US2204712A - Uniform impedance network - Google Patents

Uniform impedance network Download PDF

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Publication number
US2204712A
US2204712A US161017A US16101737A US2204712A US 2204712 A US2204712 A US 2204712A US 161017 A US161017 A US 161017A US 16101737 A US16101737 A US 16101737A US 2204712 A US2204712 A US 2204712A
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Prior art keywords
line
impedance
filter
image
exponential
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Expired - Lifetime
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US161017A
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English (en)
Inventor
Harold A Wheeler
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BAE Systems Aerospace Inc
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Hazeltine Corp
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Priority to US161017A priority patent/US2204712A/en
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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
    • H03H7/38—Impedance-matching networks
    • H03H7/383—Impedance-matching networks comprising distributed impedance elements together with lumped impedance elements

Definitions

  • This invention relates generally to uniform impedance networks and is particularly useful in matching the image impedance of an exponential line with the impedance of a-circuit or with the image impedance of another line to which the exponential line is to be connected, particularly a uniform line having a purely resistive and uniform image impedance.
  • the invention is particularly suitable for utilizing an exponential transmission line as means for coupling an antenna to a uniform line going to the terminal circuit of a modulated-carrier signal apparatus.
  • the image impedance may be kept nearly pure resistance if the rate of taper is made sufiiciently small and the line is made sufiiciently long to secure the desired impedance ratio.
  • means are provided for terminating an exponential line of any taper and length, by means of which the image impedance of the line can be exactly matched by a filter having either a mid-shunt or a mid-series termination.
  • a filter having either a mid-shunt or a mid-series termination.
  • an exponential line having terminal circuits in accordance with the invention provides an improved means for connecting two circuits having lmped'ances of different levels and provides an ideal means of connecting two circuits having resistive image impedance terminations of different levels; and provides an ideal means of connecting two circuits having resistive image impedance terminations of different impedance levels.
  • Fig. 1 of the'drawings is a schematic representation used for deriving the equations of an exponential line
  • Fig. 2 is a schematic diagram of the image impedance of an exponential line resolved into series components
  • Fig. 3 is a schematic diagram of the image impedances of an exponential lineresolved into parallel components
  • Fig. 4 is a graph of the image impedances of an exponential transmission line both below and .'above the cutoff frequency
  • Fig. 5 illustrates a circ'uit arrangement for matching the line image impedance with the corresponding constant-k filters by resolving into series components
  • Fig. 6 illustrates a circuit arrangement for obtaining an approximately constant image resistance
  • Fig. '7 shows a balanced disposition of the elements of Fig.
  • Fig. 8 is a graph of the relation between wire separation and distance from the. point of maximum separation for lines having various f given ratios of maximum separation to wire diame ter;
  • Fig. 9 illustrates as a special case one of the transmission lines, the characteristics of which are shown in Fig. 8;
  • Fig. 10 shows a cir cuit arrangement for substantially,matching the impedance of an exponential line with a doublet antenna;
  • Fig. 11 shows a circuit arrangement for matching a vertical-wire antenna with a single-wire exponential line having a ground return circuit.
  • E and I are vectors of alternating voltage and current along the line.
  • E Zflz/m- jazz/X
  • A 2w(-z/) ⁇ jZ/ v in which A is the wave length along the line at the operating frequency.
  • the steady-state velocity is variable and is greater than the limiting velocity which is nearly the velocity of light.
  • 'It is a form of distortion which is absent in the ideal uniform line,.but is found in filters.
  • group velocity or impulse velocity does not really exceed that of light, although the phase velocity or "steady-state velocity are greater, as effected by the taper.
  • the image impedance of the line at the narrow end is the driving-point impedance of the till) square root term is imaginary and negative.
  • the line is comparable with a transformer whose turns ratio is Tzw/mz arl n 2.7310. (11) in which I is the length of the line. It appears that a nominal ratio can be secured with a length only a small fraction of An.
  • Fig. 3 shows the resolution of Zn and Zb into parallel components, as follows:
  • Fig. 4 shows graphically the image impedance and its components at either end of the exponential line.
  • the abscissae are inversely proportional to frequency, to show conciselythe limiting conditions at high frequency.
  • the abscissae between and 1 thus represent the transmission band of the exponential line. It willbe seen that in this band (the image impedance of either end of the line is a constant, the image impedances being represented by-Zs' and Zb, respectively.
  • the image impedance Za can be resolved into series components Ra'and Xa, or parallel components Ra and- X8195 illustrated. Similarly, the impedance Za. in the transmission band can be. resolved into either series or parallel components, as shown.
  • the image impedance of the line at either end is a pure reactance and is shown represented by Za or X9. and Zb or Xb for the respective ends of the line.
