2899-22-T E' i^,riL Ltrary TR No. 106 ENGN UMR0012 DEPARTMENT OF ELECTRICAL ENGINEERING COOLEY ELECTRONICS LABORATORY Technical Report No. 1Q6 A Study of Traveling-Wave Directional Filters with Wideband Ferrite Tuning By: D. K. ADAMS Approved by: A. B. MACNEE ~ EL I EC 4BORATO A Under Contract With: CONTRACT NO. DA-36-039 sc-78283, DEPT. OF ARMY PROJ. NO. 3A99-06-001-01, PLACED BY: U. S. ARMY SIGNAL RESEARCH AND DEVELOPMENT LABORATORY, FORT MONMOUTH, N. J. April 1961 _E~~~~~aS.5

CoolW Eletironios Laboratory Technical Rpor t No. 106 MRRAIA Paee 15, Eq. 29 should read:, m^ Wr) J 6W - +WW ) * t <2) (,kaar l)J a)1 (krl)J(kt~rl)+ ltjakri-y j -kjp ) - kr r Jm ri)m b) mjk J (29) Pap 16, Eq, 30 should'ead:?77~~~~~~1 ~(30) Pae 16, Eqo 31 should read 1 + 1( A 2 (31) +VPage 16, 15 hl r ed Page 16, l 15 8should red: vs d b -r) te post radius and ri thic kness, t od s ctiwvly) when the Page 18, Eqo 33 should read. J1 (krl) 0 (33) Pas 18, line 10 should read: radius r2o Thus - l

THE UNIVERSITY OF MICHIGAN OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR A STUDY OF TRAVELING-WAVE DIRECTIONAL FILTERS WITH WIDEBAND FERRITE TUNING Technical Report No. 106 2899-22-T Cooley Electronics Laboratory Department of Electrical Engineering By: D. K. Adams Approved by:.. A. B. Macnee Project 2899 TASK ORDER NO. EDG-4 CONTRACT NO. DA-36-039 sc-78283 SIGNAL CORPS, DEPARTMENT OF THE ARMY DEPARTMENT OF ARMY PROJECT NO. 3A99-06-001-01 April 1961

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iv LIST OF SYMBOLS v ABSTRACT vii 1. INTRODUCTION 1 2. DISCUSSION OF COUPLING NETWORKS 3 3. ANALYSIS OF COAXIAL TRAVELING-WAVE RESONATORS 6 4. THE INTRODUCTION OF FERRITE INTO A TRAVELING-WAVE RESONATOR 10 5. THE CHOICE OF FERRITE LOCATION IN A TM1ll CYLINDRICAL CAVITY 12 6. EXPERIMENTAL RESULTS 20 7. CONCLUSION 23 REFERENCES 24 DISTRIBUTION LIST 25 iii

LIST OF ILLUSTRATIONS Figure Page 1 Desired output spectral distributions with uniform incident spectrum at port o. 2 2 A directional filter employing two resonant cavities operating in standing-wave modes 3 3 A directional band-pass, band-elimination filter employing a traveling-wave resonator 5 4 Filter transmission loss in db as a function of resonator loss and input coupling 7 5 A coaxial traveling-wave resonator 9 6 A coaxial traveling-wave resonator loaded with a ferrite ring 10 7 Minimum tuning ranges for ferrite rings and p6sts as predicted by tuning criteria adopted in text 17 8 A traveling-wave directional filter employing 3-db directional couplers 20 9 A traveling-wave directional filter employing directional coupling holes 21 10 A typical experimental insertion loss envelope for a wideband, ferrite-tunable, traveling-wave directional filter 22 11 Proposed technique for improving traveling-wave directional filter performance 23 iv

LIST OF SYMBOLS SiJ General scattering matrix coefficient aJ Incident signal at port j of network bi Emerging signal at port j of network H General tuning parameter F(f) Transmission amplitude characteristic for tunable filter 2 sin 2n Power coupling coefficient for nth directional coupler c'Up ~ Complex phase shift in traversing path of L a Dissipative component of propagation constant in filter 3z9~ Phase component of propagation constant in filter Hi, H, Hz Components of RF magnetic field in r, cp, and z directions Er, E, Ez Components of RF electric field in r, cp, and z directions r, p, z Polar coordinates m, n, p Indices of normal modes in geometry with cylindrical symmetry h Height of coaxial cavity to Angular frequency of EM wave k Guide wave number (2) for propagation in a coaxial waveguide 10o, eo Permeability and dielectric constant in free space Er Dielectric constant relative to free space Jm(x), N (x) mth order Bessel functions of first and second kind Cm(x) Linear combination of Jm(x) and Nm(x) a,b Inner and outer radii of coaxial cavity 1i1ll Polder tensor representing permeability of saturated ferrite |,

