# Патент USA US2115138

код для вставкиApril 26, 1938. s. DARLINGTON 27,115,138‘ WAVE TRANSMISSION NETWORK Filed March 20, 1955 2 Sheets-Sheet 1 FIG. / ‘ R:24 2I >> LOW-PASS HIGH-PASS LINE /5 HIGH-PASS LOW-PASS lNl/ENTOR' .S. DARL/NGTON MW“ BY ' ATTORNEY April 26, 1933. - s DARUNGTON 2,115,138 WAVE TRANSMISSION NETWORK Filed March 20, 1935 2 Sheets-Sheet 2 FIG. 7 // IN l/ENTOR 5. 0A RL/NG TON BygMw/MIK A TTORNEY Patented Apr. 26, 1938 2,115,138 UNITED STATES PATENT OFFICE . A 2,115,138 WAVE TRANSMISSION NETWORK Sidney Darlington, New York, N. Y., assignor to Bell Telephone Laboratories, Incorporated, New York, N. Y., a corporation of New York Application March 20, 1935, Serial No. 11,934 14 Claims. (01. 178-44) This invention relates to wave transmission networks and, more particularly, to networks having frequency selective transmission charac teristics. Frequency selective transmission networks hav ing the property of constant resistance character istic impedance have been known and used for some time but heretofore only in the form of two-terminal impedances or of ordinary four 10 _ terminal networks. An object of the present in vention is to extend this constant resistance prop erty to more complex structures, in particular to eight terminal networks. Networks of this type comprise four individual transmission paths and are useful for the separation of currents of ‘ diiferent frequencies as, for example, in carrier current repeaters. The problem of connecting thesenetworks into a given transmission system is greatly simpli?ed by the present invention 2 0 since, by virtue of the constant resistance char ‘ acteristic, re?ection effects at the insertion points are greatly reduced and, in many cases, substan tially eliminated. The nature and the underlying principles of the invention will be more fully understood from ' the following detailed description and by» refer ence to the attached drawings, of which Fig. 1 illustrates schematically one form of network embodying the invention; Figs. 2 and 3 illustrate alternative forms of the invention; . _ Fig. 4 is illustrative of the application of the invention to telephone repeaters; ' Fig. 5 shows a typical arrangement of one of the component networks of the system of Fig. 1; Fig. 6 is a diagram illustrating a characteristic of one of the networks of Fig. 5; Fig. 7 represents another component network of Fig. 1; and Fig. 8 illustrates a characteristic of a complete ‘7 system in accordance with the invention. The network shown in ‘Fig. 1 comprises two parallel transmission paths extending between pairs of terminals I, 2 and 3, 4, and each in cluding two component networks 9, ID, in the upper path, and II, I2, in the lower path. Re sistances of value R are connected between ter minals I, 2 and 3, 4, and also between two ad ditional pairs of'terminals 5, 6 and ‘l, 8, located between networks 9 and I0 and networks II and I2, respectively. The networks 9, II], iI and I2 are of the sym metrical’ lattice type and are similar in pairs, networks 9 and I I being alike and also networks III and I2. The line and the lattice branch im pedances of networks 9 and I I have values A and B, respectively, and the corresponding branch impedances of networks It and 12 have values C and D, respectively. These impedances are pref erably pure reactances and may be chosen to give each network a desired type of transmission characteristic. For example 9 and II may have low-pass characteristics while II] and I2 may be of the'high-pass type. _ ' The order of the component networks is re versed in the one path with respect to the other, but otherwise the two paths have the same com position and therefore have equal transfer con stants. There is also a reversal in the intercon-' nection of the two paths at terminals 3 and 4 with respect to the connections at I and 2, the‘ effect of which is equivalent to the addition of a phase 10 shift of 180 degrees in one or other of the two I paths. I This phase reversal, together with the similar 15, ity between the two paths gives rise to a condi tion of conjugacy between the resistance paths at terminals I, 2 and 3, ll, respectively. This may be demonstrated as follows: Consider the resist ance R between terminals 3 and 4 to be replaced 2o, by a short circuit and a Voltage Eu applied to, terminals l and 2. Under this condition the totalv current through the short circuit will be the re sultant of the separate transmissions through the individual paths, the output from the one path being unable to pass beyond the short circuit into the other path. From the principle of reciprocity it follows that the current in the short circuit from either one of the paths alone will not be changed if the path is turned end for end. Since such a reversal in one path would make the two paths exactly alike, it follows that the output, currents in the short circuit will be of equal mag nitude and will completely neutralize each other because of the relative reversal of phase due to the circuit connections. Since no current flows in the short circuit an impedance of any magni tude may be inserted therein without disturbing 35, the conjugacy. Similar considerations will show that the resistance paths between terminals 5 and '40’ 6 and between terminals 1 and 8 are also cone jugate. In this case the two paths include net 'works'l?, II