly the condition E~ 11 (.r. b)= E~ 1 (.r. b) gives
· in €, (yb d )
t ; eJ"b + t ;
(5)
(';nil=
00 00
L
H; 11 (.r.y)=
 """ L... r,, a .,,
s, co. a, (.t·+ a )
s 111 .
(€,, d)a 2 F ,((a).
( 15)
rlt = l
rn =O
. cos
€... (y  b  d )
Ji.2
(6)
T he condition
1 /_"" )H . ·(E+ ;> PREDICTIONS AND COMPARISON WJTll EX PE R I ~I ENT To illustrate the appl icabi l ity of lhc phasor diagram approach. we considered the HEMT equivalcm c ircui t o f Fig. I as applied to the lnA IA s/lnGaAs HEMT for which the nonlincaritics arc show n i n Fig. 2. The source [Z ,(..v)l and load IZI (....i)l tcrminations were assumed to be about 10 f2 at 94 GHz and 50 n a1 3 GHz. respectively. Th is corresponded to the conditions used for the 9t GH1. MMIC H EM T mixer presented i n [2]. T he mixing terms of ( I I ) can be expressed w ith the help of phasors in the 2D complex plane o f conversion gain as shown in Fig. 3 for an LO power level of 5 dBm. Each o f the vec1ors indicates the mixing terms from the harmonics of the three nonl inear elements. T he follow ing obser vati ons can be made from Fig. 3: I ) The major contribution to the conversi on gain a ri ~es from the fJ, 1 mi xing term.
2167
IEEE TRANSACTIONS ON MICROWAVE THEORY AND T ECHNIQUES. VOL. 43. NO. 9. SEPTEMBER 1995
25 (i)
5., (J
20
. ..
"'c
10
~
..."'
1st harm.
,
,. ..  ..     .,(
c
f
c
... 10
t: 0...
0.4
11111
"'Q
''
'' Gm i',Rdst ' '
, ,Gmt
''
0 5 10 LO Power (dBm) Fig. 4. Comparison of conversion gain calculated using the analytical phasor diagram approach with experimental and 2tonc harmonic balance si mulation results for a 94 GHz monolithic lt1Pbased HEMT mixer 121 \ :q.< = 1.0 v, 1;1. = 1.0 v. 5
an improved physical understanding of the mixing mechanisms and provides a simple method for fi rst pass desig n of IIEMT mixers. Very good agreement was shown between the analytical predictions and the measured performance of monoli thic integrated lnPbased II EMT mixers operating at 94 GHz.
,G m (2or morc)• Rds l '
I
\
[' Analyllcal ExpressiOn 2·tonc HB Analysis .... 10
5
c
0.6 > c 0 u ....
8
•
12
.
0 5 LO Power (dBm) Fig. 2. Harmonic content of G "' as a function of LO power for a 0.1 xDO 11111 lnAIAsllnGaAs HEMT.
·;;; 0
c 0
3rd harm
0  15
4 .
u ~
.
c 0 '§
..,:>
2nd harm.
5
,.~L
·a
~
/ \
15
"0
co ::::.. 0 
Oth harm.
c
"' u
4
)' .
'
RdstXGrno
"' 0.2 "' .§
c ·eo
R EFERENCES
1.2
·0.8
·0.4
Real Part of Conv. Gain Fig. 3. Plwsor diagram of conversion gain vector, showing the various mixing components (PuJ = 5 dBm). 2) The r.t.• 1 mixing term mu lt iplied by [/m 0 degrades the conversion gain due to its incoherent phase with the .'Jm 1 mixing term. 3) The intermixing term from f1,., 1 and r d, 1 is not negligible and adds in phase to the [Jm 1 term. 4) The contributi on from the intermixing terms greater than 2nd order is very smal l. 5) The image component mixing term is small compared to RF component because of the low c9 • 2 / c9 , 0 ratio. 6) Large g,., 1 /r; ..,0 and r,t,, 0 / r.t.• 1 ratios are needed to obtain optimum conversion gai n. Simulations using various LO power levels showed that the trends described in points in 1 6 apply to a wide range of LO power cases. The results discussed above identify c learly the role of each HEMT parameter on mi xer conversion gain and suggest methods of improvi ng the conversion gain throug h the control of dev ice parameters. Finally, in order to validate the formul a, the conversio n gain calculated from ( I I) is compared in Fig. 4 with the measured performance of the 94 GH z monolithic ln AIAs/lnGaAs II EMT mi xers realized by the authors 12]. A lso shown is a twoto ne harmonic balance simulation obtained for the same dev ice using EEsofLibra for the same source and load term inations. The complete HEMT equi valent circuit was used for this simu lation. Good agree ment, within I dB. can be seen over a wide range of LO drive levels. The analytic expressions presented in thi s paper can c learly be used successfu lly for first pass design and performance estimatio n of HEMT mixers. IV.
