US2963666A - Eight-terminal directional filter network with constant impedance on the four pairs of terminals - Google Patents

Description

Dec. 6, 1960 J. OSWALD EIGHT-TERMINAL DIRECTIONAL FILTER NETWORK WITH CONSTANT IMPEDANCE ON THE FOUR PAIRS OF TERMINALS Filed Oct. 21, 1957 2 Sheets-Sheet 1 9 T 2 2 Y 41 L4- Fig.7

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EIGHT-TERMINAL DIRECTIONAL FILTER NETWORK WITH CONSTANT IMPEDANCE ON THE F OUR PAIRS OF TERMINALS Filed 001'. 21, 1957 Z Sheets-Sheet 2 g L 2 is? g T 1 Fig.6

JACQUES 05 WALD United States Patent EIGHT-TERMINAL DIRECTIONAL FILTER NET- WORK WITH CONSTANT IMPEDANCE ON THE FOUR PAIRS OF TERMINALS Jacques Oswald, Paris, France, assignor to Compagnie Industrielle des Telephones, Paris, France, a corporation of France Filed Oct. 21, 1957, Ser. No. 691,248

Claims priority, application France Oct. 30, 1956 1 Claim. (Cl. 333-) The present invention relates to an eight-terminal directional filter of which the eficctive impedance on the four pairs of input terminals remains constantly equal to an ohmic resistance R whatever the frequencies of the voltages applied to the eight-terminal network.

A directional filter of this kind, which will hereinafter be called strict directional filtering, is particularly suitable for the repeater stations of carrier current telecommunication systems in which different frequency bands and a common amplifier member are used for the transmission of signals in the two direcitons.

Devices of this type have already been suggested, which consist of bridged circuits comprising reciprocal fourterminal networks. These devices, however, require very delicate balancing.

There is also a description in German Patent No. 673,- 336, and the US. counterpart No. 2,115,138 of eightterminal networks in which each of the pairs of terminals is connected to the adjacent pairs of terminals by networks with complementary pass-bands, with phase inversion on one of the pairs of terminals. Although, however, the condition necessary for the unit constituted in this way to offer a constant impedance to all the pairs of terminals is indicated in this patent, the means given do not make it possible for this condition to be strictly satisfied.

Now, a strict realisation of the constancy of the impedance on the inputs is of the greatest practical interest; in particular it makes it possible to reduce to a very low value the irregularities of attenuation on a cable comprising repeater stations, even when the number of these is high, since such irregularities become very troublesome on a long cable if this condition is not satisfied; it is also well known that in a reactive network, a very low value of the attenuation of the filters in their pass-bands cannot be obtained without reducing the impedance deviations which determine the reflection losses.

The eight-terminal directional filter network according to the invention consists of two channels placed in parallel between their opposite inputs, and each comprising a symmetrical low-pass filter and a high-pass filter, having complementary pass-bands, the two low-pass filters and the two high-pass filters respectively being identical, said filters being arranged in inverse direction on the two channels with phase inversion on one of the channels, said eight-terminal network being characterised in this that, in order to ensure on each of the inputs the same efiective impedance strictly constant as a function of the frequency of the voltage applied to said inputs, the filters of which it is composed satisfy the specifications which will hereinafter be set forth.

The attached drawings represent:

Fig. l, a general diagram of the eight-terminal directional device.

Fig. 2, the splitting up of the eight-terminal network according to Fig. 1 into two six-terminal networks.

Figs. 3 and 6, two particular embodiments of a strict eight-terminal directional filter network according to the invention.

ice

Figs. 4a and 4b, 5a and 5b, various filter sections of which the equivalence is utilised for the realisation of the eight-terminal network according to Fig. 6.

Fig. 1 represents the known diagram of a low-passhigh-pass eight-terminal directional filter of which the four pairs of access or input terminals are respectively designated by 1, 1', 2, 2', '3, 3', 4, 4. The term inpu is used herein in designating the terminals, but it is intended that this terminology describe all terminals, any pair of which may assume a similar role. This directional .filter comprises two identical low-pass filters 5 and 6, and two identical high- pass filters 7 and 8. These filters are connected in parallel, but they could equally well be connected in series.

