THE DESLGN OF A RAMO FREQUENCY MECHANICAL FLLTER Thesis for the Degree of M. S. MICHIGAN STATE COLLEGE Richard Neal Devereaux 1954 1N1"? This is to certify that the thesis entitled u ryn gne Desi~n of a deio Frequency Nechanical Filter" presented by Richard Keel Devereaux has been accepted towards fulfillment of the requirements for _£1_._S_>___ degree in _E;v__h-__ M ' {V J. A. sé’é’iz‘é’fiem ‘ Date l-"Iay 28, 1951+ O~169 THE DESIGN OF A RADIO FREQUENCY MECHANICAL FIIEER BY RICHARD NEAL DEVEREAUX M A THESIS Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering 19Su \l \{3 Six~ U‘ \_ A CKN OWLEGMEH T The author wishes to express his thanks to Dr. J. A. Strelzoff for the instruction received on the undergradu- ate and graduate level which made this thesis possible and for his patience in reading the manuscript. 3313.12 III. IV. VI. TABLE OF cor.T TEN T ‘ Section Introduction The Transmission Line Analogy A. The Filter Structure B. Longitudinal Mode of Vibration C. Torsional {ode of Vibration Transmission Lines as Bandpass Filters A. Bandwidth Equations for Lines of Diff- erent Characteristic Impedance in Cascade B. Bandwidth Equations Found by Using Approximately Equivalent Lumped Elements Related Problems in Filter Design A. Method of Drive and Take-Off B. Filter Termination C. Materials Pre-Design Considerations A. Filter Parameters and Assumptions B. Equations for Obtaining Chebishev Response Examples of Filter Design A. Campbell Type - Longitudinal Mode B. Campbell Type - Torsional Mode C. Chebishev Type - Longitudinal Mode D. Chebishev Type - Torsional Mode E. Chebishev Type - Torsional(unsymmetrical) Page 20 22 2M 27 29 32 35 37 hO VII. Conclusions VIII. Bibliography I. INTRODUCTION One of the first requirements of a good quality comp mercial, military or amateur communications receiver is a high degree of selectivity. Ordinary receivers generally obtain their selectivity by cascading several intermediate frequency tuned transformer stages, ihile the better re- ceivers incorporate the much more expensive crystal type bandpass filter. It is doubtful whether the crystal type filter can be surpassed in performance but the high cost of such a filter may make its use impractical. In recent years interest has been developing around mechanical type filters that will perform.almost as good and yet cost considerably less than the crystal filter. This type of filter is composed of a chain of resonators made of a suitable high thetal with non-resonant metal couplers joining adjacent resonators. The filter is driven electro-mechanically by magnetostriction and is terminated in a similar fashion. It is the object of this thesis to discuss the anal- ogous relationships between the mechanical filter anda chain of electrical transmission lines; to elaborate on a -1- method suggested by Burns and Roberts1 for the design of a.mechanica1 filter; to show how methods now in use for the design of transmission line micro-wave filters might be employed, and how the filter can be designed to give the so-called Chebishev response plus several actual fil- ter design examples. .OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...0.00.00.00.00. 1W.V.B. Roberts and L.L. Burns, Jr., "Mechanical Filters for Radio Frequencies," RCA Rev., Vol. 10, Sept., 19h9. PP. 3&8 ' 3650 - - 2 - 00" I... eveeobbte II. THE TRANSMISSION LINE ANALOGY A. The Filter Structure The analysis of a mechanical vibratory circuit would in general involve motion in any one or all three dimenp sions. Compared to the analysis of a simple electrical circuit with only one independent variable, such as the current, the former could become extremely complex. Due to the endless number of different shapes and modes of vi- bration possible with a mechanical structure, an endless. number of different structures must also exist which could produce filter characteristics.‘ In order to simplify the analysis as much as possible and to be able to incorporate various electrical-mechanical analogies, only one mpde of vibration will be considered for the filter structure; also, only a structure whose symp metry and dimensions are such that one mode of vibration ‘ results will be considered. The type of structure to be discussed in this paper is undoubtedly one of the simplest as far as analysis is concerned. This structure is composed of cylindrical rods, which will be called the/resonators, coupled tOgether by .... '3 .4,- cylindrical rods of smaller diameter which may or may not be of the same material as the resonators. These will be referred to as the coupling necks or simply couplers. A diagram of one section of the structure is shown in fig- ure 1. 