LIST OF SYMBOLS (Continued) i, v Components of Polder tensor.e Effective permeability of ferrite (see Eq. 21) A; v/i (see Eq. 21) kfi k Wave numbers in homogeneous media of ferrite or air (see Eq. 21). J7, N7 Bessel functions of (complex) order 7 of first and second kind Xy, ZY Particular linear combinations of J7 and N7 I7, K7 Modified Bessel functions of first and second kinds rl, r2 Inner and outer radii of ferrite ring in coaxial cavity Ho DC magnetic field used for tuning Hor Value of Ho corresponding to ferromagnetic resonance M Saturation magnetization of ferrite vi

ABSTRACT Consideration is given to the possibility of employing ferrite for wideband frequency tuning in a class of narrowband, reflectionless filters employing traveling-wave resonators. Theoretical arguments are presented for choosing a symmetrically loaded cylindrical resonator, operating in the unloaded TMlU mode, as the tunable resonator for this filter. Exact, but implicit, expressions are formulated for the resonant frequencies and field configurations in this ferrite-loaded resonator for all TM,, modes. For the cases where the ferrite geometry is a post at the center of the resonator or a ring on the outer wall of the resonator, and when ferrite losses are negligible, the previous expressions are reduced to a more convenient form. When the ferrite is unmagnetized, a numerical solution is given for the resonant frequencies of the TM110 and TM210 modes. When the effective ferrite permeability (Le) is zero, the resonant frequency of the TMllO mode is similarly found. These three frequencies have been used to estimate the useful tuning range of this filter, as a function of the ferrite dimensions, for both ferrite configurations. An experimental model of this filter has been constructed using the ring geometry. This device had a tuning envelope with less than 10 db insertion loss and more than 20 percent bandwidth at X-band. Suggestions are given regarding possible methods of improving these values, particularly the rather high insertion loss. vii

I i I 1 i i i i I. i I k1

A STUDY OF TRAVELING-WAVE DIRECTIONAL FILTERS WITH WIDEBAND FERRITE TUNING 1. INTRODUCTION During the past few years ferrite materials have been used for both wideband tuning (> 20 percent) of conventional cavities (Refs. 1, 2) and narrowband tuning (< 10 percent) of reflectionless resonant filters (Refs. 3, 4). However, a great many potential applications for tunable microwave filters require both wide tuning ranges and matched terminal conditions. The problem considered here is that of tuning narrowband filter characteristics over a wide frequency range. Specifically, the characteristics of interest are those of the four-port, reflectionless, narrowband, directional filter represented schematically in Fig. 1. In scattering matrix notation, this ideal filter has the following description. 4 bi = Sijaj; i = l, 2, 3, 4 (la) J=l ii o0 (lb) Is3ll2 = Is2412 = F[f - f (H)I (lc) Is212 = s3412 = 1 - F[f - f(H)] (ld) Here, aj is the incident signal at port j, b. is the emerging signal at port i, F[f - f (H)] is assumed to be a narrow, bandpass characteristic such as that sketched in Fig. 1, and H is a tuning parameter. When ferrite tuning is employed, H represents an externally applied magnetic field. In this case, the anisotropic nature of ferrite may cause the scattering matrix to be unsymmetrical. Therefore, the following definitions of the remaining scattering matrix elements are generally consistent

with those in Eq. 1. iS1312 = s4212 = F[f - f(H)] (2a) IS12 = I4312 = F[f - fo(H)] (2b) It is further assumed to be desirable that f and fo be linear, singlevalued functions of H. For a reciprocal filter, f = f 0 0 H -Fff- ) *' -----— fC 0 I fo f- IDEAL fDIRECTIONAL -___ _o^ _ FILTER f- fo f Fig. 1. Desired output spectral distributions with uniform incident spectrum at port @ While considerable effort has gone into the development of ferrite tunable circuits to approximate the characteristics specified above, the results too date have compromised these anticipated goals. Although large tuning ranges have been obtained, difficulty has been encountered in obtaining a satisfactory insertion loss between arms 1 and 3 over a large tuning range. This limitation on filter transmission can be attributed primarily to losses in the ferrite. In the filter structures to be described, other mechanisms can contribute heavily to insertion loss, but in principle these can be made quite small by careful design of the metallic portions of the filter. The purpose of this report then is to summarize: (1) design considerations that have been employed in this problem, (2) experimental results obtained to date, 2

and (3) theoretical considerations that could lead to improved versions of this filter. 2. DISCUSSION OF COUPLING NETWORKS A basic circuit by which the above filter characteristics might be approached is shown in Fig. 2. Here, cavities resonating in conventional standing-wave modes are used to couple a four-port network, but difficulties in tuning this circuit are obvious. Not only must both cavities be tuned simultaneously, but the phase difference between them is critical and must be maintained over the tuning range. fo fo _-_- - _ _ g - I 4 Fig. 2. A directional filter employing two resonant cavities operating in standing-wave modes. More suitable coupling networks can be constructed using a recent innovation to resonant circuits known as a traveling-wave resonator. Using this component and ideal directional couplers, as shown in Fig. 3, one obtains a novel circuit known as a traveling-wave directional filter (Refs. 5, 6, 7), whose properties will be shown to satisfy the conditions outlined in Fig. 1. In the notation of Fig. 3, where the power coupling 3

