The YIicrowavs Tleory and Techmques Society isat organization, uithm the framework of the IEEE, of members with principal professional mteiest in !he field of microwave theory and techniques, All members of lhe IEEE are eligible for membership in the Soc\ety and will receive this TR.A>”S.ACTIO\S upon payment of ~heannual .Soclety membership fee of $8.00. Affiliate membership is available upon payment of the annual affdiate fee of $24.00, Plus the Society fee of $8.00. For information onjoining write to the IEEE at the address below ADMINISTRATIVE COMMITTEE H. G. OLTMAFJ, JR., President H. HOWE, JR., P’weP~esiden~ J. E. !?AUE, Secretary-Treasurer N. w. Cox T, ITOH H. 3. ~UNO F, J. ROSENBAUhi* J. E. DEGENFORD, JR. F lVANEK S. L. MARcH C. T. RucKER’ V. G. GELNOVATCH G. JERINIC D. N. MCQUIDDY, JR. R. A. SPARKS* P.T, GREILING K S. K4CiIWADA E. (2.NIEt+ENKE B, E, SPIELMM4 R. B. HICKS R, H. KNERR J. M. ROE i, ~x ~~~jcjo (p~~~~r~~rd~~~.~J Honorary Life Members D/stirrguished Lectui-er,\ A. C. BECK W W. MLIMFORD A. A OLI~ER k TOPViTYASL S. ADA.M P.T. GREILING S, B COHN T S .SAAD L ~OIJNG S. WEINREB S-MT!- chapter Cctmirmen Albuquerque: J. F’.CASTILLO india: B. BHAT San Diego D. T, GAVIN Atlanta. G. K. HUDDLESTON Isrsel: A. MADJAR Santa C]ara Va]]ey: R. \V. WO?-JG Baltimore: f%TER D. HRYCAK Kltchmer-Waterloo Y, L. CHOw Schenectady: J. BOR<EGO Benelux: A. GUISSARD Los Angeles: F,J, BERNUES Seattle: D. G. Dow Buffalo: INACTIVE Mdwacikee: T. J. KIJCHARSKI Southeastern kllchig~n: D. P,NYQUISJ Canaveral: C.-F. A. CHUANG Montreal: J. L. LEIZEROWICZ St. Louis: CURTIS E LARSON Central Illinols: G. E. STILLMAN f’dew Jersey Coast: RUSSELL A. GILSO~ Sweden: E. L KOLLLrERG Central New England: C. D. BERGLUND New York/Long Iskmd: R. KAMIFiSKY Switzerland: F. E. G.ARDIOL Ch]cago: S. S. SAAD North Jersey M, SCHNEIDER Syracuse: B K. MITCHELL Columbus: N. WANG Orange Countj: J. C, AIJKLA.ND Tokyo T OKOSHI Connecticut: M. GILDEN Orlando: F, P.WILCOX Tucson, E, P. PIERCE Dallas: R. E. LIWMANN ott3wa:J, WIGHT Twin Ci[ies C R, SEASHORE Denver-Boulder: G-.R, OLHOEFT Phdadelph]a: f). L. JAGGARD Utah/Salt Lake City M F. ISKANDER Florida West Comt: R, E. HENNING Phoeni\: LEX AKERS Washington/V] rg]nia: Z, TURSKI ~OLiStOI): S. LONG Portland, R. C. CHEW West Germany: N, J KEEN Huntsville hf. D FAHEy Princeton WALTER SLUSARK IEEE TRANSACTIONS@ ON MICRQWAlrZ THEORY AND TECHNIQUES Associate Editors N R. DIETRICH F, lVA~EK E. YAMASHITA (Pa!errr Absrracrs) [Absrrarrs Editor—Asia) THE DW5THWTE OF ELECTRICAL AND ELECTRONICS ENGINEERS, INC. officers RICHARD J.GOVEN, President J. BARRY OAK ES. P’lcePresident, Educatlortal Actioitles CHARLES A. ELDON, President-Elect RUSSEL C. DREW. Vice President, Professloizal .4ctleities HEN RY L. BACEMAN, Executive Vice President JOSE B. CRUZ, JR., Vice President, Publication Activities CYRIL J. TUNIS, Treasurer MERRILL W. BUCKLEY, JR.. Ylte ?resident, Regional Acriclties V PRASAD KOD4LI, Secretary STEW AN KAHXE, Vice President, Teclrmcal .4cti[lties E. W. PJ.JGH,IJlcision IV Director Headquarters Staff ERIC ~ERZ, ExecutlL e Dtrector and General Murrager ELWOCrD K. GA NJNETT, Deputy General h{anager THOMAS W. BARTLETT, Controller DAVID L. STAiGER, Slaff D[rec[or, Pub[[shlng Serclces DONALD CHRKSTIANSEN, Editor of Spectrum CHARLES F. STEWART, JR., S!aff Dire< ?or, Adm[nisiratfon Services 1RVIhIG ENGELSONJ, Slaff Director, Techmcal Acttullies DONALD L. SUPPERS, Staff Director, Fteld Seruices LEO FANNING, Staff Director, Professional Activities THOMAS C W~ITE, Staj~ Director, Public Information SAVA SHERR, Staff Director, S[undards JOi+rN F WI LHEL~, Slaff Director, ProJes~ional Ac(iuities Publications Department Production Managers: ANN H. BURGMEYER, CAROLYNE T.