0 f9 7 resonators Figure 1. Due to the relatively slow velocity of radio frequenp cy vibratory waves in the system, it will be necessary to utilize the concept of distributed parameters as in the analysis of high frequency transmission lines. Since it is already known how to design bandpass transmission line filters, it would be convenient to determine if the mechan- ical structure could be considered an analogous system.and hence use the same better known concepts. -u- B. Longitudinal Mode of Vibration Consider the vibration of a resonator alone, vibrating in only the longitudinal mode. The situation is shown in figure 20 A gé—-X——D~- g k- / i i i l 7 | I / : Lee»! / A / distance to cross section at rest displacement of section from.rest position tension on section area of section d /dx = the elongation Young's Modulus mass density ‘otllOUH-SWN IIIIUII Figure 2. The equation of motion can be easily found as follows.2 Applying Hooke's Law to the slice bounded by x and sun: we have E 3 stress = Tl s sEraIn g and T = SE(d5/dx), the tension at x. The tension at x+Ax is given by the following equation. 00.00.0000...0.0.0.000...OOOOOOOOOOOOOOOOOOOOO0.0..000...... as. R. Hubbard, "longitudinal Vibrations in a Loaded Rod," Journ. Account. Soc. of Amer., Vol. 2, 1931, p. 372. -5- sins-QOIIIsaws-e000ease-o‘seeeleevo-sweetwee-w Tl: T, + d/dx (ss df/dxflix 2. or T1: SE (dg/dx+ dg/dszx) The accelerating force on the slice is Ta Tz-T,, or 2 2 Ta: SE (dg/dx )Ax . Equating this to the newtonian equation for the slice gives, SE (dzg/dx’”)nx= (ps x) dag/dtz', or dE/dtz' =: E/P (d7?/dx2') . (l) The resonant frequency of the resonator is found to be, f°= v/2l. (2) where: v=lflflo = velocity of longitudinal waves L= length of the resonator Equation (1) is exactly like the familiar transmiss- ion line wave equation.and.JI;; like I/fLC represents the wave velocity through the medium. By merely extending the well known electrical-mechanical analogies to the case of distributed parameters, the relations in table 1 are ob- tained. Electrical Mechanical L Induc./unit 1gth PS Mass/unit lgth l/C Rec.Cap./unit lgth as Elastance/unit lgth l/JLC Wave Velocity JEZS Wave Velocity .575 Char. Impedance 3&3]; Char. Impedance JEf> Intrinsic Impedance 1 Table 1. Electrical - mechanical transmission line analogies for longitudinal vibration. C. Torsional Mode of Vibration If the resonator is made to vibrate in the torsional mode, then the analogies for a torsional mechanical sys- tem.and an electrical circuit can be extended as shown in table 2 e Electrical Mechanical L Induc./unit lgth I Mom.ofIner./unit lgth n/c Rec.Cap./unit lgth GJ Rigidity/unit lgth I/IIE'Wave Velocity JEZB Torsion Wave Velocity {L75 Char. Impedance J/CF Char. Impedance JGF> Intrinsic Impedance Table 2. Electrical - mechanical transmission line analogies for torsional vibration. - 7 - G in table 2 stands for the shear modulus of elas- ticity for the material being used. .Also, in reference to table 2, for circular cross sections: J = “5732 = polar moment of inertia of the cross sectional area I = ,ofiD732 = J,o It may happen that the value for G is not known but the value of Poisson‘s Ratio for the material is known. In this case, the following relation will prove helpful.3 G== E / 2mm (3) where p = Poisson's Ratio for the material. O...COCOOOOOOOOOOOOOOOOOOOOO0.00....OOOOOOOOOOOOOOCOOOOOO.. 3L. 8. Marks, "Mechanical Engineers' Handbook," McGraw-Hill Book Co., Inc., 19hl, p. hhh- ' - 8 - III. TRANSMISSION LINES AS BANDPASS FILTERS A. Bandwidth Equations for Lines of Different Characteristic Impedance in Cascade One method of obtaining bandpass characteristics with transmission lines is to couple two identical lines by one of different characteristic impedance.LL The structure in figure 1 could represent this condition. Let the charac- teristic impedance of the resonators be ch and their ang- ular length.9,; the characteristic impedance of the coupler Z02 and its angular length 91. we would like to find the over-all characteristic im- pedance of the section. The equations for a single loss- less line in matrix form are, Pl u c (D X O...OOOOOOOOOOOOOOOCOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOO LLW. P. Mason, "Electromechanical Transducers and Wave Fil- ters," D. Van Nostrand Co., Inc., New York, New York, 19kg: P0 750 - 9 - where for the resonators, a: C05 6! ; 6:120:55” 6! C-‘j-é-o‘sin 6, , .U= cos 6. and for the coupler, a == (056,, , 6=J 20250161 .I . C=ngzsstn911 ,chos 62’ Hence, multiplying the 066.0 matrices in order gives, I l a a (056, jstinG, (05 d1 12,250.91 rcos e. JZo.sinB, = X X I [ . ' s ‘ I . s. ' -—5L 6 case ~.3 0 9 ——$m6 cos 9 C .0 J20. n I I 1201‘" 2 ‘05 2. -20, a [J (compose,- giant),51:19,)3(z,,sine,cosq+z,.sine,cose,) Fe o s 6, J'zo, sin 6, x Eonsin 9. cos G,+_'.sin 65059,) (—gss sinqsin Q+case‘cosq) Jl-ZL 5m 9, co 5 6, 20‘ 201.. 0' _ OI H Noting that the section is symmetrical, the over-all char- acteristic impedance is given by JBVC'; hence the multipli- cation need only be carried far enough to obtain this re- sult. Continuing the multiplication we have, . . Zol ' ' ' . 0 J23, 5‘ n a, (c056,c es 9,: 23.501 9. $1"6;.)