and 9, I2 respectively, with inter mediate bridging resistances, and are similar in composition to the two paths considered in the previous case. A phase reversal is also present 45. in one path with respect to the other due to the interconnections at the intermediate terminals I, 2 and 3, (I. When the four pairs of terminals are bridged by equal resistances of value R it is possible, by 50 proper choice of the impedances of the component networks, to make the input impedances of the whole network at each of the four pairs of ter minals equal to the resistance R at all fre quencies. This is the same thing as giving the 55. system a constant resistance image impedance of value R at all four of its pairs of terminals, the image impedances of a transmission network bee ing de?ned as the impedances measured at each 60 2,115,138 of’ the several pairs of terminals when the other ' A, B, C and D‘, which make up'the networks ‘pairs_ are closed through impedances which pro 1 Sto l2, Equation 6 becomes duce zerore?ection e?ects at all fre'quenciesi that iswhen" they are closedthrough their. respective image impedances. 7 r ..Infsymmetrical systems the v ‘>1_.1V'1 "-e~A"B+eTJ+E(K+B>(CtD> iii‘: 11' 7 <7), V ' V imageimpedances are equal‘a'nd are equal to the * Equations 6 and 7 set. forth. relationships or the . network impedances, and of the branch imped tionships between the Vcomponent'network' im-i' "ances when'lattice type'networks are used, which ' re pedances to achieve this constant resistancechar I’ must be ful?lled in order that the' constant sistancel requirement" may be met. A procedure ' ‘10 _ acte'ristic may be found as"fol1ows: ,w characteristic impedance. Thedneces’sary' 111619’: which may be followed in‘ choosing the imped “Consider ‘the'input impedance at terminals 1 V and 2. ances A, B, ‘C'and' D in accordance with these Since the resistance path'b‘etween 3 and requirements will be described later. If will be sufficient at‘this point to note that the require 4 is conjugate to the path betweenv l, andwg; the input impedance will not be. affected» by the value '15 ‘of the resistance in the branch 3, 4.3 "The termie J inents can always be met with physical imped 15 1 nals 3. and‘ 4v may therefore’ be assumed to' be ance elements and that these inayrbe selected fshort-circuitediin'whichrcase.the input imped ‘andproportioned 'to provide desired transmission '_ _ ahce at! and '27 is simply the" impedance of the characteristics between adjacent pairs of ter-v I a - 1 two transmission paths in‘ parallel“ ‘Let Z1 de . mingle 7' . v -. a 1 , In’ the network of Figure 1 the two transmission 20 note the‘ impedance of the" path including net 20 works 9, and II] was the joutput‘pterminals of I70. . paths are 'ofjtherbalanced type and the network snort-dreamed and let Z2 denote, the impedance ' is therefore suitedIfor direct connection to trans ' of. their-other transmission path’ra'lsoi shortecir-f Vmiss'on lines having "both sides ‘balanced I130’ Qcuite'd at the output ,re'nd,rthen; if Z denote‘the; ground. "A. modi?ed form’ of ‘the invention in _ . total input impedance: at land 2, '25 . “ " -. v"2711 1'1’1 _ . - ' ' which the‘ two transmission» ramsw are‘ of “the 257 unbalanced type 'is shown Figure; 2.] In‘ this‘? , network the component'net'works areid‘esig'natedr . ti a 19> r 9',- 59’, Ill’, and IZf and'consist offseries-shunt .'The values of Zr and Z2 may be determined from impedance , combinations with the series Vim-V ‘ '30. theope'n ‘circuit and short circuit impedances :ofv pedances inserted in only’ one line of each path. _; " ’ the component networks. "Since each of these‘ Terminals 2, 4, -6 and Bare connected’ together 30 ‘ networks isfsymmetrical' its open and short cir- ' and'to ground.‘ 7' o_ permit the grounding ‘of one cuit'impedances will be the same ;at; both ends. " . side of eachfpath a transformerTr‘isincluded ~ . Networks Sand H',‘ being similar, will have the at one end'of‘the upper. path was second trans: former T2 at the other’. end of the lower path.’ same impedances'and' networks ID and ~I_2 like j 35 wise.” Let 'the'open'circuit andshort circuit 'imf These transformers should besixnilar in‘, their _ - >7 "sedans/seer :9 and _l I" be denoted Z0 and’ Zs re . , sp'ectivel'y' and let; the corresponding impedances . V of networks’ “land 12 be denotedtbyYo and ‘Y5. be a?ected by their. inclusion and the conjugacy "'The'im'pedance Z1 is that ofthe network 9 ter 40_~lmin‘atedby.an impedance made up 'of the par of‘the opposite pairs of terminalswill not be’ -. impaired. The component networks in Figure 2 are shown as symmetrical T‘ networks, .8’ and ' allel ‘combination of resistance‘ R and-the short circuit '_ impedance of network I0 and has the value '45? ‘ II’VhaVing series impedances F and shunt im-V " ' . . Z=Zo Zr1+ZaV ’ , l . V. . .- ing'fimpedancesu H and .K. ‘t Single sec'tionnet-I W2). , ' pedances G and I6’ and 12' having correspond . Z.1+Z... 'Where» v50 characteristics and'shouldrhave unity transfor mation ratios.’ If theyare similarly poled the relative phase shifts of the two paths will not works only are shown, but it will be understood that as mans‘r sections as desired may be used; ' R’Ys Zrrrma the showing being intendedto-representsimply ' the unbalanced equivalentsuofyzthe generalized Figure 1. 