factors of the individual directional couplers are denoted by sin 21 2 and sin 02, the scattering matrices for the ideal directional couplers are: Sll S12 Sll1 S12' 0 cos e 0 jsin e S21 S22 S21, S221 cos 1 ~ jsin e1 0 (3a) Sl'l S1'2 S,1 112 0 cos e1 jsin 1 cos6 01 j sin el S2,1 S2,2 S2,1, S2,2, cos L e 0 jsin e0 0 S33 S34 S33 S34, 0 cos 02 0 jsin e2 S43 S44 S43, S44, cos e2 0 jsin e2 ~ ( S3,3 S3,4 S4,3, S3,4, 0 cos 2 0 jsin 2 S413 S414 S413 S'4 4 cos e2 0 jsin e2 0 If a2 = a3 = a4 = 0 in Fig. 3, then b2, = al, cos e + Jal sin e1 (4a) bo = jaI, sin 01 + a1 Cos 01 (4b) b = ja4, sin e2; b3, = a4, cos e2 (4c) aL, = e lb,; al, = e 2b (4d) where cp = (a + jp)L (p = 1, 2), and a and B represent the attenuation and the phase parameters, respectively, which are real. Resonance corresponds to p(L + L2) = L = 2m (m = positive integer). For the present, only traveling waves in the counter-clockwise direction are considered, but if the resonator is loaded with a nonreciprocal medium, the dependence of a and p upon frequency will differ for. clockwise propagation. Eliminating a1,, b2,, b3, and a4, from Eqs. 4 yields the following scattering matrix coefficients for the total filter: 4

b -e sin 8 sin 8 S2 = 3 = 2 (5a) 31 al1 1 -e-P cos 81 cos 82 b2 e'I cos 82 sin 21 S21 a = cos 1 - (5b) 1 1 -e cos e1 cos 02 where cp = p1 + cP2. For a = 0 (i.e., a lossless resonator) and for sin 81 = sin 82 = sin e, Eqs. 5a and 5b have the form required by Eq. 1 and Fig. 1, so that F(f - fo) becomes F [(D - Pm)L] = e —o2;T -sin —-,-i (6) [ ]J 2 cos2e 1 - cos (P - )L + sin e Inspection of Eq. 6 shows it to be symmetrical about P = Em, and to yield a fractional bandwidth approxiOIRECTIONAL 2 COUPLER mately equal to sin e for ~, -4 SIN sin << 1. However, it is inter- b b bl esting to note that F(0) = 1 re- /. \ / (I / 0 -TRAVELING WAVE gardless of the value of coupling RESONATOR Lt LI (i.e., sin e). Therefore, for a \ lossless resonator, a free choice bi. oa1 bp SIN'9, s o of coupling is available to yield b, b, any desired filter bandwidth. DIRECTIONAL If resonator loss is considerFig. 3. A directional band-pass, ed, the ideal characteristics of band-elimination filter employing a travelingFig. 1 are best approximated by wave resonator. choosing c = e 2L (7) 2 cos 02 This relation represents an optimum condition in the following sense. 5

If one of the coupling coefficients is specified, Eq. 7 makes S21 vanish at midband ( m = 6m) and also maximizes |S31| at midband, such that tt2 e l sin 02 1S 311 = (8) max sin 0 It may be noted from the symmetry of Eq. 5a that S31 is independent of an interchange of the two directional couplers. This transposition inverts the sin2e2/sin2el ratio in Eq. 8 such that the numerator of Eq. 8 always contains the smaller coupling coefficient. However, perfect band rejection will not be preserved at port 2 in this case, and consequently Eq. 7 will be taken to yield optimum filter performance for given values of resonator loss. Under the condition of Eq. 7, the bandwidth of the resonator can be shown to depend explicitly upon sin 01 (i.e., the input directional coupler), and therefore Eq. 8 has been plotted (Fig. 4) as a function of sin 0e and resonator loss. The latter is expressed in units of db per cycle, where the term cycle designates the angle PBL. In plotting Fig. 4, L1 has been set equal to zero, and therefore 8.68 oL, db must be subtracted from the ordinate when L1 is not zero. It is evident from Fig. 4 that filter performance is quite sensitive to resonator loss. 3. ANALYSIS OF COAXIAL TRAVELING-WAVE RESONATORS The coaxial traveling-wave resonator shown in Fig. 5 offers sufficient generality to warrant investigation in some detail. From the traveling-wave point of view, this resonator could be looked upon as a rectangular waveguide bent into a closed loop. Such a description is misleading, however, because it disguises the connection between traveling6