CONCLUSION
In conclusion, an analytical expression has been derived and a phasor diagram analysis method has been developed which permit s
Ill P. D. Chow. K. Tan, D. Streit. D. Garske. P. Liu. and H. C. Yen. "Ultra low noise high gai n Wband hlPbased HEMT downconvcner," in 1991 IEEE 1111. Microwal'e Symp. Dig.. June 1991 , pp. 10ll  104l. 121 Y. Kwon, D. Pavlidis, P. Marsh, G. I. Ng, and T. Brock, "Experimental characteristics and performance analysis of monolithic hlPbased HEMT mixers at Wband," IEEE Traus. Micro11m·e Theory Tech .. vol. 41. pp. 18, Jan. 1993. 131 D. N. Held and A. R. Kerr, "Conversion loss and noise of microwave and milli meterwave mixers: Part 1 Theory." IEEE Traus. J\licroll'm·e Th e01y Tech .. vol. MTI26, pp. 4955, Feb. 1978. 141 S. A. Maas, Microwal'e Mixers. Boston: Ancch. 1986. 151 Y. Kwon, D. Pavlidis, T. Brock. G. I. Ng, K. L. Tan, J. R. Velebir. and D. C. Streit, "Submicron pseudomorphic double htcp in an i!.otropic medium a!> a function of node a!.pcct ratio.
(8)
~I < _!_
0.5
(6)
)
is related to the s mallest node dimension ~ 1. formu las ( 1)(3) and (4)(6) can be rcwriuen in the forms given in [ II and 121. respecti ve ly. However. there is no particular reason for the time step to be related to the smallest node dimension. The only condition which must be met in formu lating the characteristic admi uances of the stubs when modeling a passive medium is that these admiuanccs arc nonnegative 161. Hence the value of ~I must be chosen such that none of the characterist ic admiuancc of the stubs becomes negative. Applying the condit ions l ' Or 2:: 0. l ' Oy 2:: 0. l ' o , 2:: 0. it follows from (4) (6) that If the time step
'
1.0 ...
0.2
l y lx  .lv 2.lx
2
I
=~ 2r· 1 /( ~.r):.! + 1 /( ~yf >
~I
2t·
( 13)
From the Cases hl  h3 it can be seen that for any type of grading. the maximum permi ssible time step ~I ""'' lies within the limits ~I
~ ~('
::;
~1 ,,,"
~1 J2
<  .,
which can also be expressed in te rms of
( I~ )
~('
~1 0
~I n :S ~I """ < ~lo ,12.
( 15)
T his was also noted in [51. Practica lly thi s means. that in some cases the ti me step can be chosen to be up to J2 times higher than one related to the smallest node dimension. Cases h l h3 arc illustrated graphicall y in Fig. I. The ratio 11 between the maximum permissible time step ~ I "'·". as defined by formul as (7) (9). and the time ste p ~l o re lated to the smallest node dimension is pl o!led against the aspect ratio j of the dimensions ~!/ and ~ .:: . An increase in , j can be imcrpreted as a decrease in ~ = . whereas ~ !J and ~ .r remain constant. Note that 1 is prcscmcd on a logarithmic ~ca l c in Fig. I. Cases hI and h2 ( ~ y = ~ .r) appear in Fig. I as the solid line. For 1 < 1. i.e., ~ : > ~ !J . Case hI applies and the ratio 11 remains unc hanged and eq ual to unity. When j > L Case h2 applies and the time steps ratio increases up to 'I = ,/2. Case h3 is reprc~cmcd by the broken line in Fig. I, for the example chosen ~ !I = 2 ~ .r. On ly at one point. when , j = 2, i.e .. ~ .: = ~ .r. docs the maximum permissible time step. ~1, ,;"' equal ~l o . Otherwise. when 1 < 1. form ula (9) applies leading to 'I = j8fS :::::: 1.26. wherca~ whe n 1. formu la (8) applies leadi ng to 'I  ../2.
