US1828454A - Transmission network - Google Patents

Transmission network Download PDF

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Publication number
US1828454A
US1828454A US465522A US46552230A US1828454A US 1828454 A US1828454 A US 1828454A US 465522 A US465522 A US 465522A US 46552230 A US46552230 A US 46552230A US 1828454 A US1828454 A US 1828454A
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Prior art keywords
frequencies
band
impedance
impedances
frequency
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Expired - Lifetime
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US465522A
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English (en)
Inventor
Hendrik W Bode
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AT&T Corp
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Bell Telephone Laboratories Inc
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Priority to US465522A priority Critical patent/US1828454A/en
Priority to DEI41969D priority patent/DE678435C/de
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    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H7/00—Multiple-port networks comprising only passive electrical elements as network components
    • H03H7/01—Frequency selective two-port networks
    • H03H7/17—Structural details of sub-circuits of frequency selective networks
    • H03H7/1741—Comprising typical LC combinations, irrespective of presence and location of additional resistors
    • H03H7/1783—Combined LC in series path
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H7/00—Multiple-port networks comprising only passive electrical elements as network components
    • H03H7/01—Frequency selective two-port networks
    • H03H7/0115—Frequency selective two-port networks comprising only inductors and capacitors
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H7/00—Multiple-port networks comprising only passive electrical elements as network components
    • H03H7/01—Frequency selective two-port networks
    • H03H7/17—Structural details of sub-circuits of frequency selective networks
    • H03H7/1708—Comprising bridging elements, i.e. elements in a series path without own reference to ground and spanning branching nodes of another series path
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H7/00—Multiple-port networks comprising only passive electrical elements as network components
    • H03H7/01—Frequency selective two-port networks
    • H03H7/17—Structural details of sub-circuits of frequency selective networks
    • H03H7/1741—Comprising typical LC combinations, irrespective of presence and location of additional resistors
    • H03H7/1791—Combined LC in shunt or branch path
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
    • Y10S—TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y10S84/00—Music
    • Y10S84/09—Filtering