+J cos 6,(z,swhstndcosé) I 1.3 - C ’ Jcos 6,(2l;’5in6‘ cos 91+zlzfm6:_ which can be reduced to the form, Zozz Ian191'243 tanQICOtei‘ (b1 (LL) 437“ 1241119, --Z$ tan 9. cat 5; "-1- s’_ C’ " 43" where (p: Zoz/ Zen Transmission occurs in a filter whenever the charac- teristic impedance is real or whenever the above eXpress- ion is positive. Hence, the limits of the transmission bands are given by the roots of both the numerator and de- nominator of equation A. In pass bands defined by the nu- merator roots, the characteristic impedance varies from zero at the band edges to a maximum at the middle of the band. The denominator roots define bands where the char- acteristic impedance varies from infinity at the band edges to a minimum at mid-band. Hence numerator roots give re- sults similar to the electrical "T" structure and denomi- nator roots give results similar to the "fl" structure. A general solution for the roots of equation A would be rather difficult to obtain. However, it is not too diff- icult to obtain solutions if Gland 9,are assumed to have a definite ratio. Assume Lu =Lz = 72- and Zoz<< Z... . A pass band will be centered on 9=Tr/2- . Solution of the numerator in terms of (I) gives , tan‘o, =. o’H-zt» <5) - 11 - and for the denominator, tanz 9.: $14- 3?- (6) Since 4: is very small, equation 6 must be used as the tangent must be large around 9:77/2. . Equation 6 can be re-written as, cote: 1' 45/m Since cot 6 is near a zero (9=1f/Z.), cot 92$ (%-9 ). Hence, for sufficiently small ¢ , 62%1: —Q—— or in eXpanded form W ez¥1(¢‘¢1+%¢3—~~) . Discarding terms higher than Ozgives for the angular band- width, Ae=z(¢-¢D and the fractional bandwidth ( r,- r./ r.) is given by, r..- r.=_n_9_ =g¢(l-¢) (7) a e. W Another case of possible interest is where adj-‘et/ 2 and 9, is half a wavelength. Then the passband is centered on 6:“ and the roots must be determined from the numerator of equation A since the tangent must be very small. Near 6=7r the following approximations hold (9,<‘N). -12.. tan 6.:: - (1r- 6.) COt BI/Z 5: %— ‘2’- Using these approximations, the numerator of equation A becomes, 1’ a (W'- OI) +2¢(”F-9,)( 7P9.) -- ¢ = O 2. from which it is found that 6: 11’1' is sufficiently small, equations 7 and 8 f2.- f...A9_.24>(l—%) (8) can be written more simply as 60 I ¢ 8 £39 .— 2.4) g and 6. —— .— ( a) - 13 - A third type of structure could consist of half-wave resonators coupled by eighth-wave necks. Again, the numer- ator of h is used to determine the band limits. The approx- imations to be made for 9 slightly greater than fl'are, tanez 7r-e cot—3:21 - (7r—6)/2 Substituting the approximations in the numerator of equa- tion A gives, (1r- ef—zscr— e)[l - lei] - 4? = o z z or (7r-6)- 7?; (w—e)— ‘1’, = Neglecting the ¢zterm gives, 0 “IF-9 == 24>/|+4> The expression for the fractional bandwidth becomes, n-n_Ae=g¢ (% fl' f0 0 -1“- B. Bandwidth Equations Found by Using Approximately Equivalent Lumped Elements Another possible method for designing a filter is one commonly used in designing high frequency transmission line filters and can be found in almost any text on micro-wave theory.5 Let 20 be the characteristic impedance of a length of transmission line. If the line is shorted on the far end, the input impedance becomes, 25.: j Zotan-g’v—L (10) A plot of Xx/Zo versu3¢u is then simply a tangent curve. The input impedance if the line is Opened on the far end is Zoo: ‘j ZoCOt%L (11) The plot of Xa/Zo versuscu is a negative cotangent curve. By making the slepe of one of the curves, near a par- ticular frequency, approximate the slope of a reactance curve near the same frequency, a lumped inductance, capaci- tance or a series or parallel resonant circuit can be app- roximated. The type of element or circuit to be approximated 5For example, see G. L. Ragan, "Microwave Transmission Cir- cuits," McGraw-Hill Book Co., Inc., New York, New'York, First Edition, 19h8, p. 61h.S - 1 - is naturally dependent on the length of the line. The com- plete procedure will not be given here, however the approx- imating equations are as follows: L =_Z°2Nl line shorted, < (12) wk >4F~ .L 4. 1 £51 : line Open, -%-< (13) Zeno}. +4- .0 N If the line is an Odd number of quarter-waves long, a parallel—tuned circuit can be approximated where, 20 ._. I (I +Yt7r) : line shorted (lli) " to Z 7. O O or a series-tuned circuit can be approximated where, 2L: __Z_o(I[+YlTr) : line Open (15) «Jo 2' If the line is an integral number of half-waves long, a series-tuned circuit can be approximated where, 2L = 2,, ’11 : line shorted (16) (do and a parallel-tuned circuit where, 20 _I_ M : line open (17) Zo (do The application of these equations to the neck-coupled mechanical structure requires that the resonators be appr- -16- oximately short-circuited and that the thin coupling necks be approximately Open-circuited. One type of structure that can be designed by this method is that composed Of half-wave resonators and eighth- wave coupling necks. The analogous electrical circuit is shown in figure 3. This is a simple m-derived band-pass section. For narrow band, Cg is very much larger than C, and the bandwidth ratio with respect to the resonant freq- uency of the series arm is approximately as shown. Ln/Z. 2C. 2C, 1.1/2. L.