Contains reports on two research projects

Berk, A. D.

Schetzen, M.

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XII. MICROWAVE COMPONENTSProf. L. J. Chu Prof. J. G. Yates A. OrenbergProf. J. B. Wiesner L. D. Smullin R. H. RearwinProf. H. J. Zimmermann A. D. Berk M. SchetzenProf. R. B. Adler J. R. Fontana J. SciegiennyA. FERRITES AT MICROWAVE FREQUENCIESIn reference 1, variational principles for the resonant frequencies of a cavity or thecutoff frequency of a waveguide, both completely filled with ferrite, were given. Varia-tional principles for the resonant frequencies of more complicated systems, as well asfor the propagation constant of waveguides partially or completely filled with ferrites,have also been obtained and will be briefly discussed here. To illustrate their use andto develop confidence in them, they have been applied to inhomogeneous waveguides thesolutions of which are already known. It turns out that the variational principles leadto algebraic expressions which not only obviate the need of solving transcendental equa-tions but also give excellent accuracy even with rather crude trial fields. Furthermore,since the various quantities are explicitly related by algebraic expressions, the lattermay be used as design equations.1. Variational Principles for Resonant Frequencies of Cavitiesa. A cavity partially filled with ferrite. Equation 1 of reference 1 can be used inthis case provided that E is taken within the integral sign.b. A cavity partially filled with a dielectric. Here we can either use the varia-tional principle described in the preceding paragraph (with .i a scalar) orV X E • V XE* dvViXE. vE dvl E i E - E dvThe choice depends on whether it is easier to use E or H as a trial field. The symbolsof Eq. 1 are the same as those used in reference 1.c. A cavity with inhomogeneous and anisotropic substance. Here we require E, "to be hermitian. Then we haveE VX Hdv - H •Vx Edvn v - v(2)fvE * Edv + fH * -i HdvTo utilize Eq. 2, a trial field, say E, is assumed. H is then found from one of-57-(XII. MICROWAVE COMPONENTS)Maxwell's two equations. The rest of the procedure is standard. For the method ofderiving Eq. 2 see section 4 below.The formulas of the preceding paragraphs can be adapted to the cutoff frequenciesof waveguides, provided that we replace the resonant frequencies by cutoff frequencies,the volume integrals by surface integrals over the cross section of the guide, and E, Hby their z-independent parts.2. Variational Principle for the Propagation Constant yWe haveS fH V XEds -f E V XHds + j f H ' Hds +j E .E ' dsH a xEds - E a x Hdsz z(3)where E, H are the z-independent parts of the electric and magnetic fields, E , H aretheir complex conjugates, and az is the unit vector in the direction of propagation.To utilize Eq. 3, a trial field, say E, is assumed; H is then found from Maxwell'sequationV X E -j ya E = -jw * HThe rest of the procedure is standard in variational calculation.3. Applicationa. Equation 1 has been applied to the case of the lowest mode of waveguides par-tially filled with dielectric of permittivity E, as shown in Figs. XII-1 and XII-2. A verycrude approximation of the field E gives results that are in excellent agreement with theexact solutions given in reference 2. Moreover, the variational principle enables us tocompute cases similar to that of Fig. XII-1, but with the dielectric asymmetricallyplaced. As a specific result we shall give the expression for the cutoff wavelength Xcin terms of the geometry of Fig. XII-1.Fig. XII-1 Fig. XII-2-58-(XII. MICROWAVE COMPONENTS)c = a + Xe + - sin 1 (4)c e~a + a /]b. Next, we shall give an application of Eq. 3 to the case shown in Fig. XII-2.2 _ 5 2 6 Xe Z rr6 2 rr6S + k -- sin - cos -2 a e a 2w a a2 e1T2 1/2([3 v 2 koXe 2rr 2 vb 16 (kXS sin - si + sin - (5)a a a 9 a2 2where ko w 2o . Although Eq. 5 was obtained with a rather crude trial field, itfaithfully reproduces all the curves given in Fig. 11. 11 of reference 2.4. Derivation of the Preceding Variational PrinciplesThe variational principles given in reference 1 and that given by Eq. 1 can bederived by following known procedures of which an excellent account may be found inreference 3. The variational principles given by Eqs. 2 and 3 are more complicated,since the functionals [w], [y] depend on more than one function. A brief outline of thederivation of Eq. 3 will, therefore, be given. We start withv x E +j w- H = jya X E (6)V X H - jwY" E = jya x H (7)We multiply the first by H, the second by E, integrate over the cross section of theguide, and subtract. E, H are the adjoints of E, H, to be determined presently.S- V XEds - E V X Hds + jw 2 H-. - Hds + jw s E -E Eds= J I a xEds- E a x Hds (8)Equation 8 constitutes a variational principle for y if we vary E, H. It remains, how-ever, to relate E, H to E, H in a simple manner. This can be done by suitably trans-forming Eq. 8 to a variational principle for the determination of E, H. It then turns outthat E, H must satisfy the complex conjugates of Eqs. 6 and 7 provided that E, " arehermitian, that is, when the losses are disregarded. Thus E, H equal E , H .