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Pc&mastec Send Address changes to IEEE, 445 Hoes Lane, I%wataway, NJ 08854. IEEE TRANSACTIONSON MICROWAVE THEORYAND TECHNIQUES, VOL. MIT-31, NO. 6,JUNE 1983 429 Propagation Characteristics of Striplines with Multilayered Anisotropic Media TOSHIHIDE KITAZAWA AND YOSHIO HAYASHI Abstract —Various types of striplines with auisotropic media are anatyzed. The anafytfcat approach used in this paper is based on the network analytical method of electromagnetic fields, and the formulation iiCe process is straightforward for complicated stroctnres. Some numericaf REGION(1) d, I +%+ results are presented and comparison is made with the resnlts available in ~x the literature. REGION(2) ’32t E2C0 I. INTRODUCTION REGION(3) d3 33C0 I T HE NETWORK analytical method of electromagnetic r- fields has been successfully applied to analyze the Fig. 1. Generalstructureof coupledstripshaving multilayered aniso- tropicmedia. propagation characteristics of planar transmission lines. The hybrid-mode analysis of single and coupled slots was presented by employing this method [1], [2]. Recently, the As a first step we express the transverse fields in each dispersion characteristics of single rnicrostrip on an aniso- region by the following Fourier integral: [t3ro].pic substrate have been obtained using this approach @ )=;J“{ J’p(a;z)fl(a; x) )dae-j~oY 7 Single and coupled striplines on an anisotropic substrate qo) [=1 –~ Ip(a;z)zoxj(rx; x) have been analyzed by several investigators [3]-[7], but i=l,2,3 (2) hybrid-mode analysis is available only for the single micro- where strip case [3], [7], [8]. The purpose of this paper is to outline a new approach ““JG__ Koe -j.., f2=flxzo which is an extension of the treatment used in [1]–[3] and is capable of giving the propagation characteristics of various types of striplines with anisotropic media inclu- Ko=: sively. In what follows, the formulation process is il- yo~,o K = Xorx+ K=IKI (3) lustrated using the general structure with multilayered uniaxially anisotropic media. Two methods of solution are where POis the propagation constant in they-direction, X., presented. One is based on the quasi-static approximation Yo, and Z. are the x-, y-, and z-directed unit vectors, and it derives the transfo~ation from the case with aniso- respectively, and 1= 1and 1= 2 represent E waves (Hz= O) tropic layers to the case with equivalent isotropic layers. and H waves (E, = O), respectively. Equation (2) shows The other is based on the hybrid-mode formulation and it that the field components are a superposition of inhomoge- gives the frequency dependent solutions. The numerical neous waves whose spatial variation is exp{ – j(ax + Doy)}. results will be presented for single and coupled microstrips, Substituting the above expression into Maxwell’s field coupled suspended strips, and coupled strips with overlay. equation, we obtain the following transmission-line equa- tion in each region: II. THE NETWORK ANALYTICAL METHOD OF ELECTROMAGNETIC FIELDS Fig. 1 shows the cross section of coupled strips having multilayered uniaxially anisotropic media; whose permittiv- ity tensors are where Eil 00 ~[0 = F== “)=- {i= o EiL () , i=l,2,3. (1) o 0 Cill [1 1)= Z! ManuscriptreceivedJune3, 1982;revisedJanuary18,1983. T. Kitazawaiswith theDepartmentof ElectricalEngineering,Univer- sity of Illinois, Urbana, IL, on leave from the JQtami Institute of Technology,Kitami, Japan. Y}’) = (5) Y. Hayashiiswith theKitami Institute of Technology,Kitami, Japan. 0018 -9480/83/0600-0429$01,00 01983 IEEE 430 IEEE TRANSACTIONS ON MZCROWAVETHEORY AND TECHNIQUES, VOL. MTT-31, NO. 6, JUNE 1983 SHORTCIRCUIT gion (1) can be obtained from the transverse fields accord- -, ,=,, ing to _._.!.- ~:v . v.(lfpxzo). (13) j(l.)q$,,l Substituting (2) and (12) into (13) and applying PO+O, E(l) can be obtained as z “.