US2054794A - Wave filter - Google Patents

Description

R. L DxETzoLD WAVE FILTER Fild June 9, 1954 Fla/- ATTORNEY Sept. 22, 1936.

Patented Sept. 22, 1936 WAVE FILTER Robert L. Dietzold, New York, N. Y., assignor to Bell Telephone Laboratories, Incorporated, New York, N. Y., a corporation of New York Application June 9, 1934,` Serial No. 729,733 4 claims. (ci. ris-,44)

This, vinvention relates to frequency selective networks and more particularly to the control of the phase characteristics of broad band selective systems.

`5 It has for its principal object the provision of a linear phase characteristic not only in the transmission band of a band selective system, but also .through the band limits and into the attenuation ranges as far as maybe desired.

` characteristic which is made linear in this way is transfer constant of the lter network but alsov the `wave-reflection effects at the terminals andthe overall characteristic of the filter network in combination with its terminal impedances. which in practice will generally be fixed resistances. 3 This overall characteristic involves not only the the'desi'red linearity is obtained as the'result of the proper coordination of the sum total of these 1 reilection'effects withthe transfer characteristic 'u of the lter network.

Y The nature ofthe invention will be more fully understood fromthe following detailed description and from the accompanying drawing of which: Y

Fig. 1 shows schematically a general type of network of the invention; c

Fig. 2 is a reactance characteristic used in the explanation of the invention; u

Figs. 3 and 4 illustrate vtlfiecharacter of the impedances in a particular embodiment of the network of Fig. l; and

Fig. 5 illustrates certain characteristics of the networks of the'inventio'n.- i

` Referringftoj Fig'. 1, the network illustrated comprises a symmetrical lattice having series and diagonal impedances Za and Zb respectively, connected between equal terminal resistances, R in series Vwith' one of which-is a Wave source E. The branch impedances may be 4of any degree of complexity;A but should be' substantially free `from dissipation.

The properties of the symmetrical lattice are described at length in United States Patent t 1,828,445,4,-issuedpctober,20, 1931130 H. IW. Bode This object is attained vby the use in the.

wherein it is shown that by particular allocation of the resonance and anti-resonance frequencies of the branch impedances, hereinafter designated critical frequencies, certain advantageous frequency characteristics of the image impedance and the transfer constant may be provided. The characteristics discussed in the Bode patentV are those of the lattice per se, namely, its image impedance and transfer constant,as distinguished from the overall properties of the lattice plus the impedances between which it is connected. In thelatter case the transmission characteristic Vof the system is not represented by the transfer constant alone but by this factor together with modifying factors representing the reection effects at the junctions of the lattice and the eX- ternal impedances. The'total effect, which is a measure of the ratio of the currents at the re` ceiving end of the system before and after the' insertion of the lattice is termed the insertion transfer factor.

The present invention is concerned with the phase component of the insertion transfer factor, that is, with the sum of al1 the phase shifts in the system including those produced by reflection effects. The terms insertion phase shift and insertion phase characteristic are used to designate the phase component of the insertion transfer factor. In accordance with the invention this phase shift is made to have a linear variation' with frequency not only within the transmission band but also through the attenuating ranges by Va particular allocation of the critical frequencies of the branch impedances. This allocation is such that, except at each side of the cut-off frequencies, the critical frequencies are separated by a uniform interval both in the transmission band and in the attenuating ranges, the separation at each side of the cut-olf frequencies being reduced to three quarters of the interval else- Where.

The analysis which follows is directed to the demonstration of the linearity of the phase characteristic obtained by the simple frequency arrangement of the invention and tothe determination of the lattice branch impedancesk so that the network will exhibit this linearity couple with band selective properties.

The determination is further subject tothe-f condition that the impedance arms be physically-V s PATENT OFFICE realizable. This condition imposes a certain functional form for the dependence upon the, fr quency, which may be quickly ascertained. The insertion transfer factor for the network is conveniently examined in terms of the image transfer constant, 0, and the image impedance, Z1, Which are related to the lat-tice impedances, Za and Zt, say by the equations Z tan h 2- Zb (l) ZI=J (2) Equation (l) shows that for free transmission, or for 0 a pure imaginary, Za/Zb must be negative. This result is achieved over an arbitrary frequency interval if Za and Zb are reactances unlike in sign, that is, reactances of which the alternating resonances and anti-resonances correspond, a resonance in Za to an anti-resonance in Zb and so son. Also,by Equation (l) ,the network attenuates in a frequency interval in which Za/Zb is positive, for then 0 is real. This ensues if Za and Zh are alike in sign, or if resonances in Za correspond to resonances in Zb, and so for antiresonances. Since a condition for the physical realizability of a reactance is that its resonances and anti-resonances alternate, between an interval of transmission and an interval of suppression there must occur a critical frequency, the cut-off, in one impedance arm only.