'It should'be noted, however, that the component networks are of the series Substituting 7 the; value . of Z71, in Equation '2' and ’ '_. inverting, gives ' V terminated type. 'Thisis necessary where the of): ‘ resistances'B, are connected in shunt to the com ' i=2; R(Y.+Z.)+Y.Z.' ponent networkterminals; as in Figuresxl and 2. ‘In a like manner the value ofZzjis found to be " 1 1 50.. . lattices 0 RZS+YOR+ZSV < ) ,Otherwisaia shunt path would be. provided at each" pair of terminals-which would have ‘zero \ V r r (a) impedance at some frequency and wouldthér'e- I - If the inputimpedance at'terminals l, 2 is to be - fore prevent the realizationofithe constant re sistance a constant resistance of valuetRr, then Z1 and'Zz must be so related that V 7 characteristic. ' > ' .60 . Another‘ form of the inventionis shown in Figure 3 which differs from that shown in 'Figé, ~ ures 1 and 2 in that the resistances R’ are con-' 'nected in series between the component networks .7 > _ I instead of in shunt. titre-(malt of these lattices may also be used as inYFigure’Z, ‘ ' ' In the latter case the lower line of the upper path and the upper line of the lower path would a. a "707. from which, by simpli?cation, the relationship is Hobtained. l‘ @75 ' r 1‘_"_' 1 ’ .2 The component networks. 1.. 9,10, I! and I2, are shown as symmetrical lattices ' gas in Figure -1, butthe'unbalanced equivalents '_ t j V " 1 ' _ form the common grounded, conductor.‘ Trans formers Ti and T2 inserted as in. Figure 2' permit ’ . FFZO'ZfYJf'I/UYI ‘(6) terms of the line and lattice impedancesp 'the'grounding to be effected and, with appro priate poling, maintain the conjugacy of the opposite pairs of terminals. When the unbal " 2,116,138 anced networks are used they should be of the shunt terminated type for reasons converse to those dictating the use of series terminations in the network of Figure 2. From considerations similar to those applied to the analysis of the circuit of Figure 1 it may be shown that the relationship required in this case for securing a constant resistance imped ance characteristic at each of the four pairs of 10 terminals is expressed by ‘ R2:ZoZs+2Y0Zc+Yo-Ya D . (8) . R2=AB+%;(A + B) (C+ D) + CD tem should have de?nite selective properties the component networks will preferably be composed of substantially pure reactances and will have in dividual selective characteristics corresponding to the prescribed requirements of the system. With this in mind the illustrative example will corre spond to the arrangement shown in Figure 4 in which networks 9 and II have low-pass transmis sion characteristics and networks 49 and I2 have high-pass characteristics. or, in terms of the lattice impedances A, B, C and 15 3 (9) 10 Assuming the lattice branches to be substan tially pure reactances and writing in place of the impedances A, B, C and D the corresponding re actances X5, Xb, Xc, and Xd, Equation 7 can be transformed to 15 An example of the use of the invention is il lustrated in Figure 4 which shows its application 20 to a carrier telephone or telegraph repeater of the single ampli?er type shown in U. S. Pat. 1,874,492 issued Aug. 30, 1932, to A. G. Ganz. The network illustrated corresponds to that of Figure l in which the resistances are bridged 1 1 1 1 2 (f?zafjz) +,—,,4 '(10) or across the terminals of the component networks. The two portions I4 and 15 of a transmission line in which the repeater is inserted are con where nected to terminals l, 2 and 3, 1%, respectively and furnish the requisite bridging resistances. An ampli?er i3 is connected between terminals 5, 6 and and ‘l, 8. The bridging resistances being provid ed by, suitable elements associated with the in put and output circuits of the ampli?er. In a multiplex carrier transmission system it 35 is customary to include all channels transmitting in one direction in a low frequency group and those transmitting in the opposite direction in a separate high frequency group. By making net > 1 1 1 1 1 1 1 1 2 4 F12+F22=(z+§b+§c+fd> “Fl-Q5 (11) I 25 “(232) 30 It is to be noted that the right hand side of Equa tion 11 is always positive and ranges in value be. tween 4/R2 and'in?nity, in?nite values occurring at each of the resonances of the reactances Xa, Xb etc. To- meet the constant resistance require 35 ments the reactances must be so proportioned that the sum F12+F22 varies in the same manner works 9 and i i of the character of low-pass ?l 40 ters a path through the rep-eater from line I4 as the right hand side of the equation. to line [5 is provided for the low frequency chan nels, this path including ?lter 9, ampli?er l3 and ?lter II. By making networks in and I2 of the character of high pass ?lters a corresponding 45 path through the repeater from line l5 to line I 4 is provided for the .high frequency channels. that the insertion loss between terminals I, 2 and 5, 6 is given by _ .It may be shown by ordinary network analysis F e2°‘1=1+(F—:) 2 (12) and that between terminals I, 2 and 1, 8 by These paths may be made mutually exclusive by F ' 62%: 1 the proper choice of‘the ?lter cut-off frequencies 45 2 ‘(13) and by providing adequate attenuation in the ?l ters. . The conjugacy existing between the opposite pairs of terminals because of the reversal of the interconnections at terminals 3 and 4 with re spect to those at terminals l and 2 not only per 55 mits the