Definitions

  • This invention relates to wave transmission networks and more particularly to networks having broad-band transmission characteristics such, for example, as broad-band wave filters and delay networks. It has for its principal object the improvement of the transmission characteristics of networks of this type particularly with respect to the uniformity of the delay or phase characteristics in their transmission ranges and to the maintenance of high attenuation outside the transmission range.
  • a further object is to reduce the number of impedance elements required to obtain a desired transmission characteristic.
  • Fig. 1 represents one schematic form of the networks of the invention
  • Figs. 2 and 4 show typical forms of the network branch impedances
  • Fig. 3 shows the variations of the branch impedances and illustrates the principle of band formation
  • Fig. 5 is a phase characteristic used for reference purposes
  • Figs. 6 and 8 illustrate typical impedance arrangements in accordance with the invention for securing linear phase characteristics
  • Figs. 7 and 9 are phase characteristics for the arrangements of Figs. 6 and 8;
  • Fig. 10 shows the impedance arrangements of a low-pass filter having both impedance and propagation constant controls in accordance with the invention
  • Fig. 11 shows the transmission characteristics corresponding to Fig. 10
  • Figs. 12 and 13 illustrate the application of the invention to an all-pass network
  • Figs. 14 and 15 illustrate the application of the invention to a band-pass filter
  • Fig. 16 shows a schematic of a modified form of the invention.
  • Figs. 17 and 18 are theoretical diagrams relating to the structure of Fig. 16.
  • the lattice structure of Fig. 1, comprising the input and output terminals, represents a typical schematic form of the networks of the invention.
  • Impedances Z, and Z represent terminal loads between which the network may be connected.
  • the propagation constant P and the characteristic impedance K of this network are given by may be varied without changing the value of the product Z Z
  • the impedances Z and Z be so proportional that, within the transmission band f denoting frequency and q being a constant.
  • the impedances Z and Z are pure react- .ances of general type and of any degree of complexity. They may have any of a wide variety of schematic forms, for example. a chain of anti-resonant circuits, or a parallel connected system of resonant circuits, or any combination of these arrangements. WVhatever the schematic form may be the value of the reactance may be expressed in terms of a numerical constant and a product of similar factors involving the resonance and anti-resonance frequencies. This general expression for the impedance of a two-terminal reactance is discussed in an article by R. M. Foster. A Reactance Theorem, Bell System Technical Journal, Vol. III, No. 2, April 1924, to which reference is made for further particulars.
  • the impcdances Z and Z are so aranged that their resonance and anti-resonance frequencies provide two sets of design parameters by means of which the frequency variations of the propagation constant and the characteristic impedance are independently controlled.
  • the constant coefficients of the reactant-es provide additional parameters for the control of the magnitudes.
  • Z consists of a chain of six anti-resonant circuits as shown in Fig. 2, the simple inductance being regarded as a circuit anti-resonant at infinite frequency.
  • each anti-resonant circuit is designated by its frequency of anti-resonance f f etc.
  • the impedance of this chain may be expressed as 2 2 2 2 Z -a -a -a f1 f3 f5 f1 f9 in which X is the limiting value of the reactance at frequencies close to zero and the frequencies f f f etc. are the resonance frequencies in their successive order.
  • the variation of the reactance is shown by the solid line curve 1 of Fig.
  • Equation 3 linearity of the phase shift characteristic requires that the ratio shall approximate the tangent of an angle proportional to frequency in the transmission range.
  • the design of the networks of the invention to achieve this relationship is based on certain mathematical theorems relating to the approximation of the tangent function by expressions of the form given by Equation 6.
  • the tangent of an angle may be expressed as an infinite product as follows:
  • Equation 8 may be written in the form tan mm Obviously as many independent frequency parameters as may be desired can be provided by simply increasing the number of antiresonant combinations in each branch and dividing the resonance frequencies as required between the transmission and attenuation ranges.
  • phase displacewhere the symbol II indicates that the expression is a continued product and where n is a positive integer.
  • the infinite product R can be replaced by a ratio of gamma functiono s, using a theorem given in Whittaker and Watson Modern Analysis 2nd edition, page 232. This gives In the range of a: from zero to unity the gamma functions may be replaced by their Stirling series equivalents giving.
  • a first approximation to the value of R is obviously R, J ("11%) a ing of the resonance frequencies in the region close to the cut off frequency.
  • the deslred spacings are found from a consideration of the further approximation for the expression for R taking into account additional terms of the expression given in Equation 13.
  • the resonance and anti-resonance frequencies occur at intervals proportional to the numbers 1, 2, 3, 4, 5, 5.95, 6.6, and 6.95.
  • the solid line curve represents the reactance of Z and the dotted line curve that of Z, as in the previous case.
  • the solid line curve represents the phase displacement and the dotted line curve represents the phase displacement of a corresponding simple ladder type filter.
  • Equation 20 shows that the various logarithmic terms of the expression used to approximate R are such that their series expansions are convergent for at greater than unity as well as for a: less than unity. It follows then, from Equation 22, that this same expression will approximate the value of l/R in the range m greater than unity. If the product approximation for R be denoted by G then the product of R and G will be approximate unity for values of a: greater than unity.
  • the filter impedances are designed in the manner described so that then tanh gwill be unity and P will be substantially infinite when zv 1, that is at frequencies above the cut off.
  • the impedance structure of the invention thus not only provides for linear phase shift in the transmission band but also for substantially infinite attenuation outside the band.