INPUT COUPLING (db) -16 -14 -12 -10 -8 -6 -4 -2 0 -2 -4;0 -6 -8 _ -10 z I-12-J -14 0.1 db/cy. 0.3 db/cy. 1.0 db/cY. RESONATOR LOSS -16 Fig. 4. Filter transmission loss in db as a function of resonator loss and input coupling. wave modes and the conventional standing-wave modes of a coaxial cavity. In fact, it is easily shown that, corresponding to each traveling-wave mode, there is a standing-wave mode with the same resonant frequency. As proof, consider the field configurations of the m, n, p coaxialcavity modes, which can be established by exciting the cavity through an arbitrary nondirectional hole. lit will be shown that the converse is not always true, since there are no traveling-wave equivalents for the TEM and TM coaxial-cavity modes, when m = 0. m, n, 7

H = C (kcr) cos mcp sin h e.; TEm, z m c h m, n, p F Ez jt 3H(9) m m, n, p Er = 2 - (10a) CP k2 or a-p + Jull, iar (lib) the tangential component of E to vanish at r = a and r = b which yields the following transcendental equations for k [with prime (') indicating differentiation]. -A N (ka) N (k b) m m (12a) AJ(ka) J (kab) amnp (1 m mc mc -A N'(k a) N'(k b) B J(k a) = J(k b) m, n, p (l2b) m M c m c The mode index n arises because these equations have an infinite number of roots in k c. The nth-largest value of kc corresponds to a field conc c figuration with n half-periods in the radial direction. Similarly, the index p describes the number of half-periods in the z direction. A case m, n axial-waveguide modes. In general, the resonant frequencies of all modes |~~~~~~~~~ — ~-~ ~~

are given by 2 2 Aoo -= k2 (h)2 (13) 0 o c h Consider now the introduction of a second signal at an angle n/2m from the first, and out of time phase by n/2. The field configuration of this second standing-wave pattern can be determined from H = jCm(kcr) sin me sin pz eJt TM z m c h m, n, p (14) Ez = JCm(kcr) sin me cos. eJt TE z u m' c / h m, n, p Therefore when these two inputs are present simultaneously, the resultant field configuration is given by Piz ij( t+ m+ ) TE H = Cm(kcr) sin eZ (m, n, p (15) PA z j(Wt + mg) E = C(k r)cos e _ wtm TM z = Cm(kcr) c h m n p which, for m O. describes a traveling wave propagating in the negative p direction. Thus each traveling-wave mode is equivalent to two conventional standing-wave modes whose time and angular phases differ by r/2 and n/2m, respectively. Furthermore, each higher-order coaxial cavity mode (m A O) is equivalent to the superposition of h two traveling-wave modes propagating in opposite directions. It should be noted that although the latter statement resembles the conventional description of Fig. 5. A coaxial travelingwave resonator. 9

a standing wave as the sum of two traveling waves, the idea of reflection is not implied. In many cavity resonators (e.g., rectangular cavities) waves propagating in only one direction cannot exist. Returning to the original traveling-wave directional filter, it is seen that the excitation of the resonator through a directional coupler (i.e., in a traveling wave) is equivalent to exciting it through two nondirectional couplers spaced by n/2m radians around the circumference, with signals g/2 radians out of time phase. Therefore, when a coaxial traveling-wave resonator is excited in "race track" fashion by a directional coupler (i.e., producing traveling waves in one direction only), the resonator is constrained to operating in only one of two degenerate (m, n, p) coaxial-cavity modes. 4. THE INTRODUCTION OF FERRITE INTO A TRAVELING-WAVE RESONATOR Having established the value of the traveling-ware resonator as a filter component, it is natural to consider the introduction of ferrite for tuning this circuit. One possible technique for introducing ferrite is illustrated in Fig. 6. Here, the phase-shifting effect of the ferrite ring is related to that of an infinite ferrite slab in a straight rectangular guide (Ref. 8). Three fundamental problems immediately present themselves however. 1 FFI -- - rT These are: (1) ferrite losses, (2) M ] RING higher-order modes, and (5) the a ^ r2 b;' relatively high dielectric constant of ferrite materials. Fig. 6. A coaxial traveling The effect of ferrite losses wave resonator loaded with a ferrite ring, is evident from the previous dis10