constant resistance characteristic to be obtained but also minimizes the possibility of singing in the ampli?er by the reduction or elim ination of feed-back from the ampli?er output terminals '3, 8 to the input terminals 5, 6. The general requirements for the constant re sistance condition of the networks have been set forth in Equations 6, '7, 8 and 9. It remains to be shown how the individual impedances of the networks may be determined in accordance with 65 these requirements. This will be done by devel oping design formulae for a network of the type of Figure 1, the general procedure outline being applicable to the other forms of network. Equation 7 which expresses the requirement for the constant resistance condition in terms of the branch impedances is general to the extent that it does not depend on the character ‘of the im pedances, which may be resistive or reactive or may include both resistance and reactance. How 75 ever, since it is desirable in practice'that the sys-. where a1 and a2 are the respective insertion losses. Since it is desired that the path through net 50 work 9 have a low-pass characteristic and that through network I2 a high-pass characteristic, it is apparent from Equations 12 and 13 that the ratio FZ/Fl must be small in the low-pass range and large in the high-pass range. That is, F2 55 must be small in the range where F1 is large and vice versa. . , The character of the frequency variations of F12 and F22 may readily be determined in any par— ticular case. A suitable form for the low-pass networks 9 and l I is illustrated in Figure 5 in which X3, is characterized by a single ?nite res onance and Xb by two ?nite resonances. From considerations of ordinary ?lter theory the res onance of X1; will occur at a frequency lying be 65 tween the two resonance frequencies of Xe. In terms of the resonance frequencies F12 may be expressed by A20) A460 A660 70 where w denotes 2 11' times frequency, (012, (022, and 0132 correspond to the three resonance frequencies, and A0, A2, A4 and A6 are constants. The res 75 74 r 2,115,138. onance frequencies all occur in the low-pass ~G. 'Bell and Sons, London, England, transmission range and at these frequencies F12 ‘ tion, 1895, pages 265 and 267. become in?nite. . v second edi- ' 7 . Equation 14 may rbertransformed by‘ordinary' (17) ,. algebraic‘ processes to’ the. form V where 7. ~ Esn(uK, k) . 10 r wherein the following notations ,are?used: and’ win; am, am, correspond to frequencies at V Expressions of the form sn (y,.a)'denoterelliptic;~ which F12 has zero values. ' sines of modulus'a, andargument y, the modulus 15' 7 ' 'pass 'VThese range; frequencies, but in the if real, ‘general all liecase above certain the low-'7 of i 7 being a- positive numeric of value less than unity; ' ' them'may be imaginary‘or complex. ' The general expression for F1 in the factorial 2d form of Equation 15,;hay'be written as i _' r ' . ' K is the complete elliptic integral or the ?rst kind. of modulus k; j Kris the. complete elliptic integral 'of the ?rst; ' kind,'ofmodu1usk;;_ '1 -'> J - j I (195 Q wherell'lr indicatesa product of terms of the type ‘following it andm; ispthe number of factors in . F12, ‘that is, the number of poles or: the number 7 - of zeros. ' . -‘ Bypa proper choicerof critical frequencies dee ' nf-—-'21'r_i’+1 C2, C3 - and’ C4 are‘ numerical ‘constants and: ?ning the zeros and polesiin~ Equations 15 and .16 f the'rfrequency Va51‘1atiOn__OfIF1g may. be made ‘to sn(uK,k). is the variable parameter. ~takejthe form illustrated ebyvrther, curve in?Fig. V6‘ r The quantities K, Khk, kl, are reeteem whichvis characterized by alseriesof equal mine Moreover the. values of F12 in the high frequency 7' ‘range may be made so small as tov be completely negligible. 7 ' , and 40 _ . ' I a frequencies and a .. frequency range are equal to 4/R2 and so that the descending portion ‘of the F12 characteristic in the range betweenfthe high and the low values passes ' through the value 2/R2at a predetermined fre 145 quency marking the‘pdivision of the ranges. e ' " , 4of 7 q'=e_’rF pass network so thatthe minima of F? in the low - : ' ’ V- 7 e '1' ‘#1? I . _ . _ YKI' e iminedeforv'Fzz'lhaving minima in the highfre I 1-6:“? quency range likewise equalto 4/32 andhaving its f .I' "it ' _' K1’; H11? then :KLand K1’ ‘are such that 'A complementary characteristic‘ is next deter ' ,1/1'~1.<12 respectivelyyand let, the element values of the "low-l V ‘we V - The design proceeds‘ by determining thecritircal ' r Let ‘K’ and :K1 denote the complete‘. elliptic. 1h; tegrals'ofthe?rst kind, of'modulin .tween the vZeros in the'?hig'h‘ frequencyvirange. 7 the following manner. ima'betwe'en the poles in the low frequency range and a corresponding'series' of equal maxima bee . j J’: Q1) >750 descending portion "intersect that of F12 at‘ the and qr=qn 1 ‘(2% predetermined dividing frequency, itsslo-pe at Knowing .q and n the value of'q1 is determined V ' this point being equal that of F12 ‘but of opposite and the corresponding value of 701 can be found. . sign. This ensures a minimum’ at'ethe‘ dividing" from standard tables of elliptic functions. 7 For’ the value of which ‘will be Iii/“R2. 