%MNEY, GAIL S FERENC, Associate Editor: WILLIAM J. HAGEN IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1spublished monthly by The Institute of Electrical and Electronics Engineers, Inc. Headquarters 345 East47Street,NewYork, NY 10017. Responsibility for thecontents restsupon the authors andnot upon the IEEE, theSociety, or its members, IEEE Service Center (for orders, sttbscriphotrs, address changes, Region/Section/Student Services) 445 Hoes Lane, Piscataway, NJ 08854. Tek@mnes Headquarters 212-705 + extension: Information -7900, General Manager -7910, Controller -7748, Educational Services -7860, Publishing Services -7560, Standards -7960, Technical Services -7890. IEEE Service Center 201-981-0060. Professional Services: Washington Office 202-785-0017. NY Tek?copiec 212-752-4929. Telex 236-411 (International messages only). hrdwldtral copies IEEE members $6,00 (first copy only), nonmembers $12.00 per copy. Annual subscription price: IEEE members, dues plus Society fee. Price for nonmembers onrequest. Available in mlcrof:che and microfilm. Copyright and Reprint Permission Abstracting ispermitted with credit to the source. Libraries are permitted to photocopy beyond the limits of U.S, Copyright law for private useof p~trons: (1) those post- 1977 articles thai carry acode at the bottom of the first page, provided the per-copy fee indicated in the code ispaid through the Copyr]ght Clearance Center, 29 Congress Street, Salem. MA 01970; (2) pre-1 978 articles without fee. Instructors are permitted to photocopy isolated articles for noncommercial classroom usevithout fee. For other copying. reprint or republication permmlon, write to Director, Publishing Services at IEEE Headquarters. All rights reser!,ed, Copj-right ‘G1984 by The Institute of Electrical and Electronics Engineers. Inc. I%nted in L’.S, A. Second-class postage paid at New York, NY and at additional m.xlmg offices, Postmaster: Send Address chsnges to IEEE. 445 IHoes Lane. Piscataway, NJ 08854. IEEETRANSACTIONS ON MICROWAVE THEORYAND TECHNIQUES, VOL. MTT-32, NO. 10, OCTOBER1984 1275 Finite-Element Analysis of Dielectric-Loaded Waveguides MITSUO HANO Abstract —A finite-element analysis in which nonphysical spurious solu- component finite-element methods: tions do not appearhasbeen established to solvethe electromagnetic field problem of the closedwavegnide filled with various anisotropic media. This 1) occurrence of the nonphysical spurious solution, method is basedon the approximate extremization of afunctional, whose 2) restriction on the discontinuity of either permittivity Euler equation is the three-component cnrlcurl equation derived from the or permeability of the media. Maxwell equations, with anewconforming element. Specific examples are given and the resnfts are compared with those obtained by exact solutions andIongitmfimd two-component finite-element solutions. Very closeagree- II. VARIATIONAL FORMULATION OFTHE ment was found and afl nonzero eigenvalues hiwe been proved to have MAXWELL EQUATIONS one-to-one correspondence to the propagating modesof the wavegnide. Consider an arbitrarily shaped metal waveguide im- mersed in several anisotropic and Iossless dielectrics as 1. INTRODUCTION shown in Fig. 1. This waveguide is assumed to be uniform A SA RESULT of the broad variety of practical appli- along its longitudinal z axis. ? and ~ denote the tensor cations of the closed wavegtiide filled with several permittivity and the tensor permeability without off-diag- kinds of media in microwave and optical frequency re- onal elements, respectively, and they are assumed to be gions, the development of methods to solve the associated constant in each region. electromagnetic field problems has attracted the attention Maxwell curl equations for time-harmonic fields are of many researchers. The finite-element method, which vXH=jdE (1) enables one to compute accurately the mode spectrum of a waveguide with arbitrary cross section, has been widely vXE=–jcofiH (2) used [1]–[7]. However, the two-component finite-element where the vectors E and H are the dielectric- and the solutions have been known to include nonphysical spurious magnetic-field intensity, respectively, and o is an angular modes [2], [3]. frequency. From (1) and (2), we construct Konrad has derived a three-component vector varia- tional expression for electromagnetic field problems [5], E= –(j/a){-lV XH (3) and has selected a family of functions, as a trial solution, in H= (j/u)p-lv XE. (4) which each component of the vector field is continuous By taking the curl of (3) and (4), and then substituting into along all interelement boundaries [6]. ,Therefore, the (1) and (2), the following common curlcurl equation is material parameters are restricted; either permittivity or obtained: permeability should be constant in all regions. Spurious solutions have likewise appeared as the result of this v Xp-yv x v)–arz~v=o (5) numerical calculation. We investigated his three-component formulation