=R/7r(5.'5u) °—’m‘—l I‘m—f Ca: I/1r(f,+f,)R T C2. R f, : I/zvrf/JE. I : ————fi-'f’ a: 3.9 ft C). Figure 3. The parameters of the equivalent mechanical structure can be found as follows. For an Open-circuited eighth- wave line, the input impedance is Assuming then that the reactance curve of the capacitor C1 is approximately the same as the negative cotangent curve - 17 - near 0=7T/4- , the characteristic impedance of the coupling neck can be determined as simply z,,_= l/ 2mm, (18) The characteristic impedance of the half-wave reso- nators can be found using equation 16 from which it is found that, 20' 21).”. 2 (19') 11' «tom The bandwidth ratio is thus given by, (20) This result is seen to be the same as equation 9 which was found using the first method. The terminating resistance for the section is given by, JETE- __ -..L .. 2.. <21) Cg MC . which of course is equal to the mid-band image impedance of the section. A check of equation L), for the case 0,=‘fl' and 9,, 1r/4 willgive the same result for the mid-band image im- pedance. Another structure that can be treated by either method 15 that composed of half-wave resonators and half-wave - 18 - couplers. The fractional bandwidth is found from the sec- ond method by comparing it to a constant-K band-pass sec- tion. Both methods give the same result for the fractional bandwidth and the mid-band image impedance. These are, fa" ft = 21,-2- sz (22) f0 T ZO‘ and RIF-EJ Zea Zoz (23) These can be readily verified by the reader using the meth- ods given. This fact, that the image impedances are equal, is stressed because it has been observed that unless this is so, both methods will not give the same results for the same type of structure. This does not come as a complete sur- prise since in using the second method, a definite lumped electrical equivalent is assumed. In general though, if the mid-band image impedance for a particular structure as found from equation A, is of the same form as a potentially equiv- alent electrical structure, then it may be possible to ob- tain the same result using both methods. Otherwise, the two methods will give rise to two different structures. - 19 - IV. RELATED PROBLEMS IN FILTER DESIGN A. Method of Drive and Takeoff The fractional bandwidth equations found in the last section will be sufficient for the design of the mechani- cal structure as will be shown later. The next problem is how to electrically set the structure in vibration; and once this has been accomplished, how to change the vibra- tion energy back to electrical energy. The possible use Of the piezoelectric effect has been investigated,6 but the best and simplest method for this particular application appears to be electro—mechanical drive by magnetostriction. Hagnetostriction is the phenomenon whereby a material placed in a magnetic field experiences a change in dimen- sion. Some materials undergo an expansion while others a contraction. Of the metals, nickel and some of its alloys undergo the greatest change in length. This change in length is a contraction, and for a constant field is gen- erally not more than one part in 20,000. For a rod of 6Ibid 1 b 20 - nickel or preferably Ni Span C (a nickel alloy), the re- sulting motion can be greatly increased by applying a sin- usoidally varying field with a frequency equal to the resonant frequency of the rod. In this manner, a fairly efficient conversion of electrical energy to mechanical energy can be achieved. The basic magnetostriction rod resonator is shown in figure A. A magnetic bias is required as shown, since PL (/37: ll] Figure A. otherwise the rod would contract on both half cycles of the alternating field producing a double frequency effect. With the bias, the rod undergoes first a decrease and then an increase in length. If the rod is thin enough, it may be possible to remove the biasing magnet with the rod re- taining sufficient magnetization to Operate. In the dia- gram, the coil is arranged so as to produce longitudinal vibrations. Naturally, by suitably arranging the coil, tor- sional vibrations could also be produced and as pointed out .-- 21 .. by Roberts,7 the rod could be permanently magnetized in this direction since no free poles would exist. The magnetostrictive rod can thus be used to drive the mechanical filter and it has the advantage of being one of the filter resonators. The conversion of the mech- anical energy back to electrical energy is of course accom- plished by the reverse process. A filter structure with the drive and takeoff coils is shown in figure 5. A J N {i U! I n: H U“ Figure 5. B. Filter Termination The type of filter termination to be used should de- pend on the narrowness of the pass band and whether the 0.0.0....OO0......0.0.0.0000...0.0.0.000...0.00.00.00.00... 