-59-(XII. MICROWAVE COMPONENTS)5. Mode-Expansion Solution of Anisotropic-Inhomogeneous Waveguide ProblemsIn reference 4 a general approach for the solution of waveguide problems with aniso-tropic and cross-sectionally inhomogeneous media was mentioned, and results of itsapplication were given. A brief exposition of the method follows.Let E(x,y)exp(-jyz) be the electric field, H(x,y) exp(-jyz) the magnetic field,Je exp(-jyz) the electric current, Jm exp(-jyz) the magnetic current. E, H, Je' Jm arethree-dimensional vectors depending on the cross-sectional coordinates only. Substi-tuting in Maxwell's equations, we obtainV X E - jy az X E + jwo H = Jm(9)V X H - jy a x H - jwE E= Jz o eFollowing the method used for the problem of a cavity, we wish to expand the fieldin terms of a complete set of modes, of which there are several. One possible choiceis a set comprising the usual TE, TM modes completed by a set of irrotational modes.The inclusion of the latter is necessary for the expansion of the irrotational part of thefield, as in the case of ferrites, for example. Such a set of modes is perfectly admis-sible and has the advantage that each mode has physical existence. However, we shallchoose another set which, although its individual modes have no physical existence, issimpler and real (the TE, TM set is complex).It is well known (5) that complete sets for vector fields can be generated fromsolutions of(V2 + ai 2) = 0, n 0 on boundary2 n=, = 0 on boundaryas a result of operating on 4n.n by v, V X az, V X V x azApplication of this procedure yields the setVqi Xaa n z b c 1E E a E =- V (10)n an n z n' n pn nfor the expansion of fields satisfying the electric type of boundary conditions, andHa  b Vn z c 1H =a , H n H = (11)n z n - n -- nn nfor the expansion of fields of the magnetic type. These modes satisfy a number ofinteresting relations, orthogonality being the most important. Moreover, they arenormalized, provided that 4n , 4n are normalized. We next expand E, Je in terms of-60-(XII. MICROWAVE COMPONENTS)a  b  c  a b  cE E , E H, J in terms of Hn H , H , substitute in Eq. 9, and equate coefficientsof identical modes. There results the system of equationsa a aa e + jw o h = Jm H dsn on m n-jy ea + jwo h c = J H c ds (12)n on m na a  c  E a-jwEo e + a h + jy hn = J E dson nn n e nb c b fJHbdsPe b + 1 e c + j h = J Hb dsnn n on m n-jwE eb + P h = J e E ds (13)o n nn e n-jwE e c- j h b J E c ds-o n n = e nep (p = a, b, c) are the coefficients of E P in the expansion of E, h P have an analogousn n nmeaning. The integrals are over the cross section of the waveguide. They representthe coupling between various modes.The grouping of Eq. 12 and Eq. 13 is intentional. When there are no electric ormagnetic currents Eq. 12 yields the TE modes, Eq. 13 the TM modes, of an emptyguide. For ferrites in waveguides we have Je = jwE oX eE and Jm = -jp o m - H. In thiscase Eqs. 12 and 13 reduce, after rearrangement, to a homogeneous set of equations.The eigenvalues are the values of the propagation constant y which render the determi-nant of the system zero.Equations 12 and 13 are suitable for approximate calculations. Frequently we know,on physical grounds, the small number of modes that are predominant in the unknownfield. We therefore assume the solution to be given just by these. Equations 12 and 13serve, then, to determine the relative strength of each constituent mode.6. ApplicationThe results given in reference 4 are based on the application of the preceding method.The two results given in sections 3a and 3b of the present report, and obtained by theapplication of the variational method, have also been obtained as a direct application ofEqs. 