for the where V denotes either E-or H, and j and Q are the condition required of trial solutions, and have concluded material tensors as shown in Table I. that the necessary and sufficient requirement for the trial At the interface of the region, an appropriate boundlary solution is not so strict as the one in [6], condition must be satisfied by the field vectors. The inter- In this paper, the functional describing the behavior of face continuity between two contiguous media (say the rth ,. the electromagnetic fields in anisotropic waveguides is in- and sth) requires that troduced and the set of trial functions perfectly satisfying nx(v”–v”)=o (6) the boundary conditions required in the functional, a so- (j/~) nx(fF1v xV’-j;lvxv’)=o (7) called conforming element, is derived. This approach has improved the following two serious problems which are along their common boundary, where n is a surface normal inevitable in the previous two-component and three- unit vector. On the other hand, for .$e electric wall and the magnetic wall, the boundary condltlon of the electromag- Manuscript received February 14, 1983; revised May 21, 1984. netic field requires either The author is with the Department of Electncaf Engineering, Yama- nxv=o (8) guchi University, Tofciwadai, Ube 755, Japan. 0018-9480/84/1000-1275 $01.00 01984 IEEE 1276 IEEETRANSACTIONS ON MICROWAVE THEORYAND TECHNIQUES, VOL. MTT-32, NO. 10, OCTOBER1984 A x ,2_v,,r!!rlL4 vy2 7 Vxl v., ~h. t Vyl Perfect conductor xl -------- ~ J Vzl! - ;V,2 Fig. 1. Configuration of the metal waveguide. 0 Y1 Y* Y TABLE I RELATIONS BETWEEN(j, ~) ANDMATERIAL Fig. 2. Rectangular element. TENSORSAGAINST V. v Ge trial function is sufficient to satisfy the admissibility re- E ;: quirement presented by the previous section. HH :; The vector n X V on the interface of the contiguous regions can be separated into two components: i.e., V, is a or longitudinal component and ~ is a tangential component (j/@) in the x –y plane. The trial solutions are formed by nx(F1v x J“)=0. (9) approximating u=as a bilinear form of x and y within The electromagnetic problem defined by (5) with the each element and OXand Uyas a linear function of x or y. forced boundary conditions of (6) and (8), is expressed by Within each element, the value of v=is interpolated by the vertex values of V,, and those of UXand Uyby the side 8F=0 (lo) values of VXand Vy, respectively, on the element boundary. in which F is the functional whose variation yields (5) as a Fig. 2 illustrates a rectangular element of which side lines Euler equation and (7) and (9) as natural boundary condi- are held parallel to the coordinate axes. The eight nodal tions. The functional F is determined to have the form points described in the element consist of the four corner points (corresponding to unknown values of ~) and four F=+~{(V XV*)”fl-l( VXV)-ti2V*” QV} du (11) side points (corresponding to unknown values of VX and ~). where the asterisk denotes the complex conjugate. Using matrix notation, the approximate vector func- In this paper, traveling waves of the form tional form of o is expressed as V= (ixVx+iy~y + izVz) e-~fl’ (12) V={ UX7.7YUZ} (14) are treated, where ~ is the propagation constant. By sub- where stituting (12) into (11), the particular functional is given by Ux= {Vx}q(px] (15) Uy={m%yl 02= {VZ}T[’BI 1 and +p2(px;ll~12 +Py;11L12) [ffxlT=[&f21 ) –@2(9xxlvx1+2~yylvy12+qzzlvz12~)~. (13) (17) [9YIT=[L {21 The surface integral in (13) is to be evaluated over the cross [vzlT = [{ICI (,$2 {2’$, {2’$21 / section of the waveguide. (,= (-%- x)/kx, {2= (x -x,)/hx III. FINITE-ELEMENT METHOD &=( Y2-Y)/~y, $2= (y - .YJ/hy ’18) ) The selection of a family of trial solutions for the Rayleigh–Ritz technique is facilitated if the cross section and T denotes transverse. Equation (14) can be rewritten as of the waveguide is represented by a series of finite ele- ments. If we consider the subdomain as one region, the 0= {V}qfl] (19) EMW: FINITE-ELEMENTANALYSISOFWAVEGUIDES 1277 where dimension of v,, that is four. The matrix [S] is the matrix representation of the curl operator on the space having [Q] (20) as a basis. MCfxoo Substituting (19) and (25) into (13) and performing the indicated integrations, the contribution of the particular [Q]= o 9, 0 (21) element “e” to the total values of F is obtained. The