The output connections of the filter 7 are crossed between the terminals' 9 and 2, 9 and 2. A device of this kind is used in many two-wire carrier current systems, the inputs 1, 1 and 3, 3', for example, being connected to the input and the output of an amplifier. It will be shown that it is possible to determine the elements of the filters of a directional filter of this kind in such a way that its effective impedance, seen from each of the inputs, is constant at all frequencies, and that there is no interaction between the opposite inputs, 1, 1' and 3, 3' on the one hand, 2, 2' and 4, 4' on the other.

It will be assumed that the filters of the directional device are symmetrical and the form of its matrix of admittances will be determined.

In order to obtain this matrix it is convenient to split up the eight-terminal network into two hexapoles, as seen in Figure 2, with respective inputs 1, 1', 2a, Z'a, 4a, 4'12, and 3, 3', 2b, 2'b, 4b, 4'b, a transformer of ratio --1 being inserted on the output of the filter 7 of the second hexapole, so as to obtain the arrangement of Fig. 1.

As will be seen from an examination of the admittances, this transformer ensures the suppression of any interaction between the groups of terminals constituting opposite inputs.

We shall designate as y' y' y'zz, the elements of the matrices of admittances Y' of the low-pass filters 5 and 6, and by y" y the elements of themat rices of admittances Y" of the high- pass filters 7 and 8, the prime notation identifying the elements of the matrices of admittances of the low-pass filters and the double prime identifying the elements of the matrices of admittances of the high-pass filters, whereassubscripts 11 and 12 identify admittances at the access terminals 1, 1' and 2, 2', respectively, and the subscript 12 designates mutual admittances. It is thus possible to write: 7

trix Y,, of the left hand hexapole of Fig. 2, of which the terminals are 1, 1', 2a, Za, 4a, 4a, is written;

3 Y yn "i" 1 "n @1212 1 /612 B: i 1 2 you 1}"11 and that the matrix Y of the right handhexapole, of which the terminals are 3, 3, 4b, 4'b, 2b, Zb, is expressed:

by reason of the presence of the transformer of ratio -1. In the following we shall designate respectively by U,,

by taking these relations into considerationit will be seen that the matrix of the eight-terminal network is expressed If an electromotive force U is applied to the terminals 1, 1' of the eight-terminal network, all the other inputs being closed on resistances of value R, which can be taken as unity, we have:

' J J J designating the current intensities in resistances respectively connected to the second, third and fourth pairs of terminals, and the relations can be written:

, a 4 making J equal to zero in the second and fourth Equation 12, one obtains J and I, in the function of J yf +y.i +11 M By applying these values in the first relation (12), one arrives at the desired condition of adaptation: 'n l-y' ii) -"y'12 -fy."2a

By designating by' 5 i 3 5 the conjugated quantities of J J J I, one obtains;

which expresses the preservation of the real power.

Thea-elation (14) which ensures the constancy of the efiective impedance seen from the. inputs, was certainly indicated in the aforementioned German Patent No. 673,336, which gives it a form in which the reactances of the branches of the equivalent lattice-type filter ,elements appear, but the method given in order to satisfy it is a method of approximation, based on a choice of which make it possible to determine J J L, as a functhe poles and the zeros of the mutual admittances, based tion of U on the theory of elliptical functions.

v The absence of interaction will be expressed by The approximation of this condition is all the more s a 40 satisfactory as the class of the functions increases and s= consequently the number of elements of the filters.

v In order to express the adaptation of the network on the terminals 1, .1, it is necessary to make: a a

in the Equations 7, 11 being the input current in the directional filter, and to add to it:

A system 11 is thus obtained of four linear and homogeneous equations in 11, J2, J3, J4:

(army's-1w.

The match is rigorous at the frequencies of zero and infinity, and the frequency of separation of the network for which the attenuations are equal on the two paths.

On the other hand, it should be noted that, in the network according to the US. Patent 2,115,138, the effective attenuations of the bands to be eliminated on the two paths represent, for inverse pulsations (of the product w c), equal minimum values. ,Ihis is not the same in the networks of the present invention, where the filter networks can be determined in such a manner as to impart to undesired frequencies eifective attenuations of difierent minimum values on the two paths, which can be advantageous for certain applications.