(/m aT{l)(a; z)tX(x’).e–~~(x–x’) dx’da. (14) ~ z=-(d,+d, ) —m SHORTCIRCUIT Fig. 2. Equivalent transmission-linecircuits for transversesection of Performing the integration by parts, using the equation of coupledstrips. continuity Notice that ~(i) and Kf) are the propagation constants in – j(.ou(x’) = *2X(X’) (15) the z-direction for E waves and H waves, respectively, and and applying the zero, frequency approximation u ~ O to Z(i) and Z2(i) are the characteristic impedance for these (14), we get waves. The boundary conditions to be satisfied are expressed as cosh{pl(z – dl)14} E$l)(~>z) = ‘2~7~rcm0 -~ ‘(a)~, sin~(p,d,]al) follows: ~(’)(dl)=O (6) .a(x~)e–l~(x–x’)~a~x’ (16) T“y(+o) = vp(-o) (7a) where 1~1)(+ O)-- I}z)( –O) = i, (7b) F(a) = 1 (17) ~l=coth(pldllal) +czeL ~(2)(–d2+O)=qt3)(–d2–0) (8a) I/2J(–d2+O) =lf3)(–d2–0) (8b) 1+ ~ tanh( p2d,lal) tanh( p~dqlal) L= (18) ~(3)(--d2–d3)=0 (9) tanh( pzdzlal) + ~ tanh( P3d~lal) i,= - f~ f~(a;x’)i(x’)dx’ (lo) J—W where the asterisk signifies the complex conjugate func- Pl=~ ~ze=g (19) tions, and i(x’) is the current density on the strip conduc- tors at z = O“and may be expressed as and U(X’) is the charge distribution on the strip conduc- e = xOiX(x’)+ yoiY(x’). (11) tors. The potential distribution at z = Obecomes Considering the transmission-line equation (4) together V(X) ‘J%Z(X, Z) dz with the boundary conditions (6)–(9), we can obtain the equivalent circuits in the z-direction (Fig. 2). By conven- ‘;~~G(a;xlx’)u(x’) dad~’ (20) tional circuit theory, the mode voltages ~(i) and currents aO II’) in each region can be expressed in terms of il as where 2 F(a) fi(i)(a; z)= ZjZ)(a; z)il(a) G(a; xlx’)=m. — cosax cosax’ (for even modes) [al Ip(cK;z) =Zp(a;z)i/( a). (12) 2 F(a) —— — smax sin ax’ (for odd modes). The electromagnetic fields in each region can be obtained ~ “ Ial by substituting (12) into (2). (21) III. VARIATIONAL EXPRESSION FOR THE LINE On the strip conductor a < x < b, V(x) is equal to a CAPACITANCE constant VO, that is, the potential difference between the In the quasi-static approximation, the characteristic im- strip and the ground conductors pedance and the normalized propagation constant can be obtained from the line capacitance per unit length. We will V(X) =~o =~b/~G(a; XIX’) U(X’) dadx’, .0 derive a variational expression of the line capacitance of a<x <b. (22) the general structure shown in Fig. 1. The longitudinal component of the electric field in re- From (22), the variational expression for the line capaci- EUTAZAWA AND HAYASHI: STRIPLINSSWITH ANISOTROPIC MEDIA 431 tance can be obtained [9] E. —1 .—v~ h SAPPHIRE SUBSTRATE T to CQ ir- (a) /~b~mU(X)G(a;X,X’)a(X’) ~a~X’~X . (23) &o (Jba(x)dx}2 h SAPPHIRESU8STRATE t to where Q is the total charge on the strip conductor a < x < b d C@ [ Q=/bu(x)dx. (24) @) a co Equation (23), together with (21) and (17)–(19), suggests that, in the quasi-static approximation, coupled strips with d ISOTROPICDIELECTRICOVERLAY q c, multilayered uniaxially anisotropic media can be trans- formed into the case with effective isotropic layers, of h SAPPHIRESUBSTRATE ? co [ which the effective thickness and the relative permittivity (c) me ~~” di and ~~, respectively. Fig. 3. (a)Coupledmicrostrips.(b)Coupledsuspendedstrips.(c)Cou- IV. HYBRID-MODE ANALYSIS pledstripswith overlay. The analytical method for the frequency-dependent characteristics of coupled strips shown k Fig. 1 is ex- where ‘4k are vanatiomd parameters which are determined plained here. This method is analogous to those used in so that the best approximation is obtained. [1]-[3] and will be outlined briefly. In the numerical computations, the choice of the basis The transverse electric fields, which were obtained in the functions, ~k(~) in (27) and ~Xk(x) and ~yk(x) in (25), is integral representation in Section II, must be zero on the important. It