Cohn, S.B.

7958IRE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Parak14kmpled Transmission~Line~Resonator SEYMOUR Filters* Summmy-This paper describes the synthesis of band-pass transmission-line titers consisting of series of half-wavelength resonant conductors such as strips. The design differs from the usual end-coupled strip configuration in that successive strips are parallel coup~ed along a distance of a quarter-wavelength. The resulting coupling between resonators is partly electric and partly magnetic. Several important advantages are gained by this arrangement: 1) the length of the filter is approximately half that of the end-coupled type; 2) the gaps are larger and therefore less critical; and 3) the insertionIoss curve is symmetrical on a frequency scale with the first spurious response occurring at three times the center frequency of the pass band. transFormulas are derived for the parallel-coupled-resonator mission-line filter that permit accurate design for Tchebycheff, maximally flat, or any other physically realizable response. The formulas are theoretically exact in the limit of zero bandwidth, but frequency-response calculations show them to give good results for band widths up to about 30 per cent. An experimental strip-line filter of this typ,s has been constructed, and the data given in this paper show that excellent performance has been obtained. ripple. checked found about and method paper3 resonator low-pass made resonant The by to accuracy exact of the design and for formulas they bandwidths flat in has have been been up to computation, results give excellent cent in the 20 per 30 per of for case of maximally ripple used of that types the the either by An between response ‘1’he earlier an cent in the case of equal of this other method, having of response. direct-coupledresponso or paral analysis a number filters. In to is basically lumped-constant desired seriesis [el - prototype equivalent filter a set LC arms interconnected broa,d-band quarequiva- ter-wavelength lence the then actual transformers. is established coupled-resonator approximate the latter circuit and structure. INTRODUCTION =Z2%E2Z2H22%Z+= ‘a) ~~ I .1 A this S shown in Fig. 1(a) ,1–3 multiple-coupled-resonator in strip have strips strips Parallel over reduced line been (or most other end TEM a~~ to end. in Fig, l—Coupled-resonator (b) and I (b) ~~nu (c) [ 1 band-pass transmission filters line) ~ comr$only composed In which the of half-wavelength paper, an alternative half-wavelength 1 (b) the and filter 1 (c). is coupled are arrangement is treated parallel-coupled, offers coupling: by strip-line (c) parallel filters: coupled. (a) end coupled; as in Fig. ber length of coupling end approximately a num1) the half; is at larger last eases adthe The coupled ment filter, design filter values which formulas are gl, may given @ for in Table of an m-resonator 1. They the for means utilize paral”[elthe eleDESIGN PROCEDURE of important advantages 2) a symmetrical obtained three gap times between is of with the the insertion-loss-vs-frequency first spurious frequency, strips response and response occurring center 3) a much The it . “ . , g. prototype either low-pass flat in that in a are of [respecieventhe have rellatare adjacent particular gaps is permitted. since bandwidth, tolerance. power that any maximally allow in desired be computed response by from maximally given vantage tolerance a broader the filter. Formu forward realizable larger importance, for for a given a given a higher derived to to such be have as or equal-ripple Table the I I.3 The filter of formulas in Fig. on the or permits Furthermore, rating the of the schematic diagram 2 shows resulting