  • characteristic impedance In the examples discussed above the filter branch impedances have no resonance or antiresonance frequencies outside the transmission band and, therefore, have no independeeann ent means for controlling their characteristic impedances.
  • characteristic impedance For the case illustrated in Fig. 6 t e characteristic impedance has the value f being the cut off frequency, and for the case illustrated by Fig. 8
  • the form in the first case corresponds to the mid-shunt impedance of a simple series shunt low-pass filter and the latter to the mid series im edance.
  • the variation of the impedance is ue to the cut off factor in each case. To compensate this variation it is necessary to introduce into the expression for the impedance additional factors which will substantially cancel the efi'ect of this factor. These factors involve resonance and anti-resonance frequencies above the cut oil frequency.
  • urve 4 shows the general This expression is in the same form as the expression for the prop tion constant in terms of frequency and wil be approximatel constant in the transmission band if the t s are given the same values as the frequencies controlling the propagation constant.
  • the more remote of the impedance controlling frequencies are spaced uniformly on an inverse fre uency scale while those closer to the cut 0 are spaced on this scale in accordance with the values given in the table above.
  • the values of the impedance controlling primacies when their number has been specied in any given case, are most readily calculated by determining first the corresponding frequencies in the transmission band on the assumption that they are to be used for control of the propagation constant and then inverting the values with respect to the cut off frequency.
  • Figs. 10 and 11 illustrate the character of the branch impedances and the transmission characteristics of a low-pass filter embodying the features of impedance control and control of the propagation constant.
  • the same number of impedance controlling frequencies is used and the spacings are inversely related. In all there are 9 critical frequencies the values being proportional to the numbers 1, 2, 3, 3.86, 4.22,
  • Fig. 10 shows the variation of Z with frequency and the dotted line curve shows the variation of Z
  • the several critical frequencies are designated f f inclusive, the cut off frequencies being f Above the cut oil frequency the two impedances have substantially the same value, as indicated by the substantial coincidence of the curves.
  • curve 3 shows the variation of the phase displacementwith frequency. this curve being the same as that of Fig. 7.
  • Curve 5 shows the variation of the characteristic impedance in the transmission range and illustrates the high degree of uniformity obtained by the invention.
  • All-pass delay networks Z and Z constitute inverse reactances having a constant roduct.
  • the characteristic impedance is t 1us constant at all frequencies and 1s a pure resistance. Linearity of the phase shift in a given fr uency range may be obtained with a fair iiegree of accuracy by so constructing the branch impedances that they have a large number of evenly spaced resonance and anti-resonance frequencies in this range. However, unless a very large number of critical frequencies is used the even spacing of the frequencies results in an undulatory characteristic such as is shown in Fig. 5.
  • a very high degree of linearity is obtained in accordance with the invention by using a frequency distribution similar to that used in the filter illustrated by Figs. 10 and 11.
  • the arrangement differs from that of the filter in that the cut off frequency is omitted, the upper critical frequencies of Z being stepped up one interval so that Z and Z are completely inverse with respect to each other.
  • Figs. 12 and 13 The character of the branch impedances and the resulting phase characterstic of an all-pass network obtained in this way are illustrated respectively by Figs. 12 and 13.
  • the system of critical frequencies is the same as that of the low-pass filter of Figs. 10 and 11 except that the cut off frequency f is omitted.
  • the solid line curve represents the variation of Z and the dotted line curve that of Z the two being inversely related at all frequencies.
  • the phase characteristic represented by curve 5 of Fig. 13 is linear over a somewhat greater range than that of the corresponding filter and does not exhibit the discontinuity at the cut off frequency.
  • the number of the uniformly spaced frequencies in the lower range maybe increased as desired.
  • Increasing the number of non-uniformly spaced frequencies results in a greater degree of linearity, that is, in greater freedom from undulcations of the characteristic in the ran of the uniformly 8 need f uencies.
  • e numbers of critics. frequencies below and above the hypothetical cut off frequency may be chosen at will, but it is preferable that not more than two in the range below the hypothetical cutoff should be spaced on the non-uniform basis.
  • the all-pass networks of the invention simulate in a limited frequency range the im dance and the ropagation constant of a ength of smoot line having uniformly distributed series inductance and shunt capacity. They may also be made to simulate a dissipative smooth line if each of the inductances is connected in series with a resistance such that the induc.- tance to resistance ratio is the same as that of the line and if each capacity is provided with a shunt conductance simi arly proportional in accordance with the line conductance.
  • Band-pass filters The rules for spacing the critical frequencies in a band-pass filter in accordance with the invention are similar to the rules for the spacing in the case of a low-pass filter and will be stated without proof.
  • the branch impedances of a typical band-pass filter are shown in Fig. 14, there being in this case fourteen critical frequencies lncluding the two out off frequencies.
  • the impedance Z is indicated by the solid line curve and Z by the dotted line curve.
  • the cut off frequencies are f, and f the location of the band being indicated by the shaded portion of the frequency axis.
  • the impedance controlling frequencies are f f f;.,, below the lower cut off and flg, i and i above the upper cut off. Frequencies f,,, f.,, f f f,,, and f within the band control the propagation constant.
  • the impedance controlling frequencies below the lower cut off are spaced on the same basis as the frequencies in the transmission band of a low-pass filter having this off frequency.
  • the impedance controlling frequencies above the upper cut off are spaced on the same basis as the impedance cont-rolling frequencies of a low-pass filter having a cut off at the upper cut off frequency.