cussion of lossy traveling-wave directional filters. Since ferrite losses can be lumped with cavity losses, any ferrite loss will lower |S13| and increase its dependence on the magnitude of coupling. At the same time, the effect of higher-order modes places a practical limit on useful tuning range, if tuning is to be single-valued with respect to the applied magnetic field. From the standpoint of tuning range, it is desirable to select as the operating mode the one whose resonant frequency is the most widely separated from that of neighboring modes. Since this mode will be the fundamental, the resonator dimensions should be chosen to yield the desired frequency of the fundamental mode and a maximum frequency for the first higher mode. The lowest-frequency traveling-wave resonator mode is the TM,10, whose frequency is determined by the dimensions a and b only, as are all TM modes. The first higher mode can be either TEl or m, n, 0 110 TM210, but by choosing the cavity dimension (h) small enough, the resonant frequency of the former can be made as high as one chooses. The ratio b/a can then be chosen to yield the greatest frequency ratio of the TM210 mode to the TMll0 mode; this can be shown to be 1.34 and occurs for a = O. Thus the cavity mode offering the greatest potential tuning range is the TM110 cylindrical cavity mode. In actual practice, however, the 34 percent tuning range predicted above may not be attainable because of the dielectric loading of commercially available ferrite materials, whose dielectric constant is usually 10 or greater. Although ferrite samples can generally be located where electric fields are small, the fairly large samples necessary for wideband tuning invariably introduce dielectric effects. Frequently these effects lower the resonant frequencies of higher-order modes more 11

rapidly than those of lower-order modes, and this further restricts the tuning range. The question also arises as to whether higher-order modes can be suppressed by a judicious choice of coupling. Since the TM210 and TM-10 modes have opposite symmetry about the cavity axis, there are several coupling techniques that can suppress the TM210 mode, as will be described later. However, it should be recalled that the transmission of a traveling-wave directional filter is relatively insensitive to the magnitude of coupling, and hence small imperfections in mode suppression could produce significant filter transmissions in higher-order modes. Therefore it will be assumed to be desirable to maximize the tuning range below the resonance frequency of the first higher-order mode. 5. THE CHOICE OF FERRITE LOCATION IN A TM10 CYLINDRICAL CAVITY The problem of determining ferrite configurations in a cylindrical cavity that will yield a desired tuning range, with a minimum of the defects mentioned in the previous section, is necessarily complex. One reasonable simplification is the use of a ferrite ring, as in Fig. 6, since this is the only simple geometry that avoids internal reflections. With this assumption, the cavity in Fig. 6 (with a = 0) can be analyzed for the TM modes as follows+ m, n, 0 In a ferrite medium, Maxwell's equations take the form curl E = - Jolj H (16) curl H = jaor E where |I I|| is the familiar Polder permeability tensor, and is based on the approximation that ferrite behaves as a gyromagnetic medium. Choosing 12

a cylindrical coordinate system and a dc magnetic field H0 applied in the +z direction (and eJct time dependence), I||l | relates the RF components of magnetic induction and magnetic field in the following way: Br () -jv 0 Hr B jv O0 H (17) Bz (RF) O O 1 Hz (RF) Here, A and v are well-known complex functions of both frequency and dc H0 (see Ref. 8 for example). Consider now the case of a traveling wave in the cylindrical resonator of Fig. 6, where E = E = H = 0 throughout the cavity for r cp z TM n modes. Assuming a lossy ferrite medium the c dependence is m, n, 0 taken as e jyP, where y = + m + ja, and the + and - correspond to traveling waves in the +cp and -cp directions, respectively. Equations 16 then yield, 1E yzE o [ J ] ( 18e) cp- 1.^ + -. (18) 1 F E yE] H = Jo,e -t + (19) where 2 2 r -ot% = L (21) e p1 2 2 2 2 kf e eErka k ) oCo Since Eq. 20 is Bessel's equation of order 7, Ez can be written1 z Since kf = f~7ka, and since the real part of Ie becomes negative for moderate values of H0, a convenient form for Eq. 22, when be < 0, is Ez = ALy [7(-jkfr) + B CV(-jkfr)], where I and K; are modified Bessel functions of the first aid second kinds. The subsequent equations can be similarly modified. 13

E= Ay [(kfr) + B N (kfr)] (22) = AX(kfr) Therefore Eqs. 18 and 19 yield H, - o;'e (23) Hr = [ [XI (kfr) + kr X2(kfr)] (24)'P 3o~'r 7 f Equations 22, 25, and 24 are. also valid in the air-filled regions of the cavity with CeEr = 1, ka = kf, and r = O. Therefore, for 0 < r < rl, eer a f B = 0 in Eq. 22; while for r2 < r < b, E C [J (kr) N (k b) - J (k b) N (k r)] (25) = CZ (kar) since E = 0 at r = b. Therefore, the total field configuration in Fig. 6 is as follows. z = J(kar) jo\oHP = kaJr(kar) 0 < r < rl = AXkfkr) Xr (kfr) + Xy(kfr)] r1 < r < r2 (25) = C Z (kr) = k C Z' (kar) r2 < r < b YCYZ ayy77 a 2 Applying the boundary conditions that Ez and Hp be continuous at r = r1 and r = r2 yields, (karl)J' (karl) 1 (kf ) (k (k,)1(k[r,) F(kjr ) XI(kfr) + ]( 7 a 2 [ rWf2. (27) In principle, Eqs. 26 and 27 are exact solutions that will give 14