The elements of the directional filter Which'is the object of the; presentinvention on' the, contrary strictly satisfy the condition (14), so that the directional filter according to the invention will have for allfrequencies,

an impedance exactly equal to 1 on the inputs, even for the directional filters of lowest degree, as is demonstrated by the first of the examples which will be given further on.

1 'Bycancelling the determinant H of this system, the condition of adaptation can be obtained on the terminals 1, 1', and consequently on the other terminals, given the structure of the directional filten,

One arrives at the result more simply by taking into account the condition (8), expressing the absence of interaction between opposed'access (admittances). By

Let 0: and a be the 'respectivesefiective attenuations between theinputs '1 and 2, 1 and 4. By taking into consideration (14), we have g2j 14=1+ For thje'low pass high pass directional filter which is the object of the present invention, y and consequently y' ofiers, for the zero frequency, one pole; y" and consequently y" offers a Zero (for y";; this results from (14) and from the symmetry of the filters).

It follows that we can put:

in which G and U represent the even and odd parts of a Hurwitz polynominal in relation to the variable 17:11, to being the pulsation (classic property of reactance or admittance functions) and f and h represent even polynominals.

The matrix (5) therefore takes the form (19) in which G, U, f and h are bound by the relation: 20) G U =f +h to represent the sum of the admittances of two symmetrical four-terminal networks of which and respectively are the mutual admittances. In the Caner hexapoles, of which the filters are simply tight-coupled, the condition (20) is sufificient for realisation.

It will be assumed that the mutual admittances admit, for an angular frequency w a common pole, with the respective polar portions:

'kp 's? p 16 p 16 The admittance represented by will admit, for this same pole, a polar portion:

n being, according to (14), given by the relation:

( 7 k k+ k as is shown by the cancellation of the polar portion of the second order. According to the known properties of the matrices of admittances of symmetrical quadripoles, however, '7 must be the sum of two positive terms 1/ 7''; such as:

'Y' k k 'Y" k which leads to:

( 'Y kzfl kHl "kli A comparison of (21) and (22) shows that one of the coefiieients O 0",, is necessarily nil; the poles of the mutual admittances are therefore separate and, if C' alone is different from zero, we have:

It is therefore necessary that, in the matrix (19) U shall be a divisor of 7%. This condition is sufficient, as can easily be seen.

Resuming the Equations 17', 17" which can be written:

it will be seen that the condition indicated entails a certain restriction in the choice of the even function which is simply the function (p determining the attenuations in the two channels of the directional filter by:

The choice of f and h, from which Caner determines the directional filter matrices, cannot therefore be made here, and an entirely new method has to be given in order to deal with the case of the eight-pole (or eight-terminal) network.

In the matrix (19) the sub-matrix of the second order composed of the elements of the two first lines and the two first columns is the matrix of the low-pass filter obtained by leaving connected on the terminals 1, 1 and 2, 2 of the first low-pass filter of the eight-pole network, the input terminals of the adjacent high-pass filters, respectively short-circuited on their opposite inputs 3, 3' and 4, 4.

This low-pass filter, which will be designated by S, is obviously symmetrical and we shall call g its characteristic admittance function and q, its characteristic attenuation function, defined by:

(25) q =coth I I being the transfer exponent of the filter, of which exponent the real and imaginary parts respectively represent the characteristic attenuation and phase-shift. As is well known, we shall have:

and the relation (20) shows that:

The representation of admittances by means of functions q q permits one to satisfy identically the relation (14) in such a manner that the match is rigorously realized (achieved) at all frequencies.

Functions :1 and q belong to the class of the Q functions? introduced by Caner, and for the properties of which we can refer to his work Theorie der linearen Wechselstromschaltungen (Theory of linear alternating current circuits) second edition, 1954, Akademie Verlag, Berlin, V1.3: Properties of Q-functions, pages 216 to 220. It willbe recalled that these are positive functions of p=jw, to being the pulsation, i.e. of positive real part for the values of p on the right of the imaginaryaxis, and of which functions, moreover, the square is a rational function, real on the imaginary axis (real frequencies). At the real frequencies the Q functions can therefore be either real or purely imaginary, and their determination, starting from their square, is'rnade so that they asfsume for the real frequencies positive values in their domain of reality. In the domain in which they are imfaginary they have the peculiarities of a reactance.