7'55, frequency Since the "values of F12; in the high. frequency the cases. of interest in connection'with thepresé 7 ent invention k1 is small ‘and’ maybe determined 7 7 range are negligible’ and those of . FgZJin the low with great accuracy from’ the, ?pplfoximate ‘rela-l Ye l I frequency range‘eare likewise negligiblethe sum 1 tionship of .the two‘quantities gives. a'series' orfjminima' . ‘ 7. ‘ K23) eacheof value 4/13’? and alternating in the free ‘so . quency'scale with'a series'vof in?nite. values as ‘ The value of K1 follows ‘from its de?nition. required by Equation ‘11; It' develops ‘also’ that V From Equation 18 it ‘will be seen that the queue the form ofVF13+F22 agrees with that of the right ‘ ' tity 7 hand side of Equationell and hence that the cori .' 65 stant resistance requirement is’ met. I have found that'thelneglect of. the low vvalues of ‘F1772. ‘in the 7 high frequency range'and of theecorresponding expressed as afunctioh ofjsnill; K, 70) has poles, orin?nite valuescorresponding to thelz'eror ' :values of F22 affects the resistance of ‘the system to, .values .of the .denominatorefactors; and has an . _ . an extent ofjless than one part in 5000 in‘lpractical i . equal number of related zeros which occur in a‘ different. range" of values 'of the argument sn ' 70 ' : ~' _' n 7'." ‘. a cases. - YFI‘he choiceof the critical frequencies of F1? to " l (u,.K, k). In accordance'with'the known‘prin'cie" ples 'of elliptic functions the'square'of the above . _ 7 the following expansion theorems for elliptic ‘indicated quantity vgoes through a series of equal I give therequired frequency variation is based on p functions which follow from relationshipsv given minima between the poles and-aseries of equal‘, in Elliptic Functions, by maximabetween the zeros. ’ Cayley, published by. ~' @7775 , 2,115,138 The values of the minima are given by Q1} ’ kl 5 Substituting these values in the right hand side of Equation 18 and identifying this with F1 gives V (24) and the maxima by C12k1 (25) the ratio of the maxima to the minima being equal to 7612. which agrees with Equation 15 when . = Q1 _1_ The constants C3 and C4 are found by com 10 paring the expressions in which they appear as Sn (H K, k) approaches in?nity. This gives rise to the relationship 2 ‘00171217221332 Substitution of the same values in the right hand 10 side of Equation 1'7 gives 15 Since the variation of the function 15 Cn/EsnQ‘luKb k1) corresponds to the desired variation of the quan~ 20 tity F1, the design of the low-pass networks is readily accomplished by identifying F1 with the summation expression of Equation 17 or with the product expression of Equation 18. For this purpose let a value of w, denoted by 25 wo, be chosen in the neighborhood of the desired cut-oif of the low-pass network. For the present this may be chosen arbitrarily. The critical values of w, namely wsl and (052, in Equation 16 are now chosen such that This equation corresponds to Equation 14 and en ables the constants A0, A2, etc. in that equation to be determined in terms of the chosen elliptic function parameters so that the desired varia 25 tion of F12 will be secured. Since each term in ~ Equation 14 represents the susceptance of ‘a branch path containing only a simple inductance or the combination of an inductance and capacity 30 in series the determinations of the A’s leads di rectly to the element values. where 35 20 and the variable frequency ratio wo/w is identi?ed with The required value of C3 follows from Equations 25 and 26. Equation 25 gives the value of the 'minima of F12 between the poles, which, in ac cordance with the design requirements must be 35 equal to 4-:-R2- Accordingly _21/1?1 C4—T The value of the modulus 7c is chosen with ref 40 erence to the degree of discrimination required of the network. As the value of the modulus ap proaches unity the poles and zeros of F12 move close to we and the characteristic is marked by low minima and high maxima. Reducing the 45 value of the modulus separates the critical fre quencies and at the same time increases the ratio of the minima to the maxima. In the case of a network of small degree of complexity, for ex ample one having two poles and two zeros, a 50 modulus value of 0.9 will ensure a satisfactorily sharp discrimination by the network and will re sult in a ratio of the minima to the maxima greater than 5000. For more complex networks higher values of the modulus may be taken and 55 because of the larger number of poles and zeros both the sharpness of discrimination and the ratio of the minima to the maxima are greatly increased. In the case of the network illustrated in Fig. 5, 60 for which the function F12 has three poles and three zeros, Equations 26 and 27 give the follow ing values for the critical frequencies (30) 40 and hence, from Equation 26 _ 21/EK C3_R 111(1 (31) For the cases of interest, that is where the net 45 works have relatively sharp discrimination and high attenuation, the value of K1 will not differ sensibly from 11/2 and the approximation for C3 C __ 41/21: (32) ~ 3_'R 1'11: 50 maybe used. The high pass network, for which the function F22 has to be complementary to F12, has to meet the requirement that its function F22 must have 55 negligibly small values in the low pass range, have minima equal to 4+R2 in the high pass range, and must have the rising part of its character istic intersect the descending part of F12 at the point 2+R2 with a slope equal to that of F12 and 60 opposite in sign. - These