IEEE TR ANSACriONS ON MICROWAVE THEORY AND T ECHNIQUES. VOL. 43. NO. 9. SEPTEM BER 1995
217~
;; :::1 ~ l ..l
;;l
H
Prrc
···· 
("_.=2.89 .. .,,:2 45
I. I
fly lx ••• Oy 2~>
"() o.z r
j
l
0.< f[ 0.6 
1.0 t:::===!:_~
0.2
0.5
1.0
2.0
fJ •
5.0
Al=At_,
 6l=6lo
6l ·0
o.eo',~5,J~o,~ s.~o,~5""•o
10.0
tly/ A'l.
Fig. 2. Normalized maximum ti me step in anisotropic PTFE as a function or node aspect ratio. B. Inhom*ogeneous Isotropic Media
C. Anisotropic Media When modeling ani sotropic materi als, the analysis of the formu las (7)(9) becomes more complex, so we only give an exa mple fo r a material commonly used in microwave c ircuits, namely PTFE, with ~rr,r = 2.95. ~yy, r =2.89, and ~::,•· =2.45 [8]. The time step ~ I a is related to the smallest node dimension and the lowest electromagnetic parameters as
= ill 2c
( 16)
where i. j E {.c. y. ;;}. For the example of PTFE, plotted in Fig. 2, ~I a = ill J~::.•· /(2c) . When 1J + oo, inequali ty (8) applies, leading to 17 + J2:;yy,r/~::, r :::::: 1.536.
I V . I MPACT ON THE D ISPERSION C HARACTERISTICS
The relative propagation error for the HSCN is in vestigated for di ffe rent val ues of time step by solving the general dispersion relation for the T LM SCN [9], wi th modifications to account for stubs [ I 0]. An example corresponding to Case h2, simil ar to that in [I IJ, with ~ .r = 2il y = il ;; and a hom*ogeneous medium (~ •. = I' •· = 1), is considered for propagation along the coordinate plane ;; = 0. Fig. 3 shows the relati ve propagation error 6k [ II) in the HSC N operating on different time step values, calcul ated for the spatial discretization of ~ .r/ .X = 0.1 and plotted as a fu nction of the angle o between the propagat ion vector r and yaxis defin ed by 0 = >lrctau (J.·,.jk y) · Note that two solutions of the general dispersion relation for the HSCN appear [ 10], [ II]. Fig. 3 shows that the range of propagation error is the smallest when using the maximum time step ilt, ... , .
( ')
Fig. 3. Relative propagation error for di fferent time steps and grading set at ..:J..r = 26. y = 6, ;;.
V.
When modeli ng inhom*ogeneous isotropic microwave circuits containing regions of di ffe rent penneabili ties and permitti vities, ~...,,,,. = ~yy,r = ~ ::.r = ~ •. and ll z:r,r = ll yy , •· = 11::, •· = I'··· two cases can be considered. Case if : Grading rat io over different medi a is identical. In thi s case, it can be seen fro m formulas (7)(9) that media with higher electrical and magnetic properties all ow higher time steps, but the one related to the medium with the lowest permitti vity and permeability (usuall y chosen as the background medium) must be used. There fore, the analysis performed for the hom*ogeneous Cases hl  h3 is valid for this in hom*ogeneous case too. Case i2: Grading ratio over different medi a is different. In thi s case, full an alysis of the permissible time steps for all node regions must be performed. It can be easily con fi rmed that the time step can always be chosen so as to be related to the small est mesh dimension applied to the background medium parameters, but the actual con figuration of the mesh might allow the use of a higher time step value.