the resonance frequency and attenuation for each value of H and for arbitrary rl and r2 in the range (0, b). Since this calculation is very tedious, even if loss is neglected, a complete solution has not been made. However, considerable information can still be obtained by applying these equations to certain special cases. Two cases of particular interest are (1) rl = 0, and (2) r2 = b, which place a ferrite post or a ferrite ring, respectively, in regions of strong magnetic fields and weak electric fields. It is further assumed that loss is neglected, so r = +m. Note that whenever + signs appear henceforth the upper sign corresponds to propagation in the +e direction. For case (1 Bm = O, Eq. 26 can be dropped, and Eq. 27 can be expanded as follows:,:kr2) J(kar2) 2Jm(kab) 1 (krr2)J;(kfr2) (kard Jm~ktr2)'JM a~b> 1[(kt~2) ~k~2) + M1 (28) W ^W ____________________________ ^W _______ (28) J.(2ar + nJm(kar2)LJm(kab) N'mkar2) - Jm(kar2) Nm (kb)J e LJr(kr2) Similarly, for case (2);Eq. 27 can be dropped, B = -J (k b)/N (k b), m mf Mf and Eq. 26 can be written: ik r1) J;(kfrk) T [(kfr2) J;(kfr2) 2Jm(kb) (29) __________________________________ J (29) mal m k arl) + M J (ka L Jm(kfrl) Jm(kfr2)[Jm) Nm(kr2) - Jm(kr2)Nm(kf b + Although a complete analysis of these equations would still be lengthy, considerable information can be obtained from the special cases where f = O and 1e = 0. The former occurs when Ho = 0, and the latter when H0 = H O-Mo, where H0 is the dc magnetic field corresponding r r to ferromagnetic resonance and M0 is the saturation magnetization of the ferrite. If it is assumed that Se = 0 marks the threshold of excessive loss in the ferrite, then the solutions of Eqs. 28 and 29 for f = 0 and He = 0 can be taken as the practical bounds on useful tuning range. Admittedly, this is an arbitrary choice, but since it is mathematically 15

expedient this criterion will be given further consideration. It can be noted, when tuning is very sensitive to Ho, that HO will be considerably greater than MO, and hence jie may vanish considerably before excessive loss begins. In this case, the tuning range assumed above will be conservative. On the other hand, if tuning is not very sensitive to HO, %e = 0 is a reasonable threshold. Therefore, it seems reasonable to interpret this tuning criterion as yielding a practical lower bound on tuning range. The evaluation of Eqs. 28 and 29 for r = 0 (i.e., H0 = 0) is straightforward and has been done for b = 0.75 inch, er = 12, A = 0.7, and m = 1, 2. The choice of A = 0.7 for Ho = 0 is based on experimental data, since the tensor permeability in Eq. 17 is assumed to apply above magnetic saturation only. The results in this case are plotted in Fig. 7, which shows the resonance frequencies of the TMl0 and TM210 modes vs. r1 and r2 (i.e., the post and ring radius, respectively) when the ferrite is unmagnetized. For be = 0, only the case m = 1 is of interest and Eqs. 28 and 29 simplify as follows. Below ferromagnetic resonance,.~ -.0 corresponds to ([ - v) -,0 in the Polder tensor. Therefore, for 4e nearly zero, the right-hand side of Eq. 28 behaves as - ~ 1 ( 3~ )(30) + v while the right-hand side of Eq. 29 varies as 1- + / 2b2 (31) A + v 2~ - e r b It is apparent that Eq. 30 approaches infinity for propagation in the +8 direction and 1/2p for -e propagation, while Eq. 31 approaches infinity for either direction of propagation. Therefore, for e = O, Eq. 28 reduces to 16

2.7 TM=210 2.6 TM2 Ho =0/ 2.5 2.4 2.3 TM 20 O. 0.2 0.3 04 0.5 0.3 0.2 0.1 0 POST RADIUS (CM)- rRING THICKNESS(CM) Fig. 7. Minimum tuning ranges for ferrite rings and posts as predicted by tuning criteria adopted in text. 17 ~_ 2.20- TM,,o POST RANGE Fig. 7. Minimum tuning ranges for ferrite rings and posts _7