Two functions q q are said to be conjugated if the frequency domains for which the one is real coincide with those for which the other is imaginary. The product and the quotient of two such functions always define reactance (or admittance) functions. 7

One example of conjugated functions is given by the characteristic functions of attenuation and impedance (or admittance) of a symmetrical filter, the attenuation function being real in the attenuated bands and the admittance function being real in the pass bands. In the present case, in which the functions q g relate to a low pass filter (the same'would obviously apply to a high-pass filter) these functions only present a single branching point at the cut-off frequency; as variable the standardised frequency 9. will be taken, ratio of the frequency to thecut-oif frequency in which the complex variable P==jn.

The class of a Q function" is defined as the halfdegree in $2 or P of its square. The choice of the half-degree to define the class is made with the object of assigning the class 1 to the elementary filterelement, andiclass /2 to the half-element.

The characteristic admittance function 4g, is itself standardised, i.e. related to the value 1/R of the constant effective admittances on the inputs.

In accordance. with the relation (28), /q -1 represents a mutual admittance; the function q characteristic admittance function of symmetrical low-pass, therefore also belongs to the class of high-pass antimetric filter attenuation functions. They necessarily have one pole at infinity, a branching zero at the standardised frequency 1, 'a value 1 at the zero frequency, and are of the form, if n+- /2 is their class:

IMO) and Mn(P designating polynomials in P of n degree, of which the constant term is equal to 1, and of which the zeros, all negative and higher than 1 in modulus, are overlapping, the smallest in modulus belonging to M (P The functions q employed offer, for n frequencies 0' 0' 0,, less; than 1, maxima equal to 1, and the knowledge of said frequencies, which can be arbitrarily fixed, and which will be chosen from those for which it is desired to obtain a point of infinite attenuation on the high-pass side, completely determines :1 (see, for this determination, Caner op. cit., VI. 5. VI. 6, pages 226- Equation 28 of which the second term is certainly a mutual admittance function then gives the expression or y" which obviously has the same poles as q Then we take, as characteristic attenuation function :q, of the filter S, a conjugate function of q which will be chosen so that q -1 has double zeros in P for all the polar frequencies of g including the frequency at infinity; in this way the condition is ensured, which recognised-as necessary-and sufficient, that y and y" have no common pole. The four-pole S therefore has this peculiarity of admittingthe peaks of infinite attenuation for all the poles of-the characteristic admittance. If; m be the class of q this function will be in the form':

R (P and S (P representing polynominals in P of degrees m and m-l, of which the terms of highest degree have as coeflicient 1, of which the zeros, all negative and less than 1 in modulus, are overlapping, that of .the highest modulus belonging to Rm(P Theicondition indicated, that q --1 presents double zeros in P for the zeros of M,,(P as also for the infinite frequency is expressed by 2114-1 relations which-perfectly determine q if-m equalsnfl l, The preferable mode of determination was indicated by Cauer in the passages of his work already mentioned. W

If m is higher than n+1, the determination of q, will be made for an arbitrary choice of (m-n-l) additional double zeros .of (1 -1. For these zeros, the admittance y' will be zero and there will bea peak off infinite attenuation, on the low-pass side of the directional filter.

As q; and q; have been chosen in the manner just explained, y' =q /q l will only. admit the poles of q with the same moduli of residues as q q and 3 the ,polesof g with the same moduliof residues as q q which makes it possible, merely by the development of the mutual admittancesalong their polar parts, to obtain y;, and y" The low-pass filters of the eight-terminal network, 5 and 6, and the high- pass filters 7 and 8, are thusperfectly determined. It will be noted that the admittances of the branches of the equivalent lattices, having as their respective values y i y' for the low-pass filters, y" :y" for the high-pass filters, admit for polar portions, for each of the filters, the double of the polarportions with positive residues of the mutual admittance, and the double change of sign, of the polar portions with negative residues of the same admittance.

It is in no way meant to. saythat the lattice-type diagram should preferably be used in producing filters. The peaks of infinite attenuation, on the high-pass side of the directional filter, are given by the values 1 of q and the poles of q the peaks of attenuation on the lowpass side by the poles of q and the additional values, 1 Of q 7 It would equally well have been possible to extract from the matrix 19 the sub-matrix composed of the elements of the first and fourth lines. and, the first and fourth columns, which represent a high-pass symmetrical four-terminal network S, which could be made replay the same part as S; it would then have been necessary to introduce a function 11 of low-pass antimetric filter attenuation. 7

Finally it should be noted thatby substituting matrices of impedances for the matrices of admittances, one would make the synthesis of an eight-terminal network of strict directional filtering of the series diagram on all its inputs. 7 a

Two embodiments of directional filter eight-pole networks according to the invention will now be given.