  • the frequencies within the band are spaced uniformly except at band limits where they are crowded closer together.
  • the factor 2m by which the constants given in the table are to be divided is such that 2m-1 is the number of uniform frequency intervals occurring in the band.
  • the numbers of the critical requencies in the various categories may be chosen at will.
  • Fig. 15 The forms of the transmission and impedance characteristics of the band-pass filter having the branch impedances of Fig. 14 are illustrated in Fig. 15.
  • Curve 6 represents the variation of the phase component of the propagation constant in the band. Except close to the cut off frequencies this characteristic is linear to a high degree of accuracy.
  • Curve 7 represents the total transmission loss characteristic of the filter when connected between finite constant impedances. The peaks at the impedance controlling frequencies are of the same character as those in curve 4 of Fig. 11 and do not appear directly in the propagation constant.
  • Curve 8 represents the characteristic impedance in the band.
  • the value at either cut off may be made zero or infinity as desired.
  • the impedance is substantially constant.
  • FIG. 16 illustrates a modified structure in which this duplication is avoided.
  • This network is of the bridged-T type the bridging branch having the impedance respect to its impedance and transmission characteristics.
  • T should be a perfect transformer, that is, one having infinitely great winding inductances and unity coupling.
  • the equivalence of the networks of Figs. 1 and 16 can be achieved with a very high degree of exactances by making use of a c1rcuit equivalence illustrated by Fig.
  • the unity ratio transformer T of Fig. 16 is equivalent to the bridged-T circuit of Fig. 17 the terminals of which are correspondingly identified.
  • T is a perfect transformer of unity ratio.
  • the bridge circuit connected between terminals 8 and 9 rises an inductance equal to 1/2(L+M), L eing the inductance of each of the windings W and W of the transformer T and M being their mutual inductance.
  • a capacity 6 representing the effective capacity between windings W and W
  • the centrai branch between the transformer T and terminal 10 comprises a small inductance 2(L M ⁇ .
  • the impedance of the bran'ch may take the forni shown in Fig. 18, comprising a group of parallel connected circuits one of which is a simple inductance L another is a simple capacity Coo, and the others are simple resonant combinations.
  • the negative shunt inductance may then be effectively introduced by increasing the value of L, in the proper proportion and the negat ve capacity may be introduced by diminishing the value of Coo.
  • Figs. 1 and 16 both constitute six-branch networks of the well known Wheatstone bridge or Star Delta type.
  • the load impedances are non-contiguous branches and in the bridged- T form they are branches terminating at a common point.
  • a wave transmission network comprising a plurality of impedance branches, two impedances adapted to determine the transmission characteristics of the network, said impedances each having a plurality of critical frequencies defining resonances and anti-resonances and being proportioned to provide a broad band in which transmission is substantially undistorted, the said critical frequencies being evenly spaced throughout the major portion of the band and being progressively more closely s aced towards the band limits in the remainc er of the band to provide a substantially linear phase characteristic throughout the band.
  • a wave transmission network comprising a plurality of impedance branches, two impedances adapted to determine the transmission characteristics of the network, said impedances each having a plurality of critical frequencies defining resonances and anti-resonances and being proportioned to provide a free transmission band and an attenuation band, the said critical frequencies being evenly spaced throughout the major portion of the band and being progressively more closely s aced towards the band limits in the remainder of the band to provide a substantially linear phase characteristic throughout the band.
  • a wave transmission network comprising a plurality of impedances, two impedances adapted to determine the transmission characteristics of the network, said impedances each having a plurality of critical frequencies defining resonances and antiresonances and being proportioned to provide a free transmission band and attenuation at frequencies above the transmission band, said critical frequencies bein spaced on an inverse frequency scale in t e attenuation range remote from the band limit and on a progressively closer scale near to'the band limit to provide a substantially uniform characteristic impedance throughout the band.
  • a wave transmission network comprising two pairs of e ual impedances arranged to form a symmetrical lattice structure, said impedances comprising multiple resonant reactances, the resonances of the one pair coinciding with the anti-resonances of the other pair in a preassigned broad frequenc range to provide free transmission, and t e said resonances and anti-resonances being spaced at uniform frequencies throughout the major portion of said range and being spaced at progressively closer frequencies towards the limits of the remainder of the range to provide a linear phase characteristic throughout the range.
  • a wave transmission network comprising two pairs of e ual impedances arranged to form a symmetrical lattice structure, said impedances comprising multiple resonant reactances resonant and anti-resonant at a plurality of common frequencies and being proportioned with respect to each other to provide free transmission in a broad range extending from zero frequency, said resonance and anti-resonance frequencies being spaced at equal intervals in the lower frequency range and at progressively smaller intervals in an intermediate frequency range up to a preassigned limit to provide a linear phase characteristic in the lower frequency range, and at progressively increasing intervals above the preassigned limit.
  • a low-pass wave filter network comprising two pairs of equal impedances arranged to form a symmetrical lattice structure, said impedances comprising multiple resonant reactances, the resonances and anti-resonances of the respective pairs of impedances being coordinated to provide a single lowpass band and being spaced throughout the lower part of the band at uniform intervals and at progressively closer intervals towards the band limit to provide a linear phase characteristic in the band.