il fo
0.0
~
1 ,,.,=2.9~
1.2
C ONCLUS ION
It was shown that the value of the time step is not strictly dependent on the ratio of the smallest to the biggest node dimension as in the stubbed SC N, neither has it to be related to the smallest node dimension as desc ribed in [2]. New formulas for the allowable time steps were introduced for modeling general med ia using the HSCN and analyzed thoroughly for di ffe rent problems. It was shown that in some isotropic cases the time step can be chosen to be up to V2 higher than that related to the smallest node dimension. In the ca e of ani sotropic media thi s value can be even higher, depe ndi ng on the material properties in the principal directions. It was shown that the dispersion characteristics of the HSCN are dependent on the ti me step value used in the mesh and that the max imum allowabl e time step for the HSCN yields the smallest range of propagation errors. R EFERENCES
Il l R. A. Scaramuzza and A. J. Lowery. "Hybrid symmetrical condensed node for TLM method," Electron. Lerr.. vol. 26. no. 23. pp. 1947 1949. Nov. 1990. 121 P. Berrini and K. Wu, "A pair or hybrid symmetrical condensed TLM nodes," IEEE Microll'ave and Guided \Vave Lett.. vol. 4, no. 7. pp. 244246. July 1994. 131 P. B. Johns, ''A symmetrical condensed node for the TLM method," IEEE Trans. Microwave Theory Tech .. vol. MTI35, no. 4, pp. 370377, Apr. 1987. 14 1 V. Trenkic. C. Christopoulos. and T. M. Benson, ''Genemlly graded TLM mesh using the symmetrical supercondensed node," Electron. Lett .. vol. 30. no. 10, pp. 795797, May 1994. 151 R. A. Scammuzza, "Simu lation of conducti ve environments using transmission line modeling,'' Ph.D. disscnation. Uni versity of Noningham. UK. 1993. 161 D. A. AIM ukhtar and J. E. Sitch, 'Transmissionline matrix method with irregularly graded space," lEE Proc., vol. 128. no. 6, pan H. pp. 299305. Dec. 1981. 171 Q. Zhang and W. J. R. Hoefer, "Characteristics of 3D distributed node TLM mesh with cells or arbitrary aspect ratio," IEEE Trans. Micrml'(lve 111eory Tech .. vol. 42. no. 12, part 2, pp. 2365 2370. Dec. 1994. 181 C. Bulutay and S. Prasad. "Analysis of mill imeter waveguides on anisotropic substrates using lhe threedimensional transmissionline matrix method," IEEE Trans. Microwave Theory Tech.. vol. 41. no. 6n, pp. 1119 1125, June 1993. 191 J. S. Nielsen and W. J. R. Hoefer, "Generalized dispersion analysis and spurious modes or 2D and 3D TLM fom1u lations." IEEE Trans. Microwave Theory Tech.. vol. 41, no. 8, pp. 13751384, Aug. 1993. 1101 M. CeluchMarcysiak and W. K. Gwarck, "On the effect of bilateral dispersion in inhom*ogeneous symmetrical condensed node modeli ng." IEEE Trans. Microll'ave Theory Tech.. vol. 42, no. 6. pp. 10691073, June 1994. Ill I V. Trcnkic, C. Christopoulos, and T. M. Benson, ·'Dispersion analysis or TLM symmetrical supercondensed node," Electron. Lett.. vol. 30. no. 25, pp. 21512 153, Dec. 1994.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECH IQUES. VOL. 43. NO. 9. SEPTEMBER 1995
High Isolation a nd Low Insertion Loss Switch IC Using GaAs M ESFET' s
1175
A Fin
AFoul
..... ,
Yorito Ota. Makoto Sakaku ra, Kazuhisa Fujimoto, Sh inji Yamamoto, and Hiromasa Fujimoto
AbstractA novel RF switch IC using GaAs MESFET's has been developed for digital communication systems. The new IC is composed of a threestage SPST switch and a thin film termination resistor, which realizes a high isolation and a low return loss. In addition, a high )lO\\Cr handling capability and a low insertion loss arc simultaneously realized with two kinds of pi nchofT voltages using the orientation ciTcct of GaAs MESFET's. According to th c.~e tech no l ogic.~, the excellent performance is achieved as follows: the isolation of 60 dB, the return loss of 20 dB, the I dB power comJ>ression of 27 dBm and the insertion loss of 1.6 dB at a frequ ency of 1.9 GHz with control voltages of 0/5 V. The new switch IC contributes to a variety of communication system using highcJuality digital modulation.
I.