o 0 % resulting expression with respect to the parameters gives and [0] is a zero matrix. F’=*({v}f[K]{v} -J{v}~[M]{v}) (30) The partial derivatives of [TX], [qY], and [9,] of (17) with respect to x and y are given by in which +.1 = [%1[901! &[8J=[~y][%l [KI=[SI*(JJITIP-’[ WWY)[SI’ (22) +[%l=[ffzl[%l> J8%Y % 1=[%1[9,1 I (31) where [kf]=fJ[Q]@[Q]=dxdy [%1=+[-:]> [4=+-[-;] Y –1 !1‘Bz-]1=*[-iA411 o 0 o ~ o A’& o (32) [AZ]=; x 1 0 0 M3J 0 [ and (23) [KJ=P2PY;’[QJ+J I and [K221=P2ZZ;[Q21 +Z’[Q51 [901 =[11. (24) [KSSl=P.;’IQCl+ PY;’[Q~l From (22), (23), and (24), the v x v is derived as follows: (33) VXV={V}T[S][*] [K,zl= [K2JY= -P.; ’[Q81 (25) [K,,] = [K,’]f= -jPp;~[Qg] where [W1]=9..[Q1] “]”[’! :: T] ‘2’) [~221=~yy[Q21 (34) 00 [JLs]=%z[Qs] } [q?]= ? Ipx o (27) where ~denotes the complex conjugate and transverse, and [1o 0 90 the matrices [Qj] (i= 1- 10) are given in the Appendix. By and [1] is a unit matrix. applying the Silvester’s inequality to (31), the rank of [K] On the other hand, from the commutativity of the dif- will become equal to that of [S]. ferential operators d/ax and 8/ dy, the following relation Summing the contribution of all elements over the cross is obtained: section of the waveguide yields [B.][AY] ‘[ A.][BX]. (28) Using the relation of (28), it is derived that the rank of =;({P}’[k]{P}-J{ P} ’[ Aq{P}) (35) the 8X 5 matrix [S] of (26) becomes four. This factor can be explained as follows. From (19) and (25), the curl where operator v x is a linear operator from the space having [k]=~[K] (36) [@] as a basis to the space having [~] as a basis. Therefore, e the operator is a degenerate operator with a kernel, which [M]=z[il’1] (37) is the subspace satisfying the following relation: e vlvZ + jbor = O (29) where {~} is an ordered array of the three-component where vl is a transverse operator and q is a transverse nodal variables. The matrices [~] and [~] are an adjoint component of o. The nullity of the operator is equal to the matrix. Hence, the variation of F in (35) gives the follow- 1278 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.MTT-32,NO.10,OCTOBER1984 2,0 I I 1.0 0,5 ‘Perfect ccnductor Fig. 3. Cross section ofhalfdielectric-loaded metal wavegnide; (l=(., PI=Po, ~2=4co, P2=I.L0. 0 2,0 4,0 6,0 8,0 k,H 2,0 Fig. 5. Comparison of exact solution and present three-component finite-elementanalysisresults. 1,5 1,0 )E, j HZ 0.5 0 2,0 4.0 6.0 8,0 koH Fig. 4. Dispersion characteristicsfrom two-component finite-element E. anrdysis. Fig. 6. Plots of field intensity of LSM1l mode for (a) E- and (b) ing algebraic eigenvalue problem: H-presentationat/lH = 5.0. [R]{ P}-L7[M]{F}=0. (38) The matrix [~] has components proportional to the ~“, /31, the two-component finite-element analysis. In Fig. 4, the and B2. The solution of this eigenvalue problem will pro- occurrence of the spurious modes and the difficulties at vide the required results on the angular frequency of /3/k0 = 1 can be found. Fig, 5 shows the dispersion char- various modes on a particular waveguide. From the anal- acteristics obtained from the present finite-element analysis ogy between the space of the element and the space of the for the E-formulation and from the exact solutions. On cross section of the waveguide, the rank of [~] is equal to comparing the results of Fig. 4, the spurious modes have NX+ NY, where NX and NY are the number of unknown not occurred at all in Fig. 5. And then, it is confirmed from values of {VX} and {VY}, respectively. Therefore, the alge- the numerical experiment that the algebraic system of (38) braic system of (38) has N, zero eigenvalues where N, is has the implicit zero eigenvalues, of which the number is the number of unknown values of {V,}. Other field compo- equal to that of the longitudinal nodal points. All nonzero nents can be derived from the eigenvector of (38) by (3) or eigenvalues were found to have one-to-one correspondence (4). to the propagation modes from its field distribution. Agreement between the finite-element solutions and the IV. EXAMPLESAND CONSIDERATION~ exact solutions is excellent. Fig. 6 shows the. field intensity To demonstrate the excellent quality and the accuracy of configuration of the LSMII mode taken at ~H = 5.0. These the finite-element analysis of the previous section, the field configurations are almost identical with those