The impedances Za and Zt by their frequency variations and magnitudes completely determine the transmission properties of the network and for that reason may be termed characterizing impedances.

Therefore, filter properties: are obtainable from a physically realizable lattice network if only the arms are reactances having the appropriate type of correspondence between their respective natural frequencies. This is illustrated in the case of the low-pass` lter, in which the branch impedances Za and Zb are of the types shown in Figs. 3 and 4 respectively, by the reactance ex- 1Where Ka and Kb are constants and where f1 and f2 are critical frequencies in the transmitting band, fc a cut-01T intermediate between f2 and f3, and f4 and f5 critical frequencies in the attenuating band. For convenience the critical frequencies representing resonances are termed zeros and those representing anti-resonances are termed poles. Within the transmission band of a filter the zeros and poles are inversely coincident, that is, the zeros of the one impedance are coincident with the poles of the other impedances, while in the attenuation ranges the zeros and the poles of the two impedances are directly coincident, zeros with zeros Y' and poles With poles. Plots of the impedances, showing the manner of coincidence of the resonant frequencies, are given by Fig. 2 in which full line curve l0 represents the frequency variation of the reactance of Za and dotted line curve Il represents the variation of Z. With these values for Za and Zh, the Equations (1) and (2) become It is seen that 0 is imaginary and Z1 real for f f, whereas for f f, 0 is real and Z1 imaginary, corresponding tothe case of the low-pass filter. It will expedite the discussion to confine the attention to this case, subsequently extending the results tol high-pass and band-pass filters.

The relations (la) and (2a) then indicate the form which the dependence of the image parameters upon the frequency must take in orderv that the network may be a physically realizable low-pass filter. Evidently no restriction is placed upon the number of transfer-constant controlling frequencies (fi and f2 in the example) Vnor upon the number of impedance controlling frequencies (f3 and f4 in the example). The cut-off factor,

f2 f2 Y (Jn-E2) n (lfaZn-l) f2 wel) el "fr (1 3) f2 Y I fbx,

wherein K1 and K2 are constant real quantities. The solution of (1b) and (2b) always yields physically realizable expressions for Za and Zb if the Ks are positive and The adjustment of the lattice elements accord- The insertion constant of the network is de fined by e"'Y=I-r, Where Ir and Ir are the received currents before and after theinsertion of the network.v When expressed in terms of the image parameters and the terminating resistance, R, fy is found to be a sum of the transfer constant and the reflection and interaction constants. These latter are deiinedrespectively by eh: *T2 and en: 1 Te (4) (1+ -I c Y R It may be noted in passing that the interaction constant defined by. Equation 4 represents the repeated reflection of the initially reflected part of the current or wave as it passes back and forth betweenY the terminal. impedances and infinite number of times. The convenience of this form of expression becomes manifest when one eX- amines the variation in the phase shift separately in the three intervals, the transmitting band, the attenuating band, and the transition band. In so doing is established the distribution of the critical frequencies j.,i and f i corresponding to linear phase shift.

Transmitting band-From Equation (2b) it is seen that tends to 1 as f tends toward zero, if'K2 be taken equal to R.. Furthermore, the form of the function is such that Z1 differs but little from R in this interval, the immediate vicinity of the cutoff having been set aside for the transition inter-l val. On this account in the pass band the contributions of the reflection'and interaction factors to the phase shift are negligible and the transfer constant represents substantially the total insertion loss. This is readily seen from Equations (3) and (4), the right-hand sides of which converge to the value unity as Z1 approaches the value R, corresponding to negligibly small values of the reflection and interaction constants r and Y 0i. Since this condition holds throughout the interval in question, thel phase characteristic there is determined substantially wholly by the transfer constant alone. If 6=a|7`, where i is the imaginary unit, then by (1b),

when A increases by vrias f varies from one critical frequency to the next. In order thatthe slope be constant throughout the interval, it is therefore necessary that the Vcritical frequencies fm be uniformly spaceds If this spacing is Af, then the phase shift undulates about the chord" Lr i x 2 2 Af having its ideal value at least'at each critical frequency. Y Y n VAttenuatimy band- In this interval,v 'Ithe imaginary part of the transfer constant is either Zero or 1?; while interaction'effects are negligible on account ofthe factor e26 in (4)', With 0 real. The part of the phase-shift dependent-upon-the frequency is therefore the imaginary part of the reflectionconstant, 0r. Since Z1 yis reactive in this range, the phase of the denominator of (3) isA while that of the numerator is a 2 arctan 1. R