requirements are most readily met by making the schematic form of the network such that F2 has the same number of poles and zeros as F1 and locating the poles and zeros symmetri 65 65 4031: (no-Z- ICSH(%K, k) : Log/p3 70 cally with the corresponding poles and zeros of F1 with respect to the cross over frequency, which will be denoted by wc+21r. The poles of F1 and F2 will thus occur in pairs having we as their geo metric mean and the zeros will likewise occur in 70 similarly related pairs. and 75 30 6022 = woe/(v11 “a1 = wag/mu . ' The cross over point we will not be the same as we which appears in the formulae for F1 but will be slightly lower. The mathematical determina: tion of this point is rather lengthy and only the 75 "2,115,138 6 paths connected- in parallel between a ?rst pair’ T explicit formula‘ for the ratio of t6 t9’ at willibe ' of_ terminals and a second pair‘ of terminals, a pair of symmetrical frequency selective networks having complementary transmisison bands. 'con nected in tandem in one of said paths, apair of respectively ‘similar _networksconnected in tan "dem in the other of said paths, ‘butin reverse order with respect to thenetworks in _thejlsaid In design practicewthe value‘wc will generally be ‘ ?rst paths, 'said‘networks having open circuit and assigned. Equation 33 _then permits the value short circuit 'impedances related in accordance , . (33) 310 5am for the function F1 tofbe determined and also, from the required symmetry of the two systems, ' with the equation a corresponding value which will be denoted 101i’v 7 P15 . Qq’ for the function F2. {The values of m0 and too’ are related to we by the equation 2is (34) Where Z0 and Z5 are respectively the open 'cir- , cuit andthe short circuit impedances of the one Using the ‘same principles 7 as discussed above' pair of similar networks, Y0 and Y5 are the cor formulae for F;v corresponding to Equations 28 ,respondingvalues for the other pair of similar and 29 can'be. developed,~but it'is simpler'to de . networks, and ‘R is an arbitrarily assigned re- ' termine F2 directly from Equation 29 taking ad» 6 , vantage of the symmetry of therFi and-F's char; . s-istance. 3. A wave; transmission network ~comprising 7 wacteristics, For each rvaluerof ‘m above we the func- ‘ two paths of terminalsand connected ainsecond parallelbetween pair of terminals, a firsta a _A 'tion Fzzlwill have the same value as Fizat the pair ' 7 therefore be derived from Equation 29 by the sim pleexpedient of replacing w therein by@—"oc2+w. The. negativesign appears in thistransformation .to take accountof'the difference in-rsign of the 230 ‘pair fhaving of symmetrical complementary frequency transmission selective bands networks con nected in tandem in one of ‘said paths, a pair of reactancesof coilsand of condensers whichjrepre respectively similar networks connected in tandem ' in the other of saidpaths, but in reverse order_ .30 . 1 with respect’tothe networks in the’ said ?rst 5* sent complementaryimpedances. paths, said networks having open circuit and ' ' l ' .. Thezte'rms ofithe expression lthus found in. F2 7 will‘i'ncliide two kinds; one set will directly rep ‘__resent physically realizable V susceptances and the remainder ‘will represent physically» realizable ' susceptances when the signjis changed. ,Those representing directly realizable susceptancesrare idBHti?Ed‘With'Xc andthe others with Xe.‘ The short circuit impedances _relatedin accordance with the equation ' ' fR2=ZoZs+V2YoYs+YoYs ‘ where Z; andZg are respectively the'open'circuit and the short circuit impedancesof the one pair ‘ schematic ‘form of the high-pass network thus fof similar networks; .Yt andYs are the 'corre obtained isillustrated in _Fig. '7. The reactance Xe .sponding‘ values for the other pair of similar net- 3"#40 comprises a capacity and a resonant circuit con works, and R is an arbitrarily assigned resistance. 7' ‘ nected in parallel and the 'rea'ctance Xd' com 42A, transmission network comprising two " I prises two simple resonant circuitsconnectedin ‘paths extending in parallel between'a _pair of in put. terminals and ‘a pair ‘of output terminals, Fig; 8 showsthe curves", of the two functions F12 ‘two frequency selective networks‘ having substan and F22 and their sum plotted against the logae I ‘ 1 parallel. rithmgof, frequency. .Curve h'llrepresents F12, tially complementary.’ transmission bands . diS.-‘ 7' posedin tandemvin one of said paths, and two ' curve H-representsFs2 and the looped portions : 'respectivelysimilar networks disposed in tandem oi thetwocurves together with dotted curve I2 _in reverse, order-in the other oi said paths, the represents their sum; The crossover frequency , is designated by f0, the poles of F1? by'f12,j22, . networks off one of the similar pairs being defined = and fsz, and the poles of F22 by the inversely re lated frequencies 'f'iz, ffzzgand $32.. .The sym Imetry otthetwo characteristics about fc is clear- ' by two frequency characteristics 11(40) and not) which-jointlyfdetermine'their transmission prop; . cities, and the networks of p the, other similar pair; J beinglliklewise de?ned by frequency characteris ' ‘ Whilejthe foregoing procedures give the de-' tics J°s(w) and ,f4(w),and the saidnetworks be; ; 7755' 1y shown by: this ?gure. sired networks as symmetrical lattices, these may be'reduced by known procedures to unbalanced types of: networks such, as bridged-r-T ;or ladder networks suitable for use in the unbalanced type .of networkushown' in Fig. 2.’ ‘ _What is claimed is: ' > e ‘ 1.1 i A; wave transmission networklcomprising two. pathslconnected in parallel betweena, ?rst pair :6'5 olfitermin'als and a second pairof terminals, a pair of symmetrical frequency selectivenetworks ‘ ing so proportioned that theiquantity V‘ r‘ . L [mo-Mora{sci-flan? 1 has all of, its minimum‘ values’. equal. .7 * 5. A transmission. network comprising two paths connected in parallel between a pair of input terminals and a pair of output terminals; two frequency selective networks h'a'vingsubstan tially, complementary transmission bands dis- F657 " posed in tandem in'on'e oi-"sai-d'pathsrand two ' "having; complementary transmissionlbands con: , respectively similar networksidisposed in tandem ’ nected; in tandem- in one of said paths, apair of in reverse order in the. other of said paths, said‘, , respectively_:similarf networks; connected in ta_n—, ,networks' composed‘of substantially pure:_,__; " ; ‘dam v11'1"the,(éthet..of said; pathsgbut jinrrreverse 'reactance' being ‘elements, the networks of one of the ‘7'0 ' orderwithlrespect to the;_networks in saidi?rst _ pairs being 'eacrr characterized ‘by ‘two path; andmeans for reversing the phase of the . similar reactances. Xa and Xb'eachhaving a plurality of '‘ " currents in one of, said paths with'res'pect to-the critical frequencies, and the networks ofj‘the 'currents: the other whereby said pairs of , ter'-. other, similar pair beinglikewise characterized by minals are made to be conjugate ‘to each other“ 775 ' 2. A wave transmissionnetwork comprising two . reactances Xe and Xe, the magnitude of said; " ' 2,115,138 " reactances and the values of their critical fre quencies being so proportioned, that the quantity quency range, and the quantity, Xa 1 _i)” Xei_i>2 (Xc—Xd)2 if; Xb has all of its minimum values equal. 6. A transmission network comprising two paths connected in parallel between a pair of input terminals and a pair of output terminals, 10 7 ‘uniform minimum values in a prescribed fre- ‘ two frequency selective networks having sub stantially complementary transmission bands dis posed in tandem in one of said paths, and two respectively similar networks disposed in tandem has a like plurality of in?nite values at frequen cies in an adjacent range and has minimum val ues equal to those of the ?rst mentioned quantity. 9. A transmission network comprising twov paths connected in parallel between a pair of input terminals and a pair of output terminals, '10 two frequency selective networks having substan tially complementary transmission bands dis in reverse order in the other of said paths, said posed inrtandem in one of said paths, and two 15 networks being composed of substantially pure respectively similar networks disposed in tandem reactance elements, the' networks of one of the in reverse order in the other of said paths, said similar pairs being each characterized by two networks being composed, of substantially pure reactances Xa and Xb each having a plurality of reactances X0 and X1, the magnitude of said re reactance'elements, the networksof one of the similar pairs being each characterized by two re actances Xe and X1) each having a plurality of critical frequencies, and the networks of the V20 actances and the values of their critical frequen cies being so proportioned that the quantity by reactances X0 and X , the said reactances X9. critical frequencies, and the networks of the other similar pair being likewise characterized by other similar pair being likewise characterized and Xb having poles alternately at frequencies in 25 30 has all of its minimum values equal. 7. A transmission network comprising two paths connected in parallel betweenv a pair of input terminals and a pair of output terminals, two frequency selective networks having substan tially complementary transmission bands dis posed in tandem in one of said paths, andtwo're spectively similar networks disposed in tandem in reverse order in the other of said paths, said net 35 works being composed of‘ substantially pure re actance elements, the networks of one of the ' 35 ances Xa and Xb. reactances Xa, and Xb each having a plurality of critical frequencies, and the networks of the other similar pair being likewise characterized by reactances X0 and Xd, the magnitude of said reactances and the values of their critical frequencies being so proportioned that the 10. A transmission network comprising two ' paths connected in parallel between a pair‘of ' input terminals and a pair of output terminals,‘ two frequency selective networks having substan tially complementary transmission bands dis 40 posed in tandem in one of said paths, andv two respectively similar networks disposed in tandem ‘ in reverse order in the other of said paths, said 1?; Xb has a plurality of in?nite values alternating with uniform minimum values in a prescribed fre quency range, and the quantity 50 wherein f0 is a frequency marking one end of the prescribed range, m is the total number of poles, and s is an integer taking the successive values 30 1 to m, and the said reactances Xcand. Xa have a like number of poles alternately at frequencies in an adjacent range