ob- solutions for sample problems are given and are Gompared tained by the exact solution so that the values of HX over with the conventional two-component finite-element solu- all cross sections of the waveguide are equal to zero. tions [2], [3] due to insufficient data of the three-compo- Second, a problem consisting of a rectangular metal nent one [6]. In our program, all the eigenvalues of (38) are waveguide with rnicrostrip of finite thickness in the center, obtained. as shown in Fig. 7, is treated. This waveguide geometry is First, the problem consisting of a rectangular metal given in [2] and the spurious modes were shown to be waveguide half-filled with dielectric, as shown in Fig. 3, is mixed with physical modes in the solution of the two- treated. The propagation modes in this waveguide are component finite-element method. Fig. 8 shows the disper- classified into LSM, LSE, and TE modes, asis well known. sion characteristics obtained from our method where the Fig. 4 shows the dispersion characteristics obtained from spurious modes have not occurred at all and the number of HANO:FINITE-ELEMENTANALYSISOFWAVKNJIDBS 1279 [Qd=[%][%][%]T (A4) i_f #l Air [Q,] =[~y][%o][~y]T (A5) (A6) [QJ=[BZl[~y][BJT b [QT]=[4][%][4]T (A7) (A8) [Qg]=[%l[%JIAy]T 1 [’Qg]= [CJJB.IT (A~) , \Perfect conductor ; [Qm]=[L][4]T (AlID) Fig. 7. Hrdfcrosssectionofclosedmicrostrip; a= 2b = 2W = 4H. where [uxx]=J’’Jx*[qx] [qx]”dxdy=~[: ;] (All) _o_. _._.— 13- #-0-”-” y~ x, ./.- 12- ,.0-. O,*. ‘--O--- DaIY [qy] =Jy2Jx2[9y][9yl’dxdY= y[; ;] (Al~) 11- . 10- -“OOF”Z--- —.— Present analysis y~ xl -. 9- E/EO=16 [42211 ? 5? 8. 1 , 1 0.05 0.1 ----0-’15 ((WC) 2 [qz]=Jy’Jx’[qz] [9z]’dxdy=# ; ; } ; 7.5- y~ xl .x*-”-”-”””-” 7.0- #“-”- 1224 6.5- -w”’ (A13) -...--”/ [um]=Jy2Jx2[qo] [qo]~dxdy=fixhy [l]. (A14) E/E@=9 :::. /p4- y, xl ACKNOWLEDGMENT The author wishes to thank Prof. H. Kayano for his helpful discussions and advice, and Prof. H. Matsumolto for-his helpful advice. 2.5 c/Eo= 4 IV3FERENCES 0.1 0.2 0.3 0.4 0.5 0.6 (mH/c) 2 [1] P.Silvester,“A generafhigh-orderfinite-elementwaveguideanalysis Fig. 8. Comparison of thetwo-component tmdpresent three-component program; L%&?Trans. Microwave Theo~ Tech., vol. MTT-17, pp. finite-element ar2alysi8results. 204-210,Apr. 1969. [2] P. Daly, “Hybrid-mode analysisof microstrip by finite-element methods,” IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp. the zero eigenvalues were confirmed to be equal to that of 19-25, Jan. 1971. [3] M. Ikeuchi,H. Sawrdri,andH. Niki, “Analysis of open-typedielec- the longitudinal nodal points, aswell. tric waveguides bythefinite-elementiterativemethod; IEEE Trans. Microwave Theory Tech.,vol. MTT-29, pp.234-239,Mar. 1981. V. CONCLUSION [4] C. Yeh, S.B.Dong, andW. Oliver, “Arbitrarily shapedinhomoge- neousopticaffiber orintegratedopticalwaveguide~J. Appl. Phys., In this paper, the finite-element method for solving the vol. 46,pp.2125-2129, May 1975. dielectric-loaded waveguide problems was presented in [5] A. Konrad, “Vector variational formulation of electro-magnetic fields which the nonphysical spurious solutions included in the in anisotropic media; IEEE Trunk. Microwave Theory Tech., vol. M’IT-24, pp.553-559,Sep.1976. solution of the two-component finite-element method do [6] A. Konrad, “High-order triangular finite elementsfor electromag- not appear. This program has a specific number of zero neticwavesin anisotropicmedia,” IEEE Trans. Microwave Theory Tech.,vol. MTT-25, pp.353-360,May 1977. eigenvalues. The element used in our formulation is re- [7] N. Mabaya, P. E. Lagasse, and P. Vandenbulke, “Finite element stricted to the rectangle, so that the arbitrary cross section analysis of optical waveguides~ IEEE Trans. Microwave Theo~ of the waveguide must be divided into the small rectangu- Tech.,vol. MTT-29, pp.600-605,June 1981. lar region. I m Future problems in the present finite-element analysis . Mitsuo Hano wasborn in Yamagucbi, Japan, in will be the formulation with the triangular element and the 1951. He received the B.S. and M.S. degreesin treatment of needless zero eigenvalues. electrical engineering from Yamaguchi U2river- sity, in 1974and 1976, respectively. APPENDIX From 1976 to 1979, he was a member of the Faculty of Science,Yamaguchi University. Since The [Qj] matrices in (33) and (34) are given by 1979, he has been a member of the electrical engineetig faculty atYamaguchi University. He [Q,]= [Uxx] (Al) hasbeen engagedin researchof fight modulation using the magnetooptic effect and electromag- [Q,] =[q,] “(A2) netic propagation. Mr. Hano is amember of the Institute of Electrical Engineers of Japan [Q31= [%1 (A3) and the Institute of Electronics and Communication Engineers of Japan. 