is desirable that the edge effect should be strip conductors at z = O. This gives the integral equation properly accounted for, and that the approximation to the on the current density i(x) and the propagation constant true value should be systematically improved by increasing in the y-direction & The unknown current densities iX(x) the number of basis functions. Taking these requirements and iy(x) are expanded in terms of known sets of basis into account, we adopt the following families of functions functions as follows: for basis functions: N, ix(x) = Z axkfxk(x) k=l fxk(x)=uk(’(xis)} Ny iy(x) = ~ aYkfY~(x) (25) fk-l(x) ‘k-l(’(x~s)) k=l f,k(x) = where aX~ and aY~ are unknown coefficients. Substituting } ~~ (25) into the integral equation and applying Galerkin’s procedure, we obtain a set of simultaneous equations on S=(a+b)/2, W=b–a (28) the unknown aX& and aY~. The propagation constant can be obtained by searching the nontrivial solution. where 7“( y) and U~(y) are Chebyshev’s polynomials of The definition for the characteristic impedance is not the first and second kind, respectively. By the use of these uniquely specified due to the propagation of the hybrid basis functions, the fast convergence to the exact values is mode. The definition chosen here is obtained. Preliminary computations show that N = 2 in 2.== P (26) (27) and NX= NY = 2 in (25) are sufficient fOr any case. I: VI. NUMERICAL RESULTS where 10 is the total current on one strip conductor, and P,v, is the average power flow along they-direction. Numerical computations were carried out for single and coupled microstrips (Fig. 3(a)), coupled suspended strips V. BASISFUNCTIONS (Fig. 3(b)), and coupled strips with overlay (Fig. 3(c)). In The line capacitance is calculated by applying the Ritz the open microstrip configurations of Fig. 3, the boundary procedure to the variational expression (23). In this proce- condition (6) or (9) for Fig. 1 should be replaced by the dure, we express the unknown charge distribution u(x) as radiation condition. However, the resulting equations thus obtained are the same as those for Fig. 1in which dl ~ co dx)=.lo(x)+ i Afk(x) (27) or d~ e m. These calculations were performed using the k=l same computer program with very little modification. 432 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-31, NO. 6, JUNE 1983 / !,0. ~“ ,m ,/” I /’” /’ l--/’ ,/ (a) ~:,0 .—-— /’ ,/ /’ .// s,L e., I 0 0,, 0.’ .06 0.0s “,, 1 0 0. cd ,2 .,6 Fig. 4. Dispersion characteristics of single rnicrostrip on sapphire. (c~ .,, = 9.4, ~,, = 11.6; – hybrid-mode; —-—quasi static; ——- ,0r E1-Sherbiny’s [4].) (b) 90 /“” / ,, ,//.,, 70 =.O / ,0 / //’ 1 j / ,,// 0 w ,W .,, .,6 ,,0 r’ >.0 .,> ‘.0 1[ / Fig. 7. (a) Dispersion characteristics of coupled microstrips on sapphire. (b) Characteristic impedance of coupled microstrips onsapphire. (w/h ,0 0,,05 =1, a/h = 0.25; —even mode (hybrid-mode); ———odd mode (hy- / / brid-mode); —-—even mode (quasi-static); —--—odd mode (quasi- ,, ~~ / (//./’ static).) ,, .—. 002 ,.0’ ,0, 00, 0,0 h,, ,0.0, I Fig. 5. Dispersion characteristics of the first higher order mode of single microstrip on sapphire. (— this theory; —-—TM. mode of a sapphire-coated conductor; ———E1-Sherbiny’s [4].) 1~ /.-” m --- --- :, ---- ---- : -- ./,.,. ;. .0 ---- 2.0 ----~ I +---- ‘.0 I 0 0’ ,., ,,2 .,, h,, Fig. 8. Dispersion characteristics of coupled suspended strips. ([ ~ = 9.4, c,, = 11.6, W/h = 1, a/h = 0.25; —even mode (hybrid-mode); ———odd mode (hybrid-mode); —-— even mode (quasi-static); —-- Fig. 6. Characteristic impedance of single microstrip on sapphire — odd mode (quasi-static).) ,,0 Fig. 4 shows the dispersion characteristics, the frequency ,.2==7 dependence of the effective dielectric constant C,ff = ,/,., ~~/ti2cOp0, of single microstrip on sapphire substrates, -0.0i where Cefffor the dominant mode is reported and com- pared with the results of E1-Sherbiny [8]. The agreement is ~<L /“ /’ /-’ so ,.” quite good, although some disagreement appears for wide .,..,, strips. Fig. 5 shows the dispersion characteristics of the first higher order mode, which are also compared with those from [8]. Fig. 5 also presents the dispersion characteristics of the TMO surface wave of the sapphire coated conductor Fig. 9. Dispersion characteristics of coupled strips with overlay. (c~ = 9,4, cl,= 11.6, 6.