sections center of the and bandwidth gap Ias have manner, response, is assembled containing (one-quarter their on wave. been to The for the total the n + 1 sections, The wavelength design permits been filter and structure equal quency), fied mode by n resonators. electrical parallel and length and two wave at the parallelphysically or equal- is completely coupled-resonator designed a straight- characteristic impedances—Zo,, conductors, formulas of structure flat the or the Z~” of flat odd-mode previously ing them These defined, characteristic impedances and cross always graphs section will dimensions filter * Manuscript received by the PGMTT, October 21, 195 ~. The work described in this paper was supported by the U. S. Army Signal Eng. Labs under Contract No. D.\ 36-039-sc-64625. + Stanford Research Institute, Menlo Park, Calif. 1 E, G. Fubini, W. E. Fromm, and H. S. Keen, ‘{Microwave applications of hi~h-O striu components, ” IRE CONVENTION RECORD, pt. 8, pp. 98–1 O:; Marcfi, 195~. 2 E. A. Bradley, “Design and development of strip-line filters, ” IRE TRANS., vol. MTT-4, pp. 86–93; April, 1956. J S. B. Cohn. “Direct-couded-resonator filters, ” PROC. IRE, vol. 45, pp 187-196 ~ February, 1$57. available.4 metrical In one the the should number be sym- maximally of select first equal-ripple of response will yield response. fil ker function the desired line, ” design a parallel-coupled-resollator type that and inIRE of resonators “Shielded coupled-strip 4 S. B. Cohn, TRANS., vol. MTT-3, pp. 29-38, October, transmission 1955. This differde- The curve The the exact (which bandwidth design response appear when with the the error curves later) may aid values of these relative in width coupling as the first case to compensate introduced bandwidth curves. as shown amount and. .fO I ~zo n+l (For either maximally-flat or equal-ripple response. bandwidth and last is reduced the perhaps is negligible per cent. is to for done of these in the One alter fringing of bandwidths important of the at less than step their in ends. The 2. transmission-line capacitance . Next.. puted. .t and Zao.1 the is noted () f–fo — f. 1 [ f. where AO = wavelength /0= (f. and formulas. . . Zall of the section available widths to not section. is necessary gap.. the structure is symmetrical. pass band edge of prototype filter. I f2 2- - 1 I i n line at .1. from Cohn:) (The dimensions sertion-loss may be done in Table between the function with the the I I and the in the aid pass of the and stop bands. odd-mode characteristic impedance with respect to ground of each conductor in ith section. case will. . b = prototype elements in farads and henries from Table 11 r = W’ = ~.. .–fl and show how be reduced in general. further the length design may be Zoo values resonators compensate 1 sections. in transmission +.fJ/2 gl. thin strip in the graphs and and spacings following scales approximate of the filter: prototype conductors.i = ZOO. section. the and Then.224 IRE TRANSACTIONS ON MICROWAVE TABLE I THEORY AND TECHNIQUES April FORMULAS FOR PARALLEL-COUPLED TRANSMISSIOWLINE-RESONATOR (n : FILTER t-’-t-+-’ I I $ PROTOTYPE ~ I I LOW-PASS BAND-PASS I [—. g. for of variation except (1) “ the bandwidth-error response by errors are 0. may the Z. variation a few the to This filter of thick frequency somewhat means that the more strip will difficulty parallel-coupled-resonator J . width in Fig.= right-hand load resistance in schematic of Table II. exceeds preadjusting be comof from variation ences creases for the hence resonators in the be constant. This dimensions these easily in each section should be designed This ~ may to be and strips. . It yield done with by differ of the insertion-loss formulas relationship filter and characteristic in the case of impedances. even-mode characteristic impedance with respect to ground of each conductor in ith section.) 