Thus

6,: :Fi-+2 arctan 1R (6) The significance of the constant term will appear presently. With the help of (2b), the second term is seen to be, in the attenuating band, a function of the same type as the transfer constant in the transmitting band. Hence, for the phase slope to be constant in the attenuating range, the impedance controlling factors also must constitute a chain of uniformly spaced resonances and anti-resonances. Since r increases by 1r between successive critical frequencies,l the slope will be equal to the slope in the pass band if the uniform spacing is the same constant Af in both ranges.

Transition band-It remains to determine the frequency spacings adjoining the cut-off so that the phase curves in the transmitting'and .attenuating bands are joined through the transition band by a chord of the same slope. In this interval, which we suppose to be bounded by theV last uniformly spacedY critical frequencies in the transfer constant and impedance controlling chains and to contain only the cut-olf frequency, neither the reflection nor interaction effects are negligible. In fact, for this method of decomposing the total insertion loss, these components become op-positely infinite at the cutfoff. kHowever, the interaction factor introduces no net change of phase over the interval, since it vanishes at one edge in virtue of Z1 equal to R very nearly and at the other in virtue of e2e being very small. It may therefore be ignored in evaluating the total change in phase through this interval.

In the space adjoining the cut-off on the transmitting side, the phase of. the transfer constant increases by 1f. Inthe space adjoining the cutoff on the attenuating side, the phase `of the reflection factor increases by 1r. At the cut-off, however, where the image impedance changes from real to imaginary, the reflection factor in-Y troduces 'an abrupt change in phase of jto represented by the first term of (6). Therefore the net change in the transition interval is radians, and the interval must contain 3/2 uniform spaces if the average slope is to be correct. Considerations of symmetry require that the cutoif be the center of the interval, which thus comprises two three-quarter spaces.

These observations establish necessary conditions upon the frequency pattern corresponding to the requirement of linear phase shift in both transmitting and attenuating bands. The sunlciency of these conditions, when appropriate values have been assigned to K1 and K2 in Equations (lb) and (2b), may be verified by direct computation. For this purpose the formulae for the reflection and interaction factors are not useful because of the indeterminacy at the cut-off. This difficulty is avoided by expressing Z1 and 0 in terms of the lattice impedances, in which event If y'Xa and jYb be written for Za and Zh, the insertion loss and phase shift, Afy and B'y, are given n n R R eAT= and

tan B'y which is increased by 1r radians in the threequarter interval on the attenuation side of the cut-off. The contribution of the interaction constant must, of course, remove this phase discontinuity at the cut-off. In the pass band, its imaginary part is where Z1 is real. The limiting value of this angle as the cut-off is approached through frequencies in the pass band can be determined by' expressing it in terms of the lattice impedances. At the cut-off, either Xa or Xb is either zero or infinite. Suppose that Xa is zero. Then this limit is arctan On the attenuation side of the cut-off, the imaginary part of the interaction constant is a i arc [Pe 2 lttee 1R] where Z1 is imaginary. The limiting value of this expression, as the cut-off is approached through frequencies in the attenuation band, is found by the same method to be Xb arctan )16: fc.

Thus the discontinuity in l at the cut-off is asV required to remove the discontinuity at this point introduced by the reflection effect.

The variation with frequency ofthe several phase shift components is illustrated by the curves of Fig. 5 for the case of the low-pass filter having impedances Za and Zh in accordance with Figs. 3 and 4 respectively, and having the critical frequencies spaced in the manner described. CurveV I2 represents the transfer phase shift that is, the phase component yof the transfer constant of the lattice per se, in the transmission range from zero frequency to the cut-o. This component increases by 1r in each of the intervals between the critical frequencies, including fc, and undulates about the straight line I5 departing therefrom by 1r/4 at the cut-olf. In the figure, the undulations of this curve, as well as those of the other curves are somewhat exaggerated in order that their character may be exhibited.

Curve I3 represents the reflection phase shift r in the attenuating range, this component being zero in the transmission band. By Virtue of the critical frequency spacing the general slope of this curve is that of the line I5 but it is characterized first by a departure of 1r/4 at the cutoff and a sudden change of 1r at the critical frequency f3. amounts to a reversal of phase its effect in general is not material.