= 9.6, PV\h =1, a/h = 0.25; —even mode (hybrid- which results when W = O. When the strip is not so wide mode); ———odd mode (hybrid mode); —-— even mode (quasi- compared with the substrate, the dispersion characteristics static); —--—odd mode (quasi-static).) KITAZAWA AND HAYASHI:STRIPLINBSWITH ANISOTROPIC MSDIA 433 of the first higher order mode are indistinguishable from line on a sapphire sub:trate~’ J. Inst. Electron. Conrrnun. Eng. Jap., vol. 62-B, pp. 596–602, June 1979. those of the TMO surface wave, [4] N. G. Alexopoulos and C. M. Krowne, ”Characteristics of single and The frequency dependence of the characteristic imped- coupled microstrips on anisotropic substrates,” IEEE Trans. Micrv- ance of single microstrip is shown in Fig. 6. Comparison of wave Theory Tech., vol. MTT-26, pp. 387–393, June 1978. [5] M. Kobayashi and R. Terakado, “Method for equalizing phase the results by this method and those from [8] shows that velocitiesofcoupledmicrostriplinesbyusinganisotropicsubstrate,” both results converge to the quasi-static values calculated IEEE Trans. Microwave Theory Tech., vol. MTT-28, pp. 719-722, from (23), but that some discrepancies appear at high July 1980. frequencies. For single rnicrostrip, the characteristic imped- [6] M. Homo, “Quasistatic characteristics of microstrip on arbitrary anisotropic substrates,” Proc. IEEE, vol. 68, pp. 1033–1034,Aug. ance is defined as 1980. zo=~ [7] F. J. K. Lange, “Analysis of shielded strip- and slot-lines on aferrite we (29) substrate transversely magnetized in the plane of the substrate,” o Arch. Elek. Ubertragung., vol. 36, pp. 95-100, Mar. 1982. [8] A-M. A. E1-Sherbiny, “Hybrid mode analysis of microstnp lines on instead of (26) in our calculations, whereas it is defined as anisotropic substrates’ IEEE Trans. Microwave Theoy Tech., vol. MTT-29, pp. 1261-1265, Dec. 1981. the ratio of the voltage at the center of the strip to the total [9] R. E.Collin, Field Theory of Guided Waves. New York: McGraw- longitudinal current in [8]. Hillj 1960, p. 162. The dispersion characteristics of coupled microstrips, coupled suspended strips, and coupled strips with a dielec- tric overlay are depicted in Figs. 7, 8, and 9, respectively. It should be noted that the dispersion characteristics of the Tosfsihide Kitazawa was born in Sapporo, Japan even mode of coupled suspended strips is more sensitive on December 1, 1949. He received the B.E., than that of the odd mode to the variation in d/h, there- M.E., and D.E. degrees in electronics engineering from Hokkaido University, Sapporo, Japan, in fore the frequency at which both modes have the equal 1972, 1974, and 1977, respectively, phase velocity varies largely. He was a Post-Doctoraf Fellow of the Japan Society for the Promotion of Science from 1979 VII. CONCLUSIONS to 1980. Since 1980 he has been an Associate Professor of Electronic Engineering at the Kitmni Various types of striplines with artisotropic media have Institute of Technology, Kitatni, Japan. Cut- been analyzed using the same approach, which is based on rentlv. he is a Visiting Assistant Professor of Electrical Engineering at the “University of Illin;s, Urbana. the network analytical method of electromagnetic fields. h-t Dr. Kitrszawa is a member of the Institute of Electronics and Com- this analytical approach, the derivation of Green’s func- munication Engineers of Japan. tions is based on the conventional