1 = AO/4.fz = Z. corresponding pass band edges of transmission-line filter. gz. each of the and the element in n+ terms gl. f. . g2. I 1. for strip-line construction may be obtained as function of Zo. . ! 95% Cohn: Parallel-coupled T’ransmission-Line-Resonafor TABLE 11 FOR MAXIMALLY-FLAT Filfers 225 PROTOTYPE LOW-PASS FILTER AND ITS DESIGN EQUATIONS AND TCHEBYCHEFF RESPONSE ‘@::%r=’ :Qr n Maximally-Flat r=lforalltz ODD Response n EVEN “=2 ‘in[%3 A = 10 log. L?14 .o [1 + (lO%j’O (lOA#O – 1) COS2 ]db. . A. [1 + – 1) cosh’ (n cosh-’ T s ~1 n+l filter.. n bk-1zh-1 check: g.. ‘n+t ‘T Fig. (L) ’=1 . n 3db o —— Tchebycheff r = 1 for n odd. A = 10 log..3.= cl) (J I . . P = log.. k=l.— Response (6/4) for n even r = tanh2 d! k=2.2.’. .n (n COS-’ (d’) I b~=y’+sinz () n coth ~ krr — . = glr JnnnK. g%= 2a~/y ‘i@d@ gb = ——— .. ..a (1 + co’z”)db Al db k=l.. ( ) . in db A = 10 iog.. w’) ]db. 2—Basic dimensions of the strip-line parallel-coupled-resonator . of the This designed for maximally-flat response.1 I . insertion-loss various computa- (Cf ‘/c). > .0 -1 ?. length. —=050 I 1.2 0. d. 165b.4 — fg -f. parallel-coupled designed for equal-ripple respons~vswr = 1. ” IRE TRANS.1 — [. d would be expected Cfr/e the is a stripon that For very In wided Because ivation checked tion tual mally were of to be somewhat less than elsewhere filter would where as a function of of the the approximations design formulas.2’26 IRE TRANSACTIONS ON MfCROWAVE THEORY AND TECHNIQUES April J++: I I (a) 1.1 I -10 I -08 -06 -0. ‘(Problems MTT-3. bandwidth filters an error in d would filters adjustments effects be done.2 1! $+d> Fig.75 theoretically.8 1. pp.2” — 1. Fig. This was and filter bandwidths response. this should approximately Measurements indicate (b/2). necessary their done by exact responses in the derwas acthe thickness-to-plate-spacing experimental be strips.0 cancelled needed overcome tuning or dielectric of constructional for example. AA 0.4 -0. of screws in high-electric-field resonators. t/b.4 -0.0 1 I f2 -f. 4—VSWR curves of parallel-coupled filter ances.4 — I .2 0 02 -fJ 04 06 08 1.2 0 (f-fo)/(f2-fo) six-resonator.75 be 0.0 -1. slugs the be filter I .1 E :.4 — I . -+d ~d ~d (b) 1. 5 S.4 0. I 10 12 (f -fo)/(f.6 — I . in strip transmission March.220b) of the for flat vswr networks transmission-line or Tchebycheff specified for The formulas Vol.6 -0.5 below is cut back by the VERIFICATION OF DESIGN ACCUR.. Fig.0 unimportant.3 I .6 — 1.05 1. would are already tolerby means regions be while in narrow-bandwidth by to the tuning the may error that I . accuracy of the by either by described 0. and computer maxi- computations a matrix- performed on an electronic . . Cohn. = O.8 — 1. 5—VSWR curves of six-resonator.2 — 1. 3—(a) Strip-resonat?r design+ ne@ecting f:inging capacitance at ends. 2. B. 1. 1955.0 1 I I I I I I ! — 1. ++1.8 1. The where each end (b/2).0 Ii [ 0.0 I I I . 119-126.8 -0. plotted 3(b).8 1.0 I I I [ I . — fo 10 = 0.10.4 — — f2 -f.[ 10 1.2 — I -0. —= f. (b) Suggested compensation for frmgmg capacitance at ends of resonators. as shown in Fig. line.0 W > I .0 I I 1.2 — 1. (0.6 — — —= 0.6 0.4 — I .3 — I .~CY average quantity the thin dimension (Cf ‘/e) ratio.2 cc 31..6 I .0 — I I 12 — 1. f. % ~ ‘11-1~ 40 — . of matrix algebra to filter theory. Fig.. :03 <~. and fI and . 020 025 030 the response curve.3. at (. 1946. 