Curve I4 represents the interaction phase shift. This curve is characterized by undulations of half the period of those of the other curves and by a sudden change of 1r/2 at the cut-off.

The total phase shift in the system is obtained by adding the three curves togetherin which case it will be noted that the discontinuity of curve I4 at the cut-off just neutralizes that at th-e junction of curves I2 and 32. The resultant phase shift will therefore show a smooth variation which is very close to linear through the whole range from zero to ,f3 and which continues at the same slope, subject to reversals at the critical frequencies, in the higher range.

The pattern for the transfer constant and impedance controlling frequencies which has been found is suflicient to insure only that the phase shift has its linear value at each critical frequency, or that the average slope in each space be the same. In order that the slope may closely approximate to the average at every intermediate point, it is further necessary to `determine the multipliers K1 and K2 of the transfer constant and image impedance expressions. We have already seen that K2 should be taken equal tothe terminating impedance, R, so as to obtain impedance match and vanishing interaction effects in the pass band. K1 may be evaluated from Since this latter change simply Equation of which the -principal part in the limit of smallf is i gf-af Y Y n The chord with which the phase characteristic should coincide is 5 1 11 2- 2 Af whenceV 7l' Kl-f f2, must be uniformly spaced, falling at Af and' 2M.A The cut-off, fc, is separated from f2 by three-quarters a uniform interval, and from the first cf the uniformly spaced impedance controlling frequencies by a like interval. Thus, Equations (1a) and (2a) Vbecome (Peitz) in which Af may be selected toY bring the cut-off to any desired point on the frequency scale. The solutions of these relations for the lattice impedances are then The element values for the impedances are readily found by expanding these expressions in partial fractions, after the manner described by R. M. Foster, A reactance theorem, Bell System Technical Journal, v. 3, No. 2, April, 1924.

With this choice of, parameters the greatestV deviation of the phase slope from the average is found by` computation to be of the order of l per cent'. This approximation is satisfactory for mostpractical purposes. Since 'all the parameters of the network have been determined with an eye to the phase characteristic, this is accompanied by a unique loss characteristic. The loss characteristic is marked by refiection peaks at each impedance controlling frequency, where the lattice impedances are zero or infinite together. At these frequencies the image impedance changes sign, and therefore also the constant term of Equation (6). Thus, although the phase slope is uniform throughout the attenuating range, the phase characteristic itself has discontinuities of 1r radians at each impedance controlling frequency. 'Ihis is Ythe interpretation of the constant term of Equation (6). Whether this is an increase or a decrease of 1r radians is not distinguishable fora non-dissipative network. When parasitic `dissipation of energy in'the'network elements is taken into account, the reiiection peaks of loss have finite maxima and the phase in the neighborhood increases or decreases by 1r according as the lineor cross-arm of the lattice has the smaller resistance component at the peak frequency. The infinite peak at this frequency, and the associated abrupt change in phase, can evidently be restored by adding a lumped resistance to the smaller impedance so as to bring the arms into balance. This observation is of importance in considering the effect of dissipation on the phase shift.

When the network is constructed of physical elements its performance characteristics will be somewhat changed from those computed upon the assumption of pure reactance lattice arms. However, the relations subsisting'` between the real and imaginary parts of any analytic function such as the insertion constant enable these changes to be ,readily computed so long as the dissipation can be regarded as uniformly distributed among the elements. In fact, if d is the average ratio of resistance to reactance in the elements, and AA and AB are the variations in the insertion loss and phase shift due to the introduction of dissipation, we have approximately DB Armada (9) and oA wd-; (10) where the derivatives are computed for the network of pure reactances The frequency variable w is 27d. Now the dissipation is ordinarily concentrated chiefly in the coils, so thatk wd is constant if the coil resistances are constant. Then the effect of dissipation upon the loss characteristic in a linear phase shift network is simply the addition of a uniform loss. Y

Moreover, throughout the transmitting band,

in which there is by Equation (1'0)V no first order change in the phase characteristic. But in the transition interval, when the loss is increasing, the phase curve is displaced through dissipation from the ideal straight line. This effect may be compensated in two ways. It depends upon the dissipation being uniformly distributed among the resonant combinations of which the network,

is composed, and is modified if that distribution is modified. In particular if lumped resistance be added to the meshes resonating at the first impedance controlling frequency in such a way as to balance the lattice at this frequency, the phase curve will be restored to linearity, as predicted above.