J (kb) N (k b) (32) J1kar2) Nkar2) (32) for the post with +e propagation, and Eq. 29 reduces to Jl(kar2) 0 (33) for the ring with either +e or -e propagation. The solutions shown by Eqs. 32 and 33 are also shown in Fig. 7. Comparison of Eqs. 32 and 33 with Eq. 12a shows, when He = 0, that the resonance frequency of the post with +0 propagation is identical to that of a coaxial cavity with radii r2 and b, while the resonance frequency of the ring is identical to that of a cylindrical cavity with radius r2. Thus, in each of these cases, the ferrite appears as a perfect conductor when ~e = 0 (which is a generally quoted ferrite property), except in the case of the ferrite post with -0 propagation. This result suggests that when %e = 0, a ferrite acts like a conductor to a linearly-polarized wave (as is presented to the ring) or to a positive circularly-polarized wave (as is presented to the post with +e propagation), but that it does not act like a conductor to a negative circularly-polarized wave. Since Ae is a monotonically decreasing function of H (below ferromagnetic resonance), tuning is expected to increase monotonically with H for any case where the ferrite behaves as a conductor at e = 0. Both experimental evidence and numerical solution of Eqs. 28 and 29 substantiate this statement. In the case of a ferrite post with -0 propagation, tuning is still observed, but it is not monotonic. This fact generally makes the -6 mode less desirable for tuning, but it can be used to extend the tuning range of the +0 mode if H0 is varied over both negative and positive values, since the +0 mode resembles the -e 18

mode when H0 is reversed. This technique is somewhat undesirable, however, since ferrite losses tend to be high for H0 near zero (i.e., below saturation). Also, the dependence of resonance frequency upon H0 is extremely nonlinear near saturation. Therefore, only tuning with positive H0 will be considered further. By returning now to Fig. 7, the approximate tuning ranges of the post and ring configurations can be compared. This figure demonstrates that a post thickness of 0.46 cm can be tolerated before the TM210 mode appears in the tuning band, and this thickness provides a tuning range of greater than 30 percent. In the case of the ring configuration, however, the TM210 mode enters the tuning range at a thickness of 0.2 cm and thereby limits the tuning range to about 16 percent. This suggests that the post configuration is preferable from the point of view of maximum tuning range. If one now considers ferrite losses, the question arises as to which configuration will yield the smaller insertion loss for a given tuning range. The exact evaluation of insertion loss in each of these cases is tedious, so it is expedient to assume that it is proportional to ferrite thickness, at least for small thicknesses. Some support for this assumption can be gained from Fig. 7, where it can be seen that tuning range is proportional to thickness for thin posts or rings. An analogy can then be drawn with other nonlinear reactive materials (e.g., ferroelectrics and variable capacitance diodes) where insertion loss is approximately proportional to tuning range. On the basis of this assumption, Fig. 7 suggests that the ring geometry is preferable from the point of view of minimizing midband insertion loss from terminals 1 to 3 in Fig. 3. For example, a ring thickness of 0.2 cm yields the maxi19

mum useful range of 16 percent, while a post of 0.2 cm radius yields only 8 percent tuning range. 6. EXPERIMENTAL RESULTS In view of the larger tuning rates predicted above for thin ferrite rings vs. thin ferrite posts, an experimental study of ferrite rings on the outer wall of a coaxial cavity was carried out. Attempts were made to develop a cavity with a 20 percent tuning for the TM110 mode, and with a minimum insertion loss envelope over this band. Two ways of exciting a TM1l0 traveling-wave mode have been considered. The first is with 3-decibel directional couplers, which are used to provide a t/2 phase shift between two input lines and between two output lines, all spaced by 903 around the cavity (Ref. 4). This configuration is shown in Fig. 8. The second method is to excite the cavity by a circularly-polarized field rotating about the axis of the cavity (Ref. 3) as shown in Fig. 9. In both cases, the synme- * I TRAVELING'X a3 siNit. WAYE siN', J /\ IRESONATOR l try of the exciting fields is such \, that the TM210 mode should be 2-10 / - suppressed. In Fig. 9, the coupling holes have been placed over the Fig. 8. A traveling-wave position of circular polarization directional filter employing 3-db directional couplers. in the input and output waveguides. Although this position is frequency sensitive, it was decided to employ the latter coupling technique first, with the additional plan of attempting to broadband the coupling at a later date, following the procedure 20

of Cohn (Ref. 9). In view of this plan, the waveguides were so attached that the coupling holes could be moved during the experiment. It was felt that this would allow each OUT part of the tuning band to be measured with holes of optimum directivity. In the experiments conducted, however, the ferrite losses were of such a magnitude that large coupling holes were O nea avoid excessPOSITION OF CIRCULAR POLARIZATION IN necessary to avoid excessive in- WAVEGUIDE sertion losses. When the coupling IN holes were large, their directivity Fig. 9. A traveling-wave appeared to be uniformly poor with directional filter employing directional frequency, and the coupling hole coupling holes. positions for minimum insertion loss varied very little with frequency over the tuning band. A typical experimental result for the filter shown in Fig. 9, and loaded with a ferrite ring on the outer cavity wall, is shown in Fig. 10. These data were obtained with a 1.54" OD cavity that was loaded with a 1.50" ID, TT-390 ferrite ring. The input and output coupling holes were 7/16" and 3/8" in diameter, respectively, and were centered in the top and bottom walls of the cavity. This filter had an insertion loss of less than 10 decibels from 8.4 kmc to 10.4 kmc, but the TM210 mode appeared at 10.1 kmc, which limited the useful tuning range to 8.4 kmc to 10.1 kmc in this case. It will be noted that relatively poor suppression of the TM210 mode occurred in this case, which again points to imperfections in the coupling holes. Considerable effort 21