145P-150P.fz–. Error is expressed as a percejltage of the corrected bandwidth = ~. function for greater is shown. pp. multiplication it is sufficient The insertion functions method. eBecause the filter the matrix vswr then of this matrix. and even at 0. it is important to note that these conclusions are drawn from specific cases considered.10 in the pass band.3. vol.5 the response is adequate for many applications. At 0. 34.10.2 the deviation from maximally flat is slight. 0 02 04 06 C m 0 0 02 04 06 os IO 12 14 16 lf-fol/(f. IRE. parallel-coupled filter designed for equal-ripple response—vswr = 1. e.fJ /fo = 0. 5 for six-resonator ripple vswr vswr filters designed for to have In this is obup to is inthe equal frequency. Richards.2.10 is not exceeded for relative band- In addition to the errors develop as the bandwidth in response-curve shape that is increased.f ~=oz f.l 10 !. bcmdwidth.1. while at 0. -fo) la 20 22 24 26 28 ~J--l--l. 7. _20Li-L_LLl_..1958 Cohn: Parallel-Coupled Transmission-Line-Resonafor 60 I I I I I Filfers 227 ‘0 --11 50 — I I I I I / I 1 1 I I / I I I I I I I /1 I /~1 : 8 3 ~ 30 F .g. the filter. The corresponding curves are shown case. in Fig.098.x.. ~[.1. 8 the bandwidth error in per cent is plotted vs relative bandwidth for the cases considered above. 1. the insertion and fz are the pass band of 1. and may vary somewhat with other numbers of resonators or with other equal ripple levels.l. band vswr limit In spite of this deterioration. in the case of the wider bandwidths. however. and is only about 6 per cent at 0. In Fig. 8—Bandwidth error vs relative bandwidth for six-resonator parallel-coupled filter.05. only half of each pass band loss is virtually for relative bandwidths. Because the response is symmetrical with bandwidth is slightly in error.f.. and even quite small. The actual bandwidth of the filter appears to be always less than the value assumed in the design. The relative bandwidth #2-fl is defined as (2) jo ‘ where f o= (fl +fJ /2. as computed to compute loss and of half of the elements The pass band vswr curves for six-resonator maximally flat filters of various bandwidths are shown in Fig. 7—Insertion-loss curves of six-resonator. the predicted widths up to 0.2 14 lf-fol/(f2-fo) 16 18 20 22 24 26 2. Fig. But. 6–-Insertiomloss curves of six-resonator. 6 and Fig. identical bandwidths In both with that u p to is an equal ripple-level of 1. 4. The maximally flat for relative response is seen to be truly bandwidths up to 0. fl tained 0. the deviation would be serious in the more critical applications. desired response the deviation very accurately for relative bandwidths and gradually deteriorates as the bandwidth creased further.1. figures.l 0 005 010 0[5 f2 . parallel-coupled filter designed for maximally-flat response. were is symmetrical.fz are the $db points Of Fig. It is seen that the discrepancy does not exceed 2 per cent of the bandwidth for rela tive bandwidths up to 0. The limits of the prototype 0. 8 mav be used as an approximate guide in selecting the bandwidth design value. March. and therefore. The insertion-loss curves for the above cases are plotted against a normalized frequency scale in Fig. the pass of 1.6 Fig. it would be desirable in calculating the parameters of a given filter to use a somewhat larger bandwidth than actually is required.4 or 0.. --i 20 — 10 — ‘. ” 8 P. “Applications PROC.10.10. . 110 0. and ZO to 50 ohms.10). . 