1280 IEEETRANSACTIONS ONMICROWAVE THEORYAND TECHNIQUES, VOL. MTT-32, NO. 10, OCTOBER1984 Phase Shifts in Single- and Dual-Gate GaAs MESFET’S for 2–4-GHz Quadrature Phase Shifters JYOTI P. MONDAL, ARTHUR G. MILNES, FELLOW, IEEE, JAMES G. OAKES, MEMBER, IEEE, AND SHING-KUO WANG, MEMBER, IEEE Abstract —The variation of transmission phase for single- anddual-gate peared a need to realize a phase shifter which can be easily GaAs MESFET’S with biaschangeanditsprobable effects ontheperform- integrated with the rest of the circuitry using the same anceof anactive phaseshifter have been studied for the frequency range 2 technology. The p-i-n and ferrite approaches are not con- to 4 GHz. From measured S-parameter vahtes for single- and dtud-gate venient for monolithic integration. In this paper, a phase transistors, the element values of the equivalent circuits were fitted by shifter using dual-gate MESFET’S reported recently by using the computer-aided design program SUPER COMPACT. For the normal full-gate voltage range Oto – 2 V at VD~= 4 V, the Kumar et al. [1] will be studied. This phase shifter uses the single-gate MESFET varies in transmission phase from 142° to 149° at operating principle shown in Fig. 1. Two signals, 90° 2 GHz, and from 109” to 119° at 4 GHz. However, with drain voltage out-of-phase, are presented to the two channels, namely, x varied from 0.3 to 4 V and aconstant gate-voltage bias of OV, the phase and y. The output of each channel is controlled by a shifts are much larger, 105° to 145° at 2 GHz and 78” to 112° at 4 GHz. variable gain amplifier using a dual-gate MESFET. These This suggests that large phase shifts may be expected in adual-gate device andthis isfound to be SO.With V~~= 4V and VG~l= –1.0V,Vfiation of two signals are then combined by an in-phase combiner to control (second) gate bias from O to – 1.75 V for the NE463 GaAs produce a resultant vector, as shown in Fig. l(b). The MESFET produces a transmission phase variation from 95° to 132° at 2 vector amplitude as well as the angle of rotation can easily GHz and 41” to S8° at 4 GHz. be controlled by adjusting the individual x and y compo- Such phase shifts cause both mnpfitode and phase errors in phase-sfdfter nents. In this way, one can achieve a phase shift of 00 to circuits of the kind where signsdsfrom two FET channels are combkred in quadrature with their gate voltages controlled to provide O“ to 90” phase 90° and, with four dual-gate FET’s, Kumar et al. [1] have control with constant mnpfitade. For the single-gate FET examined, the shown how one may have a full 00 -to-360° phase shift. expected ampfitude and phase errors are 0.30 dB and 6° at 2 GHz, and This type of phase-shifting technique is different from 0.36 dB and 10° at 4 GHz. If dual-gate FET’s are used in similar circuits, that studied by Tsironis and Harrop [2], where the intrinsic the distribution of errors is different. For NFA63 devices, the correspond- circuit elements are changed by changing one of the gate ing figures are 0.56 dB and 2° at 2 GHz and 1.2 dB and 3° at 4 GHz. The advantage of the dual-gate configuration is that the input impedance biases, and this in turn changes the transmission phase. conditions are more constant than for the single-gate configuration. They obtained a gain of 4 dB with 120° continuous phase shift at 12 GHz. This is suitable for narrow-band applica- I INTRODUCTION tions. The advantages of a dual-gate MESFET phase shifter PHASE-SHIFT CIRCUITS are needed in phased-array are stated in [1], to which may be added the advantage of antennas to steer the radiation direction by varying monolithic integration. Such shifters are limited to a low- the phase across the array elements. The type of phase power stage and are followed by amplification before the shifters to be used is decided by specific requirements like signal is fed to the antenna elements. The phase-shifted low VSWR, power-handling capability, insertion loss, amplitude reported in [1] shows