  • Matching the exponential line with adjacent circuits involves not only the relative magnitudes of the impedances at the junction, but also the nature of the line impedance.
  • the line impedance approximately matches a constant resistance (Rka or Rkb) only at frequenciesmuch higher than the cutofi frequency. Closer matching is secured by power-factor correction which converts the line impedance to a pure resistance variable with frequency and by means of which either end of the line can be matched exactly with a high-pass constant-k filter. An m-de rived termination of such a filter can be used to match closely a constant resistance.
  • Fig. shows the principle of power-factor correction of the line impedance.
  • the image impedance at the narrow end of the line is represented, as in Fig. 2, by Ra in series with 20a (in dotted lines).
  • the effect of reactance of 2G,; is cancelled by connecting 2L1 in parallel, while the resistance Re. is transformed to Re, which is the mid-shunt image resistance of the corresponding constantJc filter, by means of the half-section high-pass filter 3 .formed by 2C, and ZLa.
  • the wide end is also represented, as in Fig. 2, by Rb in series with -2Cb.
  • the effect of the reactance -2Cb is cancelled by connecting 201. in series, leaving purely resistive image impedance Rh, which is the mid-series image resistance of the corresponding constant-1c filter.
  • Rm is the mid-series image impedance of the halfsection'i, the mid-shunt image impedance of which is Rb.
  • Any number of half-sections may be added, according to the ordinary rules of filter designi
  • the two essential requirements of impedance matching are met by the line and filters of Fig. 5.
  • each terminal circuit presents to the line an image impedance equal to that of a continuation of the line with the same exponential taper. For example, at the wide end, 20:; and Rh in series result in the same image impedance at the narrow end as a continuation of the line at the wide end without interruption.
  • each end of theline with its power-factor correction presents to the respective filter an image impedance equal to that of the adjoining filter termination.
  • m determines the residual relative variations of the image resistances at the terminations in Figs. 6 and '7.
  • a value of m ,equal to 0.6 is suggested, which causes the image resistances to be within about 5 percent of the respective terminating resistances Rka or Rkb at all frequencies more than 15 percent above the cutoff frequency. This value of m also places the frequency of infinite attenuation 20- percent below the cutoff frequency.
  • the circuits of Figs. 6 and '7 are about the simplest that can be used to match closely either end of an exponential line with a constant-resistance circuit, such as a uniform line. If an exponential line without the matching expedients of Fig. 5 or Fig. 6 is connected directly to a constant-resistance circuit, the matchingwill not be satisfactory at frequencies less than 40% above the cutoff frequency because the power factor of the line impedance is then substantially less than '70 percent. If the circuit arrangement of Fig. 5 is used to match an exponential line with a constantresistance circuit, so that the effective image impedance of the exponential line is purely resistive but variable.
  • the constant-resistance circuit should have a resistance value which is the geometric mean of the minimum and maximum values of the nonuniform effective image resistance of the exponential line, over the required frequency range.
  • the constantre'sistance should be about 0.84 Rka to match Ra. or about 1.19 Rka. to match Re.
  • a two-wire transmission line has an exponential taper only when the separation of the wires varies correctly with distance along the wires. If the desired variation of Rk is given, in accordance with Equation 2, the actual values of R1; deterniine'fthe ratio of separation to Wire diameter at all distances. An explicit formula for the separation in terms of the distance, may be derived on the assumption of bare wires of zero resistance.
  • Equation 24 and 26 can be rewritten as:
  • Fig. 8 is plotted from Equation 36 in such a manner as to show thevariation of separation with distance along the wires from the point of maximum separation.
  • the ratio ym/d is used Interpolation is possible for values of this ratio between 100 and 10,000.
  • Fig. 10 illustrates the matching of a uniform transmission line ID with a straight-wire doublet H, 12. The following procedure is useful in designing an exponential line l3 for this use, ac-
  • Tension cords or separators ll are connected between the tapered wires, at intervals sufficiently close to maintain substantially the correct shape.
  • the straight wire of the doublet is cut out in the center at l5 for a space of about 2 meters, and the wide end of the tapered line 13 No matching filter ground return and having its constants exponentially varying along its length.
  • a wave-signal translating system comprising a transmission line having an impedance which varies exponentially alongthc length of the line and a terminating circuit for said line comprising an inductance having a value of i e la in parallel with said line at the end of the line having the lower impedance level, where we is the cutoff frequency of said line and Lia and Cm are. respectively, the inductance and capacitanceper unit length of said line at said end, saidfirst mentioned l e lh in series with said line at the end of the line having the higher impedance level, where'we is the cutoff frequency of said line and Lip and Cw are.
  • a wave-signal translating system comprising a transmission line having an image impedance which varies exponentially along the length of the line and a terminating circuit for said line comprising an inductancehaving a value of per unit length of said line at said end, said firstmentioned inductance thereby adjusting the impedance of said line at said end to match exactly over the transmission band the mid-shunt image impedance ofa constant-k filter having a seriesreactance arm' with a capacitance of 'I I 5.