7w. V. B. Roberts, "Some Applications of Permanently Magne- tized Ferrite Magnetostrictive Resonators," RCA Review, Vol. XIV, pp. 3-16; March, 1953. electrical end circuits Operate as part of the filter; also on the manner in which the filter is mounted. For a very narrow band filter, with end resonators made of relatively low Q material, satisfactory results could probably be obtained with no terminating resistance. If the bandwidth is not too narrow, the electrical and cir- cuits might be incorporated as the first and last resona- tors of the filter and the necessary terminating resistance obtained by adjusting the Q of the electrical circuit. It appears that if some form of external mechanical damping is required, a little out and try with various methods would be necessary. One method could be to connect a line which approxi- mates an infinite line, as far as resonator length is con- cerned, to the end resdnators; this line having a charac- teristic impedance equal to the mid-band image impedance of the filter. For example, the filter structure with half-wave resonators and quarter-wave couplers has a mid- band image impedance equal to the characteristic impedance of the coupler, szo The impedance of an infinite line is simply the characteristic impedance of the line, hence prOper termination should be obtained if the infinite line has a characteristic impedance equal to Zoz. The image impedance of the filter just mentioned varies from zero at the band edges to a maximum of 20* at - 23 - mid-band, hence a compromise termination as suggested by 8 Guillemin would probably give better results. The termin- ation he suggests would be equal to .707 times the mid-band image impedance. C. Materials For the magnetostrictive end resonators, two materials stand out. The choice at which is dependent on the speci- fic design requirements of the filter. The first is Ni Span C which has already been men- tioned. It is highly magnetostrictive and has a low tem- perature coefficient of eXpansion making its use desirable where frequency stability is of prime importance. The second material is more in the field of ceramics than metals. This is the so-called ferrite material. By using a suitable ferrite for the driving resonator, the Q of the coil surrounding it may be greatly increased;9 whereas if Ni Span C were used, the coil Q would be low— ered due to eddy currents, etc. Ferrites have a very high resistivity and hence eddy currents are almost non-exis- 8E. A Guillemin, "Communication Networks," Vol. 2, John Wi- ley & Sons, Inc., New York, New York, 1935, p. 309. 9Ibid 7 -2u- tent. Also, the mechanical Q of the ferrite may be of the order of a thousand or greater, whereas a nickel resonator would have a Q of only several hundred. The determining factor in whether to use the Ni Span C or ferrite lies in the temperature stability requirements of the filter since ferrite has a considerably higher temperature coefficient of expansion. The materials to be used for the interior resonators and couplers is again determined by the desired Q and tem- perature characteristics and also by the machinability of the material. It probably should be mentioned here that the mechanical Q is determined as the number of cycles of vibration required for the amplitude to die down u.32 per- cent of its original amplitude after the driving force has been removed.10 Table 311 gives the characteristics of some common materials. 10W. V. B. Roberts, "Magnetostriction Devices and Mechanical Filters for Radio Frequencies," Q. S. T., July, 1953, p.28. llIbid 1. -25- Long.vel. in Intrinsic Density thin rod x Im edancey Material 5gm/cm f 10 5 071/880 #10" gmseg/ém3 __ Q 1. Aluminum 2.73 5.11 1.39 #000 2. Brass(hard) 8.5M 3.6M 3.11 2500 3. Brass(soft) 8.50 3.52 2.99 2000 5. Ferrite h.h6 5.58 2.h9 1250 e. Nickel 8. 88 n.9h u.39 #50 7. Ni Span c 7. 99 u.ao 3.83 900 8. Steel(drill rod) 7. 86 5.13 h.03 900 9. Steel(stain- leSS) 79 9h h-97 3.95 1500 Table 3. Properties of Filter Materials. -26.. V. PRE-DESIGN CGNSIDERATICNS A. Filter Parameters and Assumptions As a preliminary to the actual design exanples, a brief review of the assumptions used and the constants to be used for the designs will now be given. It seems that in discussing mechanical filters, the best and most fundamental constants to consider are: l. fo = resonant frequency of each resonator 2. krcm-u) 9- coefficient of coupling between the r th and (r+l)th adja- cent resonators 3. dr(é£)‘= decrement of the resonator The main assumption made is that the interior resona- tors have zero decrement or infinite Q. This leads to the second assumption which is; that the coefficient of coup- ling between adjacent resonators is numerically equal to the fractional bandwidth when the two resonators are con- sidered alone. For example, consider the structure made up of half-wave resonators and quarter-wave coupling necks. The coefficient of coupling between adjacent resonators is given by equation 8 or 8a. -27.. Using equation 8 we have, ern-I) = 13‘, ¢ (1 ' ¢/2) _2_ 233. l - Zea. 71' Z0: 220: For longitudinal vibration, a k ... 