9 shows that the filter has seven sections.5 inch. total foil ground-plane was cemented spacing to the is one-half polystyrene inch. 9—Layout of parallel-coupled-resonator filter. Cohn. 37. This impedance is the width w~ of the may be obtained from a vs strip widths.085 0. the flat surfaces and a thickness of 0. ga.. are of copper designed and actual In for an 10-per equal 0.7 A further terminating graph result length quantity required 50-ohm strips. vol.55. and separations s.360 0.4 ohms ohms ohms ohms zoo. and strip thickness assumed to be zero. and gl of the prototype sevenlow-pass filter. Fig. gl = 1. I. the resonators are shortened at their ends by di to compensate for fringing capacitance. i and ZOO{are com from the formulas in Table I. 52-57. centered pass and the filter are appropriate band vswr of 1.158 0.0017 inch thick. ga. Z. The various quantities referred to above are contained in Table 111.250 inch.021 0.7 55.084 0.. In this calculation demonstrate shown v2–~1)/J”o is set equal to 0.6 43.53454.8 55. curves as calculated 5 and is Fig. the gz = 1. pp.5 58.236 0.073 0.4 with E. g~ = 1.165~ mentioned above. The two mode puted characteristic impedances ZO. The filter was at 1200 cent bandwidth. plates The with copper Dow n = 7 and Am = 0. theoretical transmission-line figure the an electronic each of characteristic and (~z :~1)/~ = 0. Then the strip widths w.8 The has six resonators.1. of the various sections are obtained from the nomograms of a previous paper. The resulting values are gl = 0:77968.163 inch inch inchinch 0.5 45. hence equals Photographs of the completed filter are shown in Fig. 10.. a polystyrene dielectric. shown curve in Fig. 0. ” IRE TRANS.346 0. “Characteristic impedance of the shielded-strip transmission line.00986 db (which corresponds put vswr of 1. This is done by means of the for Tchebycheff response in Table II. with to an in- is wT/b = 0.0976 82. B.228 IRE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES April l--’-d 7 Fig.i 0. 9.540 inch. July.3 45. parallel-coupled-resona and this tested type of in order design.6 ohms ohms ohms ohms Wi . but because of compute The first element element.1529 0. it is not given here. by ground were The spaced strips mc.449 0. Next. exact the by 7. ‘i KL-Li Zo. ripple vswr computer.361 inch inch inch inch 0.10.35921.372 inch.5 inch apart foil. labeled The of insertion-loss network.1038 0. L equal to 0. Because the formula is rather complex. si di : : 0. MTT-2. and are quite close to the value O.1. The structure is a sandwich of copper foil separated by a pair of polystyrene plates each machined to have Thus. 1954.744 or wT = 0. The details of the design procedure a discussion of the experimental results are given below. formulas the symmetry of the filter.085 inch inch inch inch DESIGN A strip-line constructed feasibility the filter of OF EXPERIMENTAL FILTER tor to As filter has in planes been the Fig. it is necessary to the parameters of only the first four sections. step in the design procedure is to compute the values gl. and its assumptions are as yet unproved./K. The section 1 is a quarter-wavelength in the dielectric. and odd- quantity and the even- 7 The values of d~in Table I I I were computed from an approximate formula.68800. and 1. Finally. TABLE III DESIGN PARAMETERS FOR EXPERIMENTAL PARALLEL-COUPLED STRIP-LINE-RESONATOR FILTER z. 8 S. equal to 2. .00 II 0 filter. . The input vswr of the filter measured with a 50-ohm termination the would highest frequency the vswr to the believed closely for to the set vswr be on the exceeds quite end. than was Le. and the center frequency was higher than the design value by 1.is construction.. EXPER. I j!29 1 1 I 1 / 1 \l 35 . value. 2 . --— I 5 // “-” . Although of 1.. copper and air dried for 45 minutes before pressing the foil ontcl the polystyrene...0”.b 1100 II ... 0.6 Y-MC 1250 I .7 db.6 per bandwidth the last the response pass may band have desired DERIVATION OF DESIGN FORMULAS ~are quite Although with The analysis of the parallel-coupled-resonator filter... of the curve Above error is for the transmission-line 2 db. 1l—Insertion I $8 loss of parallel-coupled-resonator !6 I ... rather of 1200 cases similar.0 per cent The bandwidth Corning XC-271 adhesive. occurs near of the 1.. (b) Fig.300 Fig../50 . . (b) completely assembled.. = 1207. which was applied to the seen to be only bandwicith. Filters .00 >.2s0 ..”