  • A'wave-signal translating system comprising ng a transmission line having an impedance which varies exponentially along the length of the line anda' terminating circuit for said line comprising a' capacitance having a value of i e ib in ar swith as line at the end of the line having' the higher impedance level, where we is the cutoff frequency of said line and Lib and] C11) are, respectively, the inductance and capacitance per unit length of said line at said end, said firstmentioned capacitance thereby adjusting the 'gje'impedance of said line at said end to match lyfover the transmission band the mid-series "series-reactance arm with a capacitance of i a e ib and a shunt-reactance arm with an inductance of l I 2 .2 e Clb 6:
  • Anelectrical impedance network comprisiliary filter network coupled .ing a terminal circuit including reactance tending to limit the responsiveness of said terminal
  • said filter having an m-derived termination at its other end, and a resistive-terminating circuit coupledft'o said other end of said filter and U matching the impedance thereof over the pass hand of said filter, whereby the'responsiveness of said network is maintained substantially uniform over said pass band.
  • An electrical impedance network comprising a terminal circuit including reactance ef- -fectivelyin shunt across said terminal circuit tending to limit the'responsiveness of said ter- 'minal 'c i rcuit over a wide band'of frequencies,
  • auxiliary filter network coupled to said ter-; minal ci'rcuit'and including said reactance as afull-shunt terminating element at one end of said-filter, said filter having an m-derived termination at its other end, and a'terrninating resistor coupled to said other end of said filter and matching the impedance thereof over the pass band of said filter, whereby the responsiveness of-said network is maintained substantially uniform ,oversaid pass band.
  • An electrical impedance network comprising a terminal circuitincluding reactance effectively in series in saidterminal circuit tending to limit the responsiveness of said terminal circuit 7 over a wideband of frequencieaan auxiliary filter network coupled to :said terminal circuit and including said reactance as a full-series terminating element at one end of said-filter, said filter having an m-derived termination at its other end,- and a terminating resistor coupled to said otherend of said filter and matching the impedance .thereof over the pass band of said filter, whereby the responsiveness of said netna eimpedance of a constant-k filter having a r work is' maintained substantially uniform over saidipass band.
  • An electrical impedance network comprising .a terminal circuit including an inductive ele mentefi'ectively coupled across said terminal circuit and tending to limit the responsiveness of said terminal circuit over a wide band of frequencies, an auxiliary filter network coupled to said terminal circuit and including said inductive element as a full-shunt terminating element at one end of said filter, said filter having an m-derived termination at its other end, and a terminating resistor coupled to said other end of said filter and matching the impedance thereof over the pass band of said filter, whereby the responsiveness of said network is maintained substantially uniform over said pass band.
  • a signal-translating system for operation over a wide range of frequencies comprising a pair of terminals between which there is effectively a substantial reactance tending to limit the response of said system over said range, a filter having a predetermined image impedance over said range and coupled to said pair of terminals, said filter comprising a portion of said reactance as a mid-element of said filter, and an impedance termination coupled to the dead end of said filter proportioned substantially to match the image impedance of said filter over said range, the reactive constants of said dead-end filter being so proportioned relative to said reactance and the operating frequency range that the mean value of the impedance between said pair of terminals over said range isyapproximately the limiting value that can be maintained therebetween over said range.
  • a signal-translating system for operation over a wide range of frequencies comprising a pair of terminals in series with which there is efiectively substantial capacitance tending to limit the response of said system oversaid range, a filter having a predetermined image impedance over said range coupled to said pair of terminals, said filter comprising a portion of said capacitance as a mid-series element of said filter, and an impedance, termination coupled to the dead end of said filter proportioned substantially to match the image impedance of said filter over said range, the reactive constants of said dead-end filter being so proportioned relative to said capacitance and the operating frequency range that the mean value of the admittance between said pair of terminals over said range is approximately the maximum that can be maintained therebetween over said range.
  • a signal-translating system /for operation over a wide range of frequencies comprising a pair of terminals across which there is effectively substantial inductance tending to limit the response of said system over said range, a filter having a; predetermined image impedance over said range coupled to said pair of terminals, said filter comprising a portion of said inductance as a mid-shunt element of said filter, and an' impedance termination coupled tothe deadend of said filter proportioned substantially to match the image impedance of said filtenover said range, the reactive constants of said 'deadend filter being so proportioned relative to said inductance and the operating frequency range that the mean value of the impedance between said pair of terminals'over saidrange is approxr Certificate of Correction Patent No. 2,294,912. June 18, 1940.