2(DPV(1- .5 DcPVc rum) .. .. T.___¢ <- ¢ ) W' Balk Vi) (D:f% W: and if resonator and coupler are of the same material, erV‘H) .-_ %%:(1 " 05 11%;.) R For torsional vibration, kr('+.) = %(Dc Pa ch1 " 051° Vc)) PR Va DR DP+_7\L-— and for resonator and coupler of the same material, 2.0 - -5 9%) D, D, n+ kKwH) = #Hv A summary of the symbols used is as follows: modulus of elasticity modulus of rigidity (shear modulus) poisson's ratio mass density moment of inertia per unit length .2: pJ moment of inertia of cross section area cross section area diameter length “Uqumhmm R subscript denoting resonator c subscript denoting coupler n total number of resonators in filter; this may include the electrical and circuits r resonator number in filter chain. Input resonator is no. 1. 2°. characteristic impedance of resonator (see table 1 or 2) 201 characteristic impedance of coupler ZoL/Zoi v longitudinal wave velocity v' torsional wave velocity VP voltage output at peak of response curve V, voltage output at valley level F fz-f, f; FU' Aft? fa Afv bandwidth at valley level B. Equations For Obtaining Chebishev Response The types of filters described up to this point are those made up of a chain of identical sections. This type of filter may have a fairly flat reSponse around mid-band but ripples become quite pronounced toward the band edges, especially as the number ofiilter sections is increased as shown in figure 6a. When the unloaded Q of the elements to be used is high enough for them to be considered non-dissipative, a fairly simple method worked out by Dishal12 can be incor- porated to design a filter which will have the optimum attenuation shape. This method as described by Dishal, is based on the allowable pass—band insertion loss and is particularly valuable when there are stringent requirements on the pass- band tolerance and rate of cut-off of the filter. The attenuation shape is like that shown in figure 6b. where ripples occur throughout the entire pass band but the peaks are all equal and the peak to valley ratio can be made as small as desired. However, according to Dishal, it is best to design for the maximum allowable ripple as this will give the greatest rate of cut-off. The equation for the Optimum response curve of figure 6b. is, (Xi) = l + [(191 - l]coshz' {n cosh_'A£* (214) V f, This equation was arrived at by the method of approximat- ing a constant by means of Chebishev polynomials and hence 12M. Dishal, "Two New Equations for the Design of Filters," Electrical Communication, Vol. 30, Dec., 1953, pp. 32h - 337. - 3o - the term "Chebishev Response" is used. The reSponse shape is Optimum in the sense that for a given allowable ripple in the pass band, it produces the maximum possible rate of cutoff for a given number of elements. Dishal has wOrked out two design equations to obtain the shape of equation 2h. One applies for the symmetrical filter with damping at both ends, and the other applies for the unsymmetrical filter with damping at one end only. These equations are shown in figure 7. a. Symmetrical Filter (Resistive Gen., Resistive Load) 9L =§_§__’LI_1_9_ Qz-9(n"l)=°° 2' a. lira-+0 _. iSn 4- Sin Ere) Afv/fo - hisinT2r - HGTYsin (2r+ 1W? b. Unsymmetrical Filter (Resistive Gen., Reactive Load) or vice versa g, __ sine 1 lira-+0 = (S:+ sin2r6 Kfv73‘o sea-(re ){sin (Zr-De sin r+l —L 6: 00 sh: sinh {l sinh" (v.1 - 2 _ n n V Figure 7. _ 31 - VI. EXAMPLES OF FILTER DESIGN A. Campbell Type - Longi- tudinal Mode The campbell type filter is basically that shown in figure 8. By constructing the equivalent T for a section . jjdj‘ ((12. k1 If“ . k4; id?" 3 Figure 8. of this filter, the mid-band image impedance, for M<3:L , is found to be 230:: juJM. The mechanical structure with half-wave resonators and quarter-wave couplers has a mid- band image impedance 210:: 20;. We might expect, since these forms are the same, that the same design criteria could be used for the mechanical structure. That is, the coefficients of coupling between each end resonator and its adjacent interior resonator should be .707 times the fractional bandwidth and the interior coefficients of coup- ling should be .5 times the fractional bandwidth. The decrement of each end circuit should be equal to the frac- tional bandwidth. Let the Specifications for the mechanical filter be as follows: 1. f0: llO kc 2. Af= O kc 3. = .0945 h. Resonators - half—wave length; .25" dia., Ni Span C for tempera- ture stability 5. Couplers - quarter—wave, Ni Span C 6. n.