a fluctuation of +2.5 dB; switching speed, and bandwidth, together with cost, size, this kind of amplitude variation would produce unaccept- weight, and other mechanical considerations. Switched able beam control in a phased-array antenna. One cause transmission-line phase shifters using p-i-n diodes and for such a variation becomes apparent if one considers the ferrite phase shifters are among the technologies used. way the phase shift is being carried out. For obtaining a With the rapid development of microwave integrated 450 phase shift, both the channels are switched on (i.e., 0.0 circuits on semi-insulating GRAS substrates, there has ap- V on the control gate); then, keeping one of the channels fixed, the other control gate voltage is ramped linearly Manuscnpt received February 23, 1983; revised April 30, 1984. This from OV to pinchoff. This rotates the vector resultant from work was supported in part by the Westinghouse R & D Center and by 45° to 0°, as in Fig. l(c). With the channel action inter- Carnegie-Mellon University. changed, the vector is rotated from 450 to 90°. If we J. P. Mondal and A. G. Milnes are with Carne~e-Mellon University, Pittsburgh, PA 15213. assume each channel has constant transmission phase, the J. G. Oakes was with the Westinghouse R & D Center, Pittsburgh, PA resultant amplitude will vary from (fiA ) at 450 to A at 0° 15235. He is now with Raytheon, Northborough, MA 01532. (or 900), where A is the maximum amplitude in any S. K. Wang was with the Westinghouse R & D Center, Pittsburgh, PA 15235. He ISnow with Hughes Aircraft, Torrence, CA 90509. channel with the control gate bias at OV. This will cause a 0018 -9480/84/1000-1280$01 .00 01984 IEEE MONDAL et a[.: PHASESH3FTSIN SINGLE- AND DUAL-GATE MESFET’S 1281 X.Channel , ‘“’”m” I 3dBW’ ;Two;hannels ~In-Phase~ I Coupler 1w!ihAmplifiers[CombinerI 1 ‘in (a) ———— ———. — Y R SP -- ‘ .\ @=ian-l (AylAx) L G \ R2= AX2+Ay2 SP \ Source \ Where,ITXI =Ax T \ 1~1 =Ay ‘Y \ (a) H h x G Intrinsic Elements:Cg~=t42PF (b) Rin=6 n Y Cdq=0.031pF ————--—— gmo=U.8mmho A7 lncusofRa, Ay-O T= lZ8pS &- Rd$=293Q ~ Ifilmax= IXI ~ax =A Mrinsk Elements: Cd~=0.126PF T Lgp=0,1nH Iv kL.._.&x ‘s’ =ao’ n“ >:;;;” 9P Ax %P=0“4 o (c) ‘d’= Z** Fig. 1. Quadrature phase shifter operation: (a) Schematic diagram of (b) active phase shifter showing input and output couplers and x- and y-channel amplifiers. (b) Resultant output vector composed of x- and Fig. 2. Equivalent circuit model of the single-gate FET. (a) The circuit y-channel components. (c) Phase\arrplitude pattern used by Kumar model and elements. Intrinsic elements inside the dotted line may et al. [1]. change with bias. (b) Typical element values for V~~ = 4.0 V, VG~= O V, and 1~~= 34mA. maximum deviation of amplitude at 450, with respect to the amplitude at phase shifts of 0° and 90°. This accounts for 3-dB variation from the minimum value, occurring at II. TRANSMISSION PHASECHARACTERISTICSIN A N x 90° phase shifts, N being an integer. During the SINGLE-GATE FET subsequent analysis, we will point out that phase variation with gate voltage will add to the amplitude and phase A single-gate GaAs MESFET (LN1-5 # 2B) fabricated fluctuation. at Westinghouse was chosen for this investigation. The gate Before considering phase shifters using dual-gate length was 1 ~m and the gate width was 4X 75 pm. The MESFET’S, it is interesting to examine the probable per- source–gate distance was 1pm and the gate–drain distance formance of a phase shifter using single-gate FET’s in the was 1 pm. The channel doping was 1.1X 1017cm –3and the two channels, instead of dual-gate MESFET’S, The gate pinchoff voltage was just under – 2 V on the gate. The voltages of the single-gate FETs will be varied to change S-parameters of this transistor were measured from 2 to 4 the amplitudes in the two channels. GHz at different gate-bias points with the drain voltage Section II describes the variation of the single-gate FET fixed. They were then used to determine the equivalent intrinsic elements with gate-bias change and their effect on circuit model of Fig. 2. The typical element values after the transmission phase characteristics. Tha change in the computer fitting using SUPER COMPACT are given in intrinsic elements with drain-bias change in a single-gate the caption. The bias-voltage variation affects only the