circuit theory, therefore the formulation for the complicated structures is straight- forward. Computations have been carried out by employing, the efficient method based on the Ritz and Galerkin procedure to calculate the propagation characteristics of single and Yosfdo Hayashi was born in Tokyo, Japan, on October 28, 1937. He received the B.E. degree in coupled rnicrostrips, coupled suspended strips, and coupled electncaf engineering from Chiba University, strips with overlay. Numerical results of single microstrip Chiba, Japan, in 1961, and the M.E. and D.E. were compared with other available data. degrees in electronics engineering from Hokkaido University, Sapporo, Japan, in 1965 and 1972, respectively. REFERENCES He served in the Japan Self-Defense Air Force [1] T.Kitazawa,Y. Hayasfti,andM. Suzuki,“Analysis ofthedispersion from 1961 to 1969. He was aVisiting Scholar of characteristicof slot line with thick metaf coating,” IEEE Trans. Electncaf Engineering at the University of 11- Microwaue Tlzeo~ Tech., vol. MTT28, pp.387-392,Apr. 1980. linois, Urbana, from 1981 to 1982. Currently, he [2] T. Kitazawa and Y. Hayashi, “Coupled slots on an anisotropic is a Professor of Electronic Engineering at the Kitami Institute of Tech- sapphire substratefl IEEE Trans. Microwave Theory Tech., vol. nology, Kitami, Japan. MTT-29, pp. 1035-1040, oct. 1981. Dr. Hayashi is amember of the Institute of Electronics and Commutti- [3] Y. HayashiandT. Kitazawa, “Analysis of rnicrostrip transmission cation Engineers of Japan. / 434 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-31, NO. 6, JUNE 1983 Resonant Frequencies, Q-Factor, and Susceptance Slope Parameter of Waveguide Circulators Using Weakly Magnetized Open Resonators JOSEPH HELSZAJN, MEMBER, IEEE, AND JOHN SHARP, MEMBER, IEEE Abstract —A useful quantity in the description of junction circulators is the difference between the spfit frequencies of the magnetized ferrite resonator. A knowledge of this quantity aflows the loaded Q-factor of a junction using aweakfy magnetized resonator to be determined. This paper derives an exact description of the former quantity in the case of the open quarter-wave long (partial-height) dkk resonator used in the construction of commercial turnstile wavegnide circulators. This is done by employing duality between aferrite-filled circofar waveguide having ideal electric wafl boundary conditions and one having ideal magnetic wall boundaries. The effect of an image wafl on the open flat face of the open resonator is -—- —- considered separately. The paper includes some remarks about the srrscep- tance slope parameters of disk and triangrdar open resonators. I. INTRODUCTION A USEFUL MODEL of a junction circulator is in terms of a magnetized ferrite or garnet resonator symmetrically coupled by three transmission lines. An im- portant quantity in the synthesis of this class of device is its \ Dielectric Spacer loaded Q-factor. For a weakly magnetized junction it is completely determined by the frequencies of the mag- ~’ ?~L ~ Ferrite Resonator “ netized and demagnetized resonator. The mode spectrum of magnetized resonators for use in the realization of junction circulators is therefore of considerable interest. An important class of commercial waveguide circulators (a) is that using quarter-wave coupled quarter-wave long open triangular or circular resonators open-circuited at one end and short-circuited at the other, or half-wave long open resonators open-circuited at both ends [1]–[6]. Fig. 1 de- picts schematic diagrams of three typical arrangements (b) using disk resonators. Introduction of an image plane in the configurations in Fig. 1(a) and (b) indicates that they are dual in that a single set of variables may be used to describe both geometries. The device in Fig. l(c) is also equivalent to the former ones, except that its