12 and 13. This is to be expected, since the trial field in the variational method-61-(XII. MICROWAVE COMPONENTS)was assumed to contain the same modes as the field used in Eqs. 12 and 13, and sinceit is known that orthogonality of a set insures finality of its coefficients of expansion (6).7. Integral Equation Method of Solving Anisotropic-Inhomogeneous WaveguideProblemsThe essence of this method is the formulation of the problem in the form of anintegral equation by means of suitable Green's functions.From Eq. 9 we have the inhomogeneous equationV 2 H+ (k 2 - 2 )H = (J J )  (14)whereP = -jwo J - V X J + jya X J - i v(v Jm ) - Y a V • Jo m e z e Lo m Wo z mO O2_ V(a J ) + j a z(a . J )o z m Lo z z mA similar equation can be written for the E-field. Next we define a magnetic Green'sdyadic Gh by the following relation where 6 is the delta function depending on the trans-verse coordinates, and I is the idem factor:V2Gh + (ko2 - y 2 )Gh = 16 (15)Physically, Gh is the magnetic field caused by a filamentary distribution of currentwith exp(-jyz) dependence and unit amplitude in an otherwise empty waveguide. Propercombination of Eq. 14 and Eq. 15 yieldsH = fGh P ds (16)where the integration is over the cross section of the guide. A similar equation isobtained for the E-field. In the usual case where J e Jm are given in terms of E, H,Eq. 16 and its companion for the E-field constitute a pair of integral equations and theusual tricks for approximate solutions can be applied. In particular, they are verysuitable for perturbation calculations, provided that the Green's dyadics are expandablein a set of modes. It can easily be shown that Eqs. 10 and 11 are just the sets we need,and that Gh is given byHa H a  Hb Hb Hc HCn - k + an y -k + Pn y - k 0 + P-62-(XII. MICROWAVE COMPONENTS)with a similar expression for the electric Green's dyadic. This method has been workedout in detail. In addition to giving an integral equation formulation of the problem itcompletely contains the results of the mode-expansion method. In fact, Eqs. 12 and 13are directly obtainable by this method. Note that the Green's dyadics defined here arefundamentally different from those usually defined in waveguides.8. RemarksAll three methods, the variational, the mode-expansion, and the integral-equationmethods, have been successfully applied to the following problems, in addition to thosealready mentioned: rectangular waveguide with a dielectric layer perpendicular to theelectric field, ferrite slab in a rectangular waveguide with transverse steaay magneticfield, ferrite rod concentric with circular guide and with longitudinal steady magneticfield, eccentric rod in circular guide with transverse steady magnetic field. The resultsare very promising and will be published soon.After having completed the development of the mode-expansion method we havefound that a somewhat similar investigation was made by Schelkunoff (7), who treatedthe problem from the coupled-transmission-lines point of view.A. D. BerkReferences1. Quarterly Progress Report, Research Laboratory of Electronics, M. I. T. ,Oct. 15, 1953, p. 57.2. Radiation Laboratory Series, M. I. T., Vol. 8, pp. 386-387.3. P. M. Morse and H. Feshbach, Methods of theoretical physics, Part II, chap. 9-4,(McGraw-Hill Book Company, Inc., New York, 1953).4. Quarterly Progress Report, Research Laboratory of Electronics, M.I. T.,Jan. 15, 1954, pp. 53-55.5. J. A. Stratton, Electromagnetic theory (McGraw-Hill Book Company, Inc.,New York, 1941) chap. VII.6. A. Sommerfeld, Partial differential equations in physics (Academic Press, Inc.New York, 1949) p. 5.7. S. A. Schelkunoff, Bell System Tech. J. , 31, 784-801 (July 1952).B. STRIP TRANSMISSION SYSTEMA Fourier integral analysis of the strip transmission system that was describedearlier (1) yielded results that compared favorably with experiment. However, due tothe complexity of the integral equations derived, the field pattern and the dependence of-63-(XII. MICROWAVE COMPONENTS)the propagation constant upon the various parameters of the system are obscured anddifficult to determine, making them impractical from an engineering standpoint.It has been observed that for a given system the phase velocity as a function of fre-quency is essentially constant over an extended range of frequencies; the variation isless than 2 percent over a frequency range extending from 2 kMc/sec to 10 kMc/sec.The constancy of the phase velocity would seem to indicate that a homogeneous planewave assumption is a good approximation to the dominant mode within this frequencyrange. However, since in the strip system the medium is only regionally homogeneous,the dominant mode is actually a TE-TM mode, and thus the validity of this assumptionrequires justification. A theoretical and experimental study along these lines is beingmade.M. SchetzenReferences1. Quarterly Progress Report, Research Laboratory of Electronics, M.I. T., April 15,1953, p. 86.-64-

Microwave Components

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Microwave Components

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