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US161017A 1937-08-26 1937-08-26 Uniform impedance network Expired - Lifetime US2204712A (en)

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Cited By (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2567235A (en) * 1939-06-20 1951-09-11 Int Standard Electric Corp Impedance matching arrangement for high-frequency antennae
US2643296A (en) * 1949-09-28 1953-06-23 Betsy R Hansen High-frequency energy dividing apparatus
US2720627A (en) * 1951-08-11 1955-10-11 Bell Telephone Labor Inc Impedance matching networks
US2757343A (en) * 1950-11-25 1956-07-31 Philco Corp Coupling network for television tuners
US2835872A (en) * 1953-09-01 1958-05-20 Bell Telephone Labor Inc Interstage coupling network
US2973488A (en) * 1958-11-03 1961-02-28 Collins Radio Co Impedance matching device having a folded tapered line

Cited By (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2567235A (en) * 1939-06-20 1951-09-11 Int Standard Electric Corp Impedance matching arrangement for high-frequency antennae
US2643296A (en) * 1949-09-28 1953-06-23 Betsy R Hansen High-frequency energy dividing apparatus
US2757343A (en) * 1950-11-25 1956-07-31 Philco Corp Coupling network for television tuners
US2720627A (en) * 1951-08-11 1955-10-11 Bell Telephone Labor Inc Impedance matching networks
US2835872A (en) * 1953-09-01 1958-05-20 Bell Telephone Labor Inc Interstage coupling network
US2973488A (en) * 1958-11-03 1961-02-28 Collins Radio Co Impedance matching device having a folded tapered line

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