susceptance slope parameter is twice that of the other two [6]. A (:) quarter-wave long magnetized ferrite resonator short- Fig. 1. Schematic diagrams of waveguide circulators using partiaf-height disk resonators. circuited at one end, and open-circuited or loaded by an image wall at the other, is therefore a suitable prototype for spectrum of this type of resonator has been understood for the construction of this class of device. Although the mode some time [4], [5], only the resonant frequencies of the demagnetized disk and triangular resonators have been Manuscript received July 22, 1982; revised December 17, 1982. determined in closed form [6]–[8], [19]. This paper gives an J. Helszajn is with the Department of Electrical Engineering, Heriot- exact derivation of magnetized disk resonators with the Watt University, Edinburgh EH 12HT, Scotland. J. Sharp is with Napier College, Edinburgh, Scotland. open flat face idealized by a magnetic or an image wall. 0018-9480/83/0600-0434$01.00 01983 IEEE HELSZAJN AND SHARP: WAVEGUIDE CIRCULATORS 435 This is done by employing duality between a ferrite-filled termined. This is done assuming that the resonator consists circular waveguide with electric wall boundaries, and one of a quarter-wave long demagnetized or magnetized ferrite with ideal magnetic walls. A complete modal description waveguide with ideal-magnetic walls open-circuited at one has also been recently utilized to evaluate the overall end and short-circuited at the other reflection coefficient of this class of circulator but the split cot(&Lo)=o (2) frequencies of the magnetized resonator have not been cot(p&Lo)=o. (3) implicitly determined [16]. 1 The experimental results in this paper indicate that The first of these two equations determines the length of junction circulators using weakly magnetized open-disk the open resonator from aknowledge of I@ and frequency and triangular ferrite resonators exhibit similar relation- ships between the magnetization of the resonator and its loaded Q-factor. The choice of resonator shape is therefore where primarily determined by its susceptance slope parameter. However, the ripple level in the circulator specification has (5) a significant influence on the impedance level of the gyra- tor circuit so that the resonator configuration is not as and critical aspreviously supposed [14], [15]. Since aknowledge of the loaded Q-factor and the susceptance slope parameter ko=:. (6) is sufficient for the synthesis of this class of circulator, Lo is the length of the open resonator (m), R is the radius some remarks about the latter quantity for disk and side and apex coupled triangular resonators are included for of the resonator (m), k. is the free-space wavenumber completeness. (rad/m), Cfis the relative dielectric constant of the garnet or ferrite resonator, p,ff is the relative permeability of the magnetized garnet or ferrite resonator, w is the radian II. SPLIT FREQUENCIES OF QUARTER-WAVE LONG frequency (rad/s), and c is the free-space velocity (3 X 108 OPEN-DISK RESONATORS m/s). Circulators using weakly magnetized resonators for which The second boundary condition may be solved for the the in-phase eigennetwork may be idealized by a short- relationship between the split frequencies of the resonator circuit boundary condition, exhibit 1-port equivalent gyra- in the neighborhood of the demagnetized one and the tor conductance at their operating frequency which may magnetic variables by forming the characteristic equation be described by for ~. and using the boundary condition in (3) with Lo ‘=fib’(a+u) fixed by (4) (1) (7) g is the normalized gyrator conductance, b’ is the normal- The split phase constants ~ ~ may be exactly evaluated ized susceptance slope parameter of the complex gyrator using duality between a magnetized ferrite filled circular circuit, and U. and u + are the