I Vshunt Vlhrough GND
GND
GND
Fig. I. Circuit diagram or the developed threestage switch IC.
I NTRODUCTION
R
F s.witch technologies using GaAs MESFET' s have advanced as sohd state T/R modules for phased array radars in Xband to K/Kaband II], [2]. Recent popularization of mobile telecommunication systems has accelerated the GaAs switch technologies in Lband. Low insenion loss SPOT switches are applied to digita l personal communication systems such as PHS (Personal HandyPhone System; in Japan) and DECT (Digital European Cordless Telephone}, because their high speed and low power consumption characteristics arc suitable for the systems [3], [41. On the other hand, a high isolation propeny of a RF switch is required for a base station unit using TDMA (Time Di vision Mu lti ple Access) communication and for a frequency synthesizer system . The high isolation switch reponed 151 was a hybrid IC module composed of shielded microstrip lines and dualgate GaAs MESFET' s. In order to apply these switch IC's to various communication systems, the reduction of s ize and weight is one of the most imponant factors. We have introduced a onechip highi solation IC in order to reduce them drastically. The newlydeveloped switch IC using GaAs MESFET' s is fabri cated with a novel structure and a simple process. Figs. I and 2 show a schematic circuit diagram and a top view photograph of the switch IC. The features of the switc h IC are as foll ows: I) threestage SPST switch is integrated on one chip in order to achieve high isolation. 2) the gate orientation effect is introduced to the FET arrangement in order to realize two kinds of pinchoff voltages and consequently obtain high power handling capabil ity and low insenion loss, 3) a 50 n terminati on composed of a shuntFET and a thin film resistor is employed in order to improve return loss. In this paper, the fa bricat ion process and the device design of GaA s MESFET's suitable for the switch IC are first presented. The circuit design for isolation, return loss. inscnion loss and power handling capability is investigated through the evaluations of singlestage SPST switch IC's and a package. Then, the characteristics of the new switch IC are described.
Manuscript received January 20, 1995; revised May 25. 1995. Y. Ota and K. Fujimoto arc with the Semiconductor Research Center. Matsush*ta Electric Industrial Co.. Ltd .. Moriguchi . Osaka 570, Japan. M. Sakaura is with the Materials and Devices Laboratory. Matsush*ta Electric Industrial Co., Ltd .. Moriguchi , Osaka 570. Japan. IEEE Log Number 9413432.
200 J..Lm Fig. 2.
Photograph of the threestage switch IC: the throughFET toward 10 1 11 and the shuntFET toward 1011 j on a ( I00) GaAs substrate. II.
FABRICATION PROCESS AND D EVICE D ESIGN
GaAs MESFET's in the switch IC's were fabricated by ionimplantation and recess etching process with A I gate metal [6 1. The process introduces shallow ionimplantation (acceleration energy of Si + = 80 KeY) and shallow recess etching (recess depth 100 KeY) and deep recess etching (> I00 nm). The channel region with this shallow process exhibits a steeper carrier profile, a higher current dens ity and a shall ower pinchofT voltage. These features arc suitable for lowvoltage operated switch IC's. All FET's in this study have the same device dimensions as follows: the gate length of I Jtm. the finger length of 200 Jt m and the gate width of 1.2 mm. The gatetodrain and the gatetosource distances were minimized up to 0.5 Jtm in order to reduce a turnon resistance and consequently reduce insertion loss. The 2 Kn gate resistors in Fig. I and n+ contact regions for drain and source in the MESFET's were simultaneously formed by a single ionimplantation (S i"' = 150 KeY) in order to decrease processi ng steps. A 50~~ termination whose value directly innuences the return loss propcny in switch IC's needs to be formed precisely. We fabricated it with a shuntFET and a thin film NiCr resistor. because the value deviation of the NiCr resistor was less than 5% while that of the ionimplantation resistor was around 20%. Since the turnon resistance of the shuntFET was evaluated to be 5 n. the thin fi lm resistor was set 10 45 n for the 50 n termination. The wiring on the switch IC was formed with a thick metal of 2 Jim using a goldplating technique. The wiring was minimized in
0018 9480/95$04.00 © 1995 IEEE
IEEE TRANSACTIONS ON 1\IICROWAVE THEORY AND TECIINIQUES. VOL.
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