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  • Engineering & Computer Science (AREA)
  • Power Engineering (AREA)
  • Filters And Equalizers (AREA)
US465522A 1930-07-03 1930-07-03 Transmission network Expired - Lifetime US1828454A (en)

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US465522A US1828454A (en) 1930-07-03 1930-07-03 Transmission network
DEI41969D DE678435C (de) 1930-07-03 1931-07-01 Elektrisches Netzwerk, insbesondere Filter oder phasenverzoegerndes Netzwerk, bestehend aus einem Kreuzglied oder einem ihm aequivalenten Netzwerk

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Cited By (10)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2629049A (en) * 1942-03-02 1953-02-17 John M Miller Filter
US2760167A (en) * 1952-10-29 1956-08-21 Hogan Lab Inc Wave transmission network
US2883534A (en) * 1954-10-20 1959-04-21 Ferranti Ltd Delay stages for electrical pulses
US2907960A (en) * 1954-04-26 1959-10-06 Rca Corp Signal transfer apparatus
US2980872A (en) * 1958-06-13 1961-04-18 Hughes Aircraft Co Bandpass filters
US3146292A (en) * 1954-03-08 1964-08-25 Don L Bonham Electrical vibrato and tremolo devices
US3263019A (en) * 1964-03-18 1966-07-26 Hurvitz Hyman Randomization of phases and frequencies of musical spectra
DE1287225B (de) * 1963-10-31 1969-01-16 Toyo Tsushinki Kabushiki Kaish Verfahren zum Linearisieren des Phasengangs eines Kreuzglied-Reaktanzfilters im Durchlassbereich
US3449696A (en) * 1965-04-12 1969-06-10 Claude C Routh Dual section all pass lattice filter wherein nonlinearities of two sections cancel
US3514727A (en) * 1969-02-18 1970-05-26 Toyo Communication Equip Filters having low delay and attenuation distortions

Cited By (10)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2629049A (en) * 1942-03-02 1953-02-17 John M Miller Filter
US2760167A (en) * 1952-10-29 1956-08-21 Hogan Lab Inc Wave transmission network
US3146292A (en) * 1954-03-08 1964-08-25 Don L Bonham Electrical vibrato and tremolo devices
US2907960A (en) * 1954-04-26 1959-10-06 Rca Corp Signal transfer apparatus
US2883534A (en) * 1954-10-20 1959-04-21 Ferranti Ltd Delay stages for electrical pulses
US2980872A (en) * 1958-06-13 1961-04-18 Hughes Aircraft Co Bandpass filters
DE1287225B (de) * 1963-10-31 1969-01-16 Toyo Tsushinki Kabushiki Kaish Verfahren zum Linearisieren des Phasengangs eines Kreuzglied-Reaktanzfilters im Durchlassbereich
US3263019A (en) * 1964-03-18 1966-07-26 Hurvitz Hyman Randomization of phases and frequencies of musical spectra
US3449696A (en) * 1965-04-12 1969-06-10 Claude C Routh Dual section all pass lattice filter wherein nonlinearities of two sections cancel
US3514727A (en) * 1969-02-18 1970-05-26 Toyo Communication Equip Filters having low delay and attenuation distortions

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Publication number Publication date
DE678435C (de) 1939-07-28

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