FET and with control gate-bias change in a dual-gate FET intrinsic elements of the equivalent circuit. The parasitic are discussed with their effects on the transmission phase due to bonding wires were, therefore, removed from the in Section III. Section IV shows the overall effects on the model. The resulting circuit has a transmission phase given ‘ performance of single- and dual-gate FET phase shifters by @= and discusses a possible correction for the amplitude error. @fl4-1)–(tan-l (3) (1) While variable gain amplifier-shifters using single- or Y dual-gate FET’s must include matching networks, it is where believed that the variation of transmission phase with bias in the FET itself is the primary source of phase errors. This tiT2 +sin~r study, therefore, focusses on the FET intrinsic phase re- A= sponse. Cos6.)’7+’ (/.?7172 1282 IEEETRANSACTIONS ONMICROWAVE THEORYAND TECHNIQUES, VOL. MTT-32, NO. 10, OCTOBER1984 Gd,+ GL G, GL + Gd, 1.51 1 I 1 1 X=(I) T3 + ll-~ Single Gate FEI ( gmo gmo Gcis 1.4’ LN1-5, +— G, Cdg + cd, ——cdg +Costi’r 1.3 1.2i gmo Gds G(i. ) G~ + Gd~ 1.1 cd, —14)3T1T2— — u2r1r2 sin a7 ~. Gds () Gds 1.0’ z G~ i- Gd, U2 : 0.9 y=G, - = (’~) cdgG;,cds s ( Gdsgmo ) z: 0.8 _-/4 GL+ Gd, ~o~~~ 0.7 —6)2T1T2 () Gd, 0.6 6J2 c 0.5 —— c ~+Cd~r3 +ti#sinu7 gmo ( gs Gd$ ) ds 0.4 in which 0.3 I I I I -2.0 -1,5 -1.0 -0.5 GateVoltage r = transit time in the gate region, Fig. 3. Amplitude and angle of S21are shown against the gate voltage at ~1= RinCg~, 2, 3, and 4 GHz. The values are calculated from the equivalent circuit derived from the measured S-parameters. Cdg ~2. — “~ i3m0 r 1 [ I 1 1 I 73 = (:dg + cgs)/Gds~ I 6.0 G,= ~ and G~ = l/R~. 0.45 s -& 5.0 0.40 2 In the above equation, we have identified three im- c = 0.35 & portant time constants rl, Tz,and r~, which are dependent a- 4.0 Q0,30 I on elements that change with bias. The variations of transmission phase and amplitude with gate bias are given in Fig. 3 for 2, 3, and 4 GHz. The phase variation near maximum amplitude (i.e., near V~= O V) -Cn=.0.05 I~cdq r I 1 I increases as the frequency increases. In a quadrature phase- u o I I I I shift circuit, this will effectively increase the amplitude EzzzzI variation of the resultant vector tip. A figure of merit can be defined as the ratio of the slope of the magnitude of Szl versus gate bias to the slope of the phase of S21with gate bias, i.e., -2.0 -1.5 -1.0 -0.5 0 (lAS211/AV~) GateVoltage Figure of merit= (A <S,l/AV~) “ Fig. 4. The variation of the elements in the equwalent circuit of a single-gate FET with change in gate-to-source voltage. The values are The higher the figure of merit is, the less the amplitude obtained through the fitting of the measured S-parameters. fluctuations with phase change, as will be seen from the discussion in Section IV. The variation of the FET ele- ments with the gate bias is shown in Fig. 4, As the FET The fall in Fig. 4 of transconductance g~O when ap- approaches pinchoff, the depletion region under the gate proaching pinchoff is expected. The channel resistance R~$ increases. This results in a decrease in both R,n and Cg,, as shows a small decrease near %~ = OV and then steadily shown in the upper curves of Fig. 4. The transit time ~ in increases as the channel is pinched off. As these changes the gate region also follows the same pattern. occur, the time constants of the device also change. As shown by Engelmann and Liechti [3], the pinchoff of The effect of each of three previously identified element the channel reduces the dipole region. We believe this groups or time constants [RznC~~, Cd~/gmo, and (Cg, + reduction of charge in the dipole modifies the electric field C~g)/Gd,l On the transmission phase of the FET is shown and so reduces the transit time. The variation of Cdg is in Table I. These time constants have been varied as the small. Others who have considered the bias dependence of gate-voltage changes. All FET elements were fixed at the the elements for the large-signal case include Willing et al. Vg~= – 1.0 V values, except those in the time constant [4] and Tajima et al. [5]. In our analysis, we have ap- being examined. As the time constant was varied, the phase proximately maintained R,nC@,,proportional to transit time of Szl was recorded and the maximum and minimum [6]. values at 2 and 4 GHz are noted in Table I. The R,nC~,