was expended to reduce the insertion loss envelope and to obtain better suppression of the TM210 mode, but since these two requirements conflict, little progress was made over that shown in Fig. 10. -5 TMo -10 z -15 -TM IV) -20 d | 0 500 1000 1300 o300 600 1200 1800 2100 2600 Ho -u 8 9 10 II f — (KMC) Fig. 10. A typical experimental insertion loss envelope for a wideband, ferrite-tunable, traveling-wave directional filter. Filter bandwidth approximately 5 Mc. At the time the measurements described above were being made on the ring configuration, results were published by Whirry and Nelson (Ref. 4) on the ferrite-post configuration. They reported a tuning range of about 5 percent and an insertion loss of less than 2 decibels. Comparison of their work with work at this laboratory (CEL) supports the previous assumption that tuning range and insertion loss are roughly proportional for a given tuning medium, since the larger insertion losses measured in the CEL filters did accompany tuning ranges of 20 percent or more. However, a more qualitative comparison of these two methods would have required the fabrication of a thinner ferrite ring. Such a compari22

son was not made because of the expense of machining extra ferrite rings. 7. CONCLUSION The usefulness of traveling-wave resonators in matched filters has been demonstrated, although the experimental development of a ferrite tunable matched filter was only partially successful. A new mode of attack would be to try ferrite geometries different from those described here, since the latter were motivated in part by ferrite phase-shifter techniques that had been demonstrated prior to 1958. More recently, a new ferrite phase-shifter principle has been discovered by Reggia and Spencer (Ref. 10), which offers strong phase shifts and low insertion losses below ferrite saturation. This phase shifter operates with a ferrite sample (a rod or slab) centered in a waveguide and magnetized in the longitudinal direction. One technique for employing this phase shifting principle in a OUTPUT?- ~ <COUPLER traveling-wave directional filter _ is suggested in Fig. 11. The FERRITE Reggia-Spencer phase shifter has R an additional advantage of re- CNER quiring relatively small tuning fields, which would facilitate Fig. 11. Proposed technique for improving traveling-wave rapid tuning. However, this pro- directional filter performance. posed filter has not been constructed, because it appears to call for considerable "cut and try" development procedures, which the author feels to be generally unjustifiable in University research. 23

REFERENCES 1. G. R. Jones, J. C. Cacheris, and C. A. Morrison, "Magnetic Tuning of Resonant Cavities and Wideband Frequency Modulation of Klystrons," Proc. IRE, Vol. 44, No. 10, October 1956, pp. 1431-1438. 2. C. E. Fay, "Ferrite-Tuned Resonant Cavities," Proc. IRE, Vol. 44, No. 10, October 1956, pp. 1446-1448. 3. C. E. Nelson, "Ferrite-Tunable Microwave Cavities and the Introduction of a New Reflectionless, Tunable Microwave Filter," Proc. IRE, Vol. 44, No. 10, October 1956, pp. 1449-1455. 4. W. L. Whirry and C. E. Nelson, "Ferrite Loaded, Circularly Polarized Microwave Cavity Filters," Trans. IRE, Vol. MTT-6, No. 1, January 1958, pp. 59-65. 5. F. S. Coale, "A Traveling-Wave Directional Filter," Trans. IRE, Vol. MTT-5, October 1956, pp. 256-260. 6. S. B. Cohn and F. S. Coale, "Directional Channel-Separation Filters," Proc. IRE, Vol. 44, August 1956, pp. 1018-1024; also 1956 IRE Convention Record, Pt. 5, pp. 106-112. 7. C. E. Nelson, "Circularly Polarized Microwave Cavity Filters," Trans. IRE, Vol. MTT-5, April 1957, pp. 136-147. 8. B. Lax, K. J. Button, and L. M. Roth, "Ferrite Phase Shifters in Rectangular Waveguides," J. Appl. Physics, Vol. 25, November 1954, pp. 1413-1421. 9. S. B. Cohn, "Impedance Measurement by Means of a Broad-Band Circularly Polarized Coupler," Proc. IRE, Vol. 42, October 1954, pp. 1554-1558. 10. F. Reggia and E. G. Spencer, "A New Technique of Ferrite Phase Shifting for Beam Scanning of Microwave Antennas," Proc. IRE, Vol. 45, November 1957, pp. 1510-1516. 24

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