= five resonators The resonator lengths are, [R : V/2f= 1.892(105' =086" 2Xl10x10y The coupler lengths are, lo = Lil/2 = M3" The interior coefficients of coupling should be, k1: F/2 = .0273 Thus,the coupler diameters are given by, Deg: DR (1- Il-kafi )J I.) = .25 (l -\/l -‘1|'x .0273) = .0512" By making the end resonators contain half the energy of the interior resonators, the proper coefficient of coup- ling is obtained and the first and last coupling necks can be made the same diameter as the interior couplers. The cross sectional area of the end resonators will then be made equal to half the area of the interior resonators -33.. giving for the diameter of the end resonators, D... = -707 D.u == .177" An infinite line termination consisting of a few feet of OOpper wire can be used with characteristic impedance equal to .707 Zen. The diameter Of this wire is given by, D..== Dc mow/)8 where 2 [JV of coupler ._ 1.15 y” pv‘OfIcOpper — hence, DL='.Oh62" which is about the size of no. 16 or 17 wire. The lines can be coiled in some manner to make them more compact and coated with a viscous substance to adjust for the exact amount of damping required. The complete structure is shown in figure 9. /— a ll reson atom f .m" |-—.56'—.1 i =+=T_ .715" . - L *1“ h . .0512' COf/Oer (‘1er .0462” ,I77 all couplers Figure 9. B. Campbell Type - Torsional Mode Let the specifications be as follows: 1. f.‘-' 100 kc 2. Af=Likc 30 F: 00“. u. half-wave resonators, quarter-wave couplers 5. Use electrical circuits as end resona- tors and .25" dia. ferrite for magne- tostrictive resonators and .25" dia. dural rod for other resonators. 6. Couplers - dural rod 7. n»: 7, including electrical circuits The Q of the electrical end circuits should be, Q = I -_-_- 35.3 0707 X .0}... The coefficient of coupling between the electrical cir- cuits and the ferrites should be, k3,; = k‘lq == .028 The others should be, 1:1: .02 Using equation 8a, the coefficient of coupling between ad- jacent resonators is, km“) == 3... (12:9.LL 1’ DR Although this is not strictly correct for km: and k5, , the error introduced is not great. The diameter of the coupling necks is thus found to be, -35.. .25 (TX .02 )3? 2 = .105" The aluminum resonators and couplers can be turned out from a single quarter-inch dural rod with extensions for attaching the ferrite resonators. The resonator lengths are: Ferrite: L ._ 1 G _, l n.96x10‘ cm. " " 35-2710 Lula/980 i: 1065 cm. = .65" Dural: Lam. l 2.6x10a cm. ‘ 2350’“ m = 1.505 em. = .593" The length of the couplers is: LC. ‘-= .5 x .593 = .2965" A schematic of the filter is shown in figure 10. The coils surrounding the ferrite resonators must be wound so as to provide a toroidal field (not as shown), and their positions over the ferrites adjusted to give the prOper coefficient of coupling. By drilling holes thru the ferrites as shown in the diagram, they can be permanently magnetized with a toroidal field by passing a current carrying wire through the hole. Thus the biasing magnets can be eliminated. -36- rs-“H was rt-m'fi r‘m‘fl ”"1 l l E \— ggu'fi:“+°\ a“ coupler 5 .zss:"ton3 ‘ ’05. u did” L 2:303} Lofh and circuits: A 'vi n___“ fi 1v 1 4 1 Figure 100 C. Chebishev Type - Longitudinal Mode The Specifications are as follows: 1. r, = 200 kc 2. Af = 8 kc 30 ‘ = 00“. L1,. vp/v... = 1.05 5. Symmetrical type filter using electrical end circuits as resonators, n = 7. 6. Half-wave resonators, quarter-wave coup- ‘ lers 7. .25" dia. ferrite and .25" dia. dural rod as in previous example The lengths of the resonators are, Ferrite: in = .55" Dural: in .503" The coupler length is, t. = .2565" The first set of equations in figure 7 is now used to find the required end circuit Q's and the required coeffi- - 37 - cients of coupling. a = 9OO/n = 12.860 and Sn: .27 The required coefficients of coupling are, k”, : km = .0290 k2,3 = k5“ = .0220 k3fl_ k,fi.I= .0213 For k1,3 and k5“ , the coupler diameter is, Dc ==.0k65" and for k3”. and k”; , D. = .0157" The filter structure is shown in figure 11. h—fif—vl $5031.! L l‘—.$‘0.3"-" # l-s—Joa'lv'i f——.a;{p Fort-ii“ Fer-rill U T0465” $457” T0457‘ .0465" J i all resonai’om .zsfldt'a. 1: all Couplers $565,110” i L (3:36.? fbofk and circuits: k: .019 Figure 11. D. Chebishev Type - Torsional Mode Use the same specifications as for filter A, except for torsional mode of vibration.and.Vb/V&i= 1.05. The velocity of torsion waves for Ni Span C is approximately 3 x 105 centimeters per second. -38- The The The The length of the resonators is, [R —.: 3 x 10" = .538" 2.2 X 10? X. 2051‘. coupler length is, LC == .5 x .538 = .269" required end Q's are, am = 2 sine = 2 sin 18° = 29.7 ’ Sn AfV/fo .352 x .0515 required coefficients of coupling are, k“; = k‘h" = .038 kz,3 = k3’4 : .031 coupling neck diameter for kbl and k4fi- is, .1. _ Dc = .25 (IE x .038)4 — .121. and for kz’3 and k3’4 , DC = .25 ( ‘rr x .o31fi .117" ’2 