. due design In value the was to three obtained more could be made adjustments to conform could provided resonators.20. .5 I .. ... The than of this model is 1207 mc.5 to 0. the filter was reconstructed with center frequency the cent.315 . The image impedance ZI and image phase shift p of this section are given in a paper by Jones and .7 per cent. curves respons~ been Fig.27!> I .. 1.. (a) Fig. that output the is shown in Fig ripple most band band 12.4 FILTER resonator uncompenwas the lower opena dis- In the pass band. uniformly with by acceptable in the it in the vswr if tuning These frequency where the applications. disagreement is inevitable because of the finite Q of the strip-line resonators. Finally.10. DATA A preliminary filter sated. ( THEORETICAL FOR 1... MENTAL \ 30 —- ! . however. The strip circuit then was cut on one surface with a sharp knife and the unwanted foil peeled off. it theoretical for pass pass reaches level parallel-coupled resonator ends Then.1958 Cohn: Parallel-Coupled Transmission-Line-Resonator 40. “m. The center per cut frequency cent.. the value of (Cf’/e)/(b/2) for very thin strips. back vswr peak a value circuited were elsewhere theory center is under were also filter exactly tance d = 0. +/ ‘--”-- 40 -. 0 ~ ... Nevertheless. : I ~ .REOUENC I .00 .”. about 1. 10-–Photographic views of parallel-coupled resonator filter: (a) with upper plate removed. The low l[t is more serve on the while di = O. the tested with design strip FOR EXPERIMENTAL model first value ends all of the the 4. 13(a). the agreement for a center of 1207 mc and neglecting is excellent. accurate me—an the error of only and best it the case.. the measured pass band insertion loss is quite uniform at 0.5 — I 20 -— II \ .- — >.33.220b.2. adjustments of the the values of di given in Table III. insertion frequency loss of the filter and the exact theoretical network computed dissipation. 12—Measmed vswr of parallel-coupled-resonator filter. between measured 11 shows a comparison based upon the characteristics of the individual section of Fig. - -. ’23 . . The image impedance.b impedance of a transmission X 1 ~sin~ 20 Cos + [+’:::]g E. 13(a).lpi INPUT LINE Z. length with of the coupled respect to ground transmisimpedfor the sion lines and Zo. 1947. Inc. o A B CD matrix A B CD matrix characteristic parameters. of the section means cos @ ~ZO sin+ [1[ CD— AB o x –j –jK- = ~sin+ 20– Cos cp 1[ 7.. and Zoo are the characteristic respectively. 13(b) is approximately equivalent to that of Fig. ” IRE TRANS. LA –jK o“ 1 Cos /8 = (::3’0s” where ances even -1 O is the electrical of each and odd conductor modes. VOI. t—-++ Kol Z. —-Z. cot # “at (d) zE2iii=-yO (e) K12 * Kol -90” ‘r 1 1 1 1 —— ’23 K n. – = Cos’ +]’/’ (3) substituting Zo = K. 1956. T. April. “Very High Frequency Techniques.j L -.230 IRE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES April = (b) --. line 26. Z. Therefore. K 12 Z. o (4) –. M. T. ” McGraw-Hill BoolI Co. pp. Bolljahng Zr as follows The matrix zoo)’ – (Zoe + 2 sin + zoo)’ of the ideal inverter @ = — 90° and is obtained as follows from this by [(2. K.fl = A to the matrix ele- and. 10Radio Research Laboratory Staff. 13—Development of equivalent circuit of the parallel-coupled-resonator filter. “Coupled-strip-transmissionfilters and directional couplers. “ ~. Z. -90” . New York.. Jones and J. __ -90” . vol.10 line of length @ 20 is Zr. by and image of /3. NT.4. The box represents an ideal impedance inverter having a constant image impedance. . and a phase shift of — 90° at all phase the and The frequencies. 