The effect of dissipation on the phase characteristic in the transition interval Amay be otherwise corrected for by small variations in the ide-al frequency pattern. By diminishing slightly the two three-quarter space intervals in the transition band, the non-dissipative phase slope may be caused progressively to increase through the band so that change due to parasitic dissipation displaces the characteristic toward, rather than away from, the ideal straight line. Since to shorten the cut-off spacing increases the selectivity of the network, the lattenuation characteristic is improved by increase of dissipation in the impedances together with compensating modification of the frequency spacing in this way. The appropriate variations of vthe critical frequencies from their theoretical locations are best determined by trial.

It is possible in other ways to obtain a measure of control over the loss characteristic by means of slight variations in the ideal values of the parameters. For eXample, the loss may be increased at the cost of some degradation of the phase Vproperty by varying the constants K1 and K2.

Since the spacing of impedance controlling frequencies must be uniform over that portion of the attenuating band in which the phase slope is to be uniform, the extension of this condition over the infinite attenuating band of a lowpass filter would result in an infinite network. In practice the phase slope is seldom of interest very far into the attenuating band, so that the chain of uniformly spaced impedance controlling frequencies may be soon terminated. If the phase requirement ends at a frequency fd, uniform spacing must be maintained through fd. Then the infinite chain of uniformly spaced critical frequencies greater than fd may be replaced by one or more critical frequencies so located that the corresponding factors approximate in the range below fd to the factors associated with the omitted infinite sequence. The numerical determination of the terminating critical frequencies is simple, since a close approximation is obtained by use of one, or at most two, of them at somewhat extended spacings.

The foregoing discussion has for simplicity been confined to the -case of the low-pass lter. Similar observations may be made in respect to band-pass and high-pass filters. For the bandpass filter we must have a chain of uniformly spaced critical frequencies in the pass band with cut-offs at three-quarter spacing at both edges. Uniform spacing of impedance controlling frequencies in both attenuating bands is resumed after three-quarter intervals beyond the cutoffs. Since the lower cut-off factor replaces the factor,

ofthe transfer constant expression in the frequency range above the lower cut-off, the theoretical constant multiplier is unity. The multiplier of the image impedance expression is determined to make the impedance R at the mean of the cut-E frequencies.

The high-pass filter may be regarded as the limiting case of the band-pass filter as the upper cut-off recedes toward infinity. The preservation of linear phase shift over this infinite pass band would require an infinite network on account of the necessity of uniform spacing of transfer constant controlling frequencies, but if there is a frequency, fd, beyond which the phase shift is not of interest, the high-pass filter may be realized in a nite'netwcrk by terminating the chain of critical frequencies beyond this point in the manner described above for impedance controlling frequencies.

What is claimed is:

1. In a broad band selective system comprising a symmetrical reactance network having multiple resonant characterizing impedances Za. and Zh, and equal resistive terminal impedances connected to the input and the output terminals of the network, the method of producing a linear phase shift throughout the band and beyond the limits thereof which comprises spacing the critical frequencies of the characterizing impedances at uniform intervals throughout the greater portion of the transmission band and in a-portion of an attenuation range beyond a band limit and spacing the critical frequencies on each side of said band limit at intervals therefrom substantially equal to three-quarters of the uniform interval elsewhere.

2. A broad band selective system comprising a symmetrical four-terminal reactance network having characterizing impedances Za and Zb, and equal resistive terminal impedances connected to the input and the output terminals of said network, said characterizing impedances each having a plurality of critical frequencies which are spaced at uniform intervals throughout the greater portion of the transmission band and in an attenuation range beyond a cut-off frequency, and which on each side of the cut-off frequency are spaced at intervals substantially equal to three-quarters of the uniform spacing elsewhere whereby the insertion phase characteristic is linear throughout the band and a portion of the attenuation range.

3. A broad band selective system comprising a symmetrical four-terminal reactance network having characterizing impedances Za and Zb,

and equal resistive terminal impedances connected to the input and the output terminals of said network, said characterizing impedances having a plurality of critical frequencies certain of which lie within the transmission band and others of which lie outside the band and one of which determines a band limit, said critical frequencies being spaced at uniform intervals throughout the transmission band and in a portion of the attenuation range beyond said band limit and having a spacing on each side of said band limit substantially equal to three-quarters of the uniform .spacing elsewhere whereby the insertion phase characteristic of the network is a substantially linear Yfunction of the frequency throughout the band and through the cut-off frequency.

4. A system in accordance with claim 3 in which the poles and zeros of the impedance Za are inversely coincident with the poles and zeros of the impedance Zb within the band and are directly coincident with the poles and zeros of impedance Zh outside the band.

ROBERT L. DIETZOLD.

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