having values inversely re lated by a common constant to those of react similar pairs being each characterized by, two quantity 45 a prescribed range having the values i_i>” X: Xd has a like plurality of in?nite values at frequen cies in an adjacent range and has minimum 55 values equal to those of the ?rst mentioned quantity. networks being composed of substantially pure reactance elements, the networks of one of the 45 similar'pairs being each characterized by two re actances Xe and Xt, each having a plurality of critical frequencies, and the networks of the other similar pair being likewise characterized by reactances X0 and Xd, the susceptances 50 .1- and put» terminals and a pair of output terminals, two vfrequency selective networks having substantially complementary transmission bands dis 55. i 8. A transmission network comprising two paths connected in parallel between a pair of in , X,’ Xb having poles alternately at frequencies in a pre-, scribed range having the values 60 posed in tandem in one of said paths, and two respectively similar networks disposed in tandem’ in reverse order in the other of said paths, said 65 networks being composed of substantially pure reactance elements, the networks of one of the similar pairs being each characterized by two reactances Xa and Xb each having a plurality of critical frequencies, and the networks of the wherein f0 is a frequency marking one end of the prescribed range, m is the total number of poles, and s is an integer taking the successive values 1 to m, and the susceptances 1 70 other similar pair being likewise characterized by reactances Xc and Xd, the magnitude of said, reactances and the values of their critical fre quencies being so proportioned that the quantity , (Xa-—Xb)2 75 has a plurality of in?nite values alternating with 65 v X0 and L '70 X4 have a like number of poles alternately atfreé " quencies in an adjacent range having values in-v ‘ 2,115,138 ' 8 nals of each ofjsaid networksbeing connected-to 7 i versely related by a common constant .to those’ a common‘ pair ofrinputiterminals, equal resist: 1 ‘ of terminals Vances connected to the other‘pairs and additional ter of said network's respectively, Xa 1 ‘ minating impedances connected ineparallel with >_ UL susceptances ‘ and = ' ' ; V said resistancesthe' additional impedance con _1_ -V 1 nected' to each network being equal X1’ , to the Jshort- 7 circuitimpedance of the other network. 7 1 f11.'A transmission network comprising ‘two pathsrextending in parallel between a pair of in put terminalsand a pair of output terminals, : two frequency s'elective'netwo'r‘ks having substan tially complementary transmission bands disposed E10 7 13. A constant resistance system in accordance’ with claim 12 in which the said frequencyseléc 10 tive' networks are proportioned in accordance" with the relation 7» ' in'tandem mom of said paths, and two respec “ :15; tively similar networks disposed in tandem in re verse order in the other of said paths, the net V l @“ZZ‘LYiZ?Ym 1 1" , z where R' is the'rvaluel of: the resistances connected 7 by two frequency"characteristics J‘1(w)f and f2(w) to theinetwork terminals, iZa and ‘Z5 are res’peb tively the’ open circuit andshort-"circuit'imped- ' liewithin a prescribed frequency'rangeand occur nect each par of terminalswith eachother pair; ' frequency selective networks having substantially ' “works of one of thejsimilar pairsbeing de?ned of one networkand‘Yo and 'Ys'are respec which jointly determine their transmission prop-,7 ances tively the'open circuit and shortécir'c'uitimped- '1 networks of the other similar pair. ertiés, and the ances of the other’ network‘. " V 20, ~ being’ likewise de?ned by frequency characteris '14. A wave transmission network having six tics f3(w)' and5,f4(w)', andithe said networks being terminals arranged in ,three pairs, comprising 7 so proportioned that the poles of the quantity three transmission‘ paths» arranged to intercon- V ‘; . :25 7 complementary transmissionbands ‘included rel-,5 at frequencies having substantially the values ‘ l 2s ' ' raj/E3142”l +13, k) V V_ spectively in vtwo of said paths and a networkv 7' ‘ comprising two tandem< ' connected portions ‘cor responding to‘sa'id complementary networks -in-,j wherein fo is a frequency marking one end ofthe eluded infth'e third of said paths,'said complei prescribed range, mf'is the total number of poles, ‘mentary vnetworks being‘ofzsymmetrical stru’c; and s is an" integer’ takinglthe' successive, > values V ‘I V 'ture and "being proportioned ‘so that l tom, and the quantity 7 7 , V r ' V ' iacéolqtwnf V 7' 375'; ' V a . has a like nnmberlof poles in an f?’ ' adjacent range it ~ _ V V arbitrarily assigned resistance; ' wherein‘ R is an Z0 and 25' are respectively the open circuit and‘ l '7 f common constant toethose of the ?rst mentioned ' short-circuit ‘impyedances of one complementary’ network and Y0 and Y5 are "the corresponding’ occurring'at frequencies ‘inversely related by a ' ~ ' quantity. ' ' ' " ' ‘ I '12.‘ A‘constan resistance network combination 549 comprising a pairof symmetrical frequency se-‘ ' elective networks having substantially comple “lm'entary transmission bands, one V pair‘ or termi impedances of the other complementary network. SIDNEY 'DAaLINGroN'. e

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