2. .n+l TC j 7 -90° ‘r TCr ~ -90° :H —- -90” (f) Fig. (c) X=-z. Bolljahn. tion of Fig. will be derived .. shift. the A B CD matrix 13(b) is of the complete filter sec- It is now shown that the filter section of Fig. MTT. ch. ~ (K ~~ o sinz ~ Cos’ d ) (K K E’ The ments image by parameters ZT = \/B/c are related and cos . hence.. 75-81. 1’... Examination of (3) to (6) that three. in through z. (15) do not form a consistent set. bc noted that. i+ I . :+$– 0 42. 13(f) are small near 6 = 180° and are negligible in comparison with the high characteristic impedances of the inverting elements. z.= (Zo. – zoo = Zo. 2wl’gn (16) (13) Table directly from (13) z . “’” w K ‘“” ‘J 42. error effect on the insertion-loss response.n+I I follow ( 1 3rwr — . no ~+~. 13(f) Fig. (12) is satisfied the effect bandwidth. – zoo (7) portance When sembled. —. Zo by 1/20. +g “d KOI – The formulas (16). the third relation (10) and is the (11) least are important simulTherefore. analysis the 11(c) while the latter the except filter that the circuits may will be shortened of Fig.2 — K 4z03 y () K ~+ 0 z. a judicious choice be made among them. z. Thus in relating is negligible for narrow the individual of the approximations the sections () $+: 0 42.. K sin’ + — 4z0’(:+:)c0s2+ — ]“2 ! K ~+:– 0 Cos /3 = () ~+~ o cosq5. is particularly the equivalent convenient circuit This exact equivalent for 6’ near has been transformer 180°.—– The presence a sin may of sin@ in the left in by side the of each equality vicinity with of the above makes However. and between sections equivalent of Fig.. (6) zoo —. + zoo)’. + zoo sin+= z. in which transmission lines approximately &/2 long are separated by inverters. resonator as the bandwidth is increased. (9) omitting the phase-reversing has. . impossible./K<< involved therefore. 13(c).2 (8) K sinz @ equivalent to the circuit of Fig. 7rw 2w’dgEg%+l ‘ functions equations three must of the taneously may be found. solved W is the relative bandwidth defined and last section of the filter. the circuit is approximately equivalent to that of of Fig. Thus. 1–. the parallel-coupledis approximately it is seen that the original filter of Fig. – zoo circuit the latter shown (lo) (11) Fig. (12) replaced changed. + zoo z“. and Zoo/Zo as and since the where the first z. .3 — –1 1 2 sin+ sin@ K Z. the series reactance of Fig. for are duals. A lumped-constant equivalent of a line of length cuit 13(e) O is shown in Fig. For indicates to yield z.— T w z. 1 (b) or 1(c) K = (Zo./Z~ Only purpose. of Z~/K are needed formulas for this for Zo. which In simplified cirFig. frequency-independent @ is stationary be replaced bandwidth. negligible a moderate :+:= 0 22. by () : + 5 K . 13(d). would tions shows between that the (3) and (5). but of approximately 13(a) and 13(b) increases filter in imare as- be electrically were met. 13(c). for narrow bandwidth.1958 Cohn: (7 Parallel-Coupled Transmission-Line-Resonator Filters (Z. Also. 13(a) and if the following (4) and 13(b) condi- Jt may now 1 and.+:. be used za ~ = (z”. —— ‘ — – 2WI’gl K. + zoo)’. and if L and C are inter- From these equations.. Thus. by I/A”. arms of a previous to be the arms. of @ = 90°. and filter. by (2). same. obtained if -K is parallel-resonant formulas analysis series-resonant present Hence.I+. Ki. sections of Fig. paper.. – zoo)’ Z. by (13) and (14). two of the three i=lton —1.3 z. (14) A comparison (6). Z..’ former contains and in the at 13(f) They contains the this with are point that seen by comin hence to the original parallel-coupledresonator The paring anc[ hence over unity Thus. 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