operating frequency of the waveguide with ideal electric wall boundary conditions and circulator and the spli~ frequencies of the magnetized reso- one having ideal magnetic walls. The former problem is a nator. Any two of the above variables are sufficient to classic result whose solution is given as [10]–[12] define the gyrator equation. In the case of waveguide circulators using open partial- height disk resonators, analytical descriptions of these quantities are still somewhat incomplete. However, some experimental data [17] and one approximation [8] are avail- able on the split frequencies, and some experimental and semiempirical data is available on the susceptance slope parameter [9]. A knowledge of the split frequencies of such a resonator also leads to the description of its loaded Q- factor and to nearly exact synthesis of this class of circula- tor. The relationship between the off-diagonal component of the permeability tensor and the ratio of the difference between the split frequencies of the magnetized and that of the demagnetized open-ferrite resonator will now be de- -:[k’’f(B2++k’’f(:r ’10) lA recent paper, not available at the time of writing, giving theoretical data on the split frequencies of partial-height resonators in radial catities is given in [20]. The masmetic variables, assuming a saturated material, 436 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-31, NO. 6, JUNE 1983 05 ti f. 02 e 1 kR 082 01 M, 01760T 50 100 150 kA/m Fig. 3, Experimental split frequencies of loosely and tightly coupled, 07 08 09 10 11 magnetized open ferrite disk resonators, koR Fig, 2. Split frequencies of magnetized open-ferrite disk resonator. 15 are defined in the usual way by /J=l (11) ykfo ~=— (12) 10 /4.6) Q, /Leff ‘1 – K2. (13) MO is the saturation magnetization (Tesla); POis the free- 5 space permeability (47r x 10–7H/m), y is the gyromagnetic ratio (2.2 1x 105 (rad/s)/(A/m)), and p and K are the relative diagonal and off-diagonal components of the tensor permeability. Equations (9) and (10) may occasionally have imaginary I 07 08 09 10 11 roots which require modified Bessel functions in equation koR (8) [11]. Fig. 4. Loaded Q-factor of open ferrite resonator versus koll for para- Equation (8) may be solved for 6. or k. with n = + 1 metric values of ~/p. subject to the boundary condition in–(7). T–hecalculation indicates that the splitting between the degenerate modes is The most important quantity in the theory of quarter- a function of kOR (Fig. 2). The result for a typical value of wave coupled circulators is its loaded Q-factor. This kOR equal to 0.8 is parameter is usually expressed in terms of the split fre- ‘0°6’6(;)3-0061(q;uernci+es 0of t6he m2agn1etize(d:r)esonator using (l). The result ~+—~– is Ldo Q.=[’r+i]’--’) (15) () 0< ~ <0.5. (14) P The loaded Q-factor of the junction is therefore readily Fig. 3 gives one experimental plot of the split frequencies evaluated from a knowledge of the split frequencies. Fig. 4 for a disk resonator using a garnet material with a dielec- depicts the relationship between the loaded Q-factor and tric constant of 15.3, a magnetization of 0.1760 T, for kOR for parametric values of the magnetic variable K. which kOR = 0.82 at 9 GHz. The lack of symmetry in the Although the calculation of the split frequencies is exact, splitting at magnetic saturation is partly due to the form of the derivation in (1) disregards the influence of higher K in (12). The resonator is in direct contact with one order modes (in a strongly magnetized resonator) on the waveguide wall to minimize the effect of the image wall on description of the gyrator circuit. The results in (1) and the result. The agreement between theory and experiment (15) therefore apply to a weakly magnetized resonator is in this instance within 3percent. only. For the purpose of this paper, this condition is