Enough damping must be added to the end resonators from the to reduce their Q to 29.7. Using the infinite COpper line termination, its characteristic impedance can be found expression, 201. = “NZOI 29. and the line diameter is found to be, D = .25( 7r x 8.0 x M); L 2 x 2907 809/ 307 = .25 (.0615)%5 - 39 - which gives DL=.125" . A comparison of this design with the first shows that less disparity is required between the resonator and coup- ler diameters when the torsional mode of vibration is used. A diagram of this structure is shown in figure 12. J24 .117” .H7" 1z4” i 1% 5512:: all resonators .24" Jim, .536, lon1 copper wire .12.)" it". all couplers .2‘3” long both, 67116 Figure 12. E. Chebishev Type — Torsional (unsymmetrical) This filter will be unsymmetrical; i.e., with damp- ing at one end only. The second set of equations in figure 7 will be used. The specifications are as follows: 1. f. = 100 kc 2. Af: l kc 3. F = .01 u. v./v.-= 1.03 5. All Ni Span C construction; .25" dia. half-wave resonators, quarter-wave couplers. 6. Torsional mode of vibration 70 n: ’4. The required Q for the first resonator is, Q = sin 22_.§° =69 .555 x .01 -L10- The required coefficients of coupling are, k = .01 k = .0072 k3.4 = .00725 The coupler diameters are, .1._ . Dc“: .25 (3% x .01)4 -— .0883" DCK,3 = .25 (_72T_ X .0072)3f : .0815" 1363’4 = .25 (4; x .00725)‘3‘2 = .0818" The length of the resonators is .59" and the coup- ler length is .295". I Enough damping must be added to the first resonator to reduce its Q to 69. For the infinite line termination, the diameter of copper wire needed is, D = .25 ’H’ x 2!} )‘i‘ (2 X 69 19.7 =3 .102" This is about the size of no. 10 wire. r1533" .0615” .0815” a ' ll L copper wire all resona‘l'ors .25 cluz.’ .59 Ian? 'Ioz" 4"“ all coulD lens .295” Ian, Figure 13. A diagram of the complete filter structure is shown in figure 13. - H1 - z-mL- LAX“ . -.t..‘ .‘L 414.;- F A » fi’t "1 VII. CONCLUSIONS This paper has discussed a problem of prime interest in the communications field today. Namely, that of de- signing a cheap, practical bandpass filter with a selec- tivity curve closely approximating that for the ideal filter. The mechanical filter with its high Q elements m...“ ‘x-mm-Jfi' m—n-u—m. n- has been proposed as a possible answer to this problem. It has been shown how the valuable concept of the' electrical-mechanical analogy has enabled a simple design procedure to be worked out. Although far from rigorous; from a practical vieWpoint, it provides a simple, workable method of analysis. Two closely related methods were de- scribed both of which compared the mechanical filter to a cascaded chain of electrical transmission lines. The various structural considerations were discussed and equations were given whereby the filter could be de- signed to produce the Chebishev reSponse or maximum rate of cutoff. Several examples were given for both the long- itudinal and torsional modes of vibration. Naturally, the particular filter structure described has some drawbacks. One doesn't have to raise the fre- quency very much higher than in the examples before the -142- elements become too small to handle. Flimsiness of the structure may also result for an extremely narrow band- width. This may be overcome by using the torsional mode of vibration; however, if the frequency is very high, then the lengths of the elements may be too small since the torsional mode requires shorter elements than the longitudinal. These problems can undoubtedly be overcome in some cases by changing the element shapes and vibra- ‘ tion modes, etc. 6 It is regretted that the actual construction and 3 testing of the mechanical filter could not be included in this paper, but time would not permit such a complete treatment. -m- 10. BIBLIOGRAPHY Roberts, w. V. B., and L. L. Burns, Jr., Mechanical Filters for Radio Frequencies, RCA Rev., Vol. 10, pp- 3E8-§5§. Septo. 19h9- Hubbard, B. R., Longitudinal Vibrations ig'g Loaded Rod, Journ. Accoust. Soc. of Amer., Vol. , p0 372: 19310 Marks, L. 8., Mechanical Engineers' Handbook, McGraw- Hill Book Co., Inc., l9h1.. Mason, W. P., Electromechanical Transducers and Wave Filters, D. Van Nostrand Co., Inc., New York, New York, l9u2. Ragan, G. L., Microwave Transmission Circuits, NoGraw- Hill Book Co., Inc., New York, New York, 19h8. Roberts, w. V. B., Some Applications 9: Permanently Hagnetized Ferrite Resonators, RCA Rev., Vol. 1k, pp. 3 - 16, March, I953. Guillemin, E. A., Communication Networks, Vol. 2, John Wiley & Sons, Inc., New York, New York, 1935. Roberts, N. V. B., Magnetostriction Devices and Mech- anical Filters for Radio Frequencies, QST, July, 1953. Dishal, M., Two New Equations for the Desigg‘gf Filters, Electrical Communication, Vol. 30, Dec., 1953. Adler, R., Compact Electromechanical Filter, Electronics, Vol. 20, April, 19h7. ‘//. w“ I t 1“. ‘ «v 2A.; 4 f 1 -. tJ 1 1 2.2....) ‘3, A__ ..‘ _ A ‘4 4i