The eight-pole networ-ks will be designated by the classes of the functions q and q q, is of integral class and q of semi-odd class, as has been seen.

The eight-pole network (m; n+ is an eight-pole network for which the function q is of class m, the function (12 of class n+l.

First example.-Eight-pol e network of the fyp'e (1; 0.5)

which leads to will be an odd function, with pole at zero freat infinity of q 1, considered as a function of P Therefore:

and consequently:

I 1 l I/ 1 y12= 11 21 The filters are reduced to series reactances, inductances for the low-pass filters, capacities for the highpass filters. Fig. 3 represents the diagram of the eightterminal network obtained in this way. The function (p in this case equals 2P Second example.Eight-ple network of the type (2; 1.5)

We take a function g of class 1.5; this is a high-pass antimetric filter attenuation function, taking the value 1 at the zero frequency and for a pulsation 52",, which will be defined by:

(in which a f3 a 1).

a and 5 are defined by the condition that q --1 admits a double zero for: P =m 1, which gives:

2 2(fi -5 2+ (1 [6: i

whence it follows that:

1 m z +fl a-l It is obviously necessary to take e=+1, hence: v v

- The function q is determined by its values 1, for the frequencies corresponding to the poles of q i.e. the infinite frequency and the frequency 9' defined by:

It is convenient to take as intermediary in the calculation the function q =coth to which q; is bound by the relation:

q taking the value 1 at the infinite frequency and at the frequency 9' has the form:

10 a being defined by:

The sign must be chosen, as a is less than 1; therefore, expressing a as a function of m 1+m (36) 1 +2111 and taking (35) into consideration:

and consequently, given the relation between the residues of the admittances and the mutual admittances:

As has previously been recalled in a symmetrical fourterminal network of admittances y y the admittances a l, l E 1 of the branches of the equivalent lattice-type network are respectively equal to y +y and y --y It will therefore be seen that for the low-pass filter, the reactances A', B of the branches of the equivalent lattice are given by:

G representing a reactance;

Now it is known that generally a lattice-type network of-hranches X+Y, iX-l-Z, .is equivalent to a latticefltype network of branches Y: and Z inserted'betweentwo impedances X arranged in series on the two inputs. Figs. 4a and 4b illustrate this well known equivalence.

If Y is nil, the lattice element degenerates into a shunt impedance, so that the lattice element of branches X, Y-l-Z is equivalent to a T network of which the longitudinal branches are equal to X and the middlebranch to I P P(l+2m 2H ,1 b 1 m and consist of inductances equal to 5 1+2m wo l+m whilethe vertical branch consists of a resonant circuit of capacity:

and inductance:

It will be seen that L' C' w (1+m =1, i.e. the frequency of'resonance of this branch, for which there is a pole of attenuation, corresponds well to the zero-of the function q as was foreseen.

The case of the high-pass filters will be treated in the same way and it is recognised that they' are equivalent to symmetrical T-type networks of which the longitudinal branches consist'of capacities-C" of'value equal to and the vertical branch of a resonant circuit of induc tance L112: R; H 7 2( 2) e and capacity: 7

"2 "z= o 2 V which shows that the resonance pulsation is equal to the pulsation 9",, of the a'tte nuation pole correspond to the value 1 of q Fig. 6 gives the complete diagram of this eight-pole network." 1 V I 7 An eight-terminal directional filter network-comprising two groups of pairs of opposite access terminals 1, 1-3, 3' and 2, 24, 4 and'twopairs' of symmetrical identical filters, a first pair of filters consisting of two low-pass filters the matrix of admittance of which receives y' as the admittance at the access and y' as the mutual admittance and the second pair of filters being constituted by two high-pass filters, having a passing band complementary to the band of the low-pass filters,--the matrix of admitta'nce .of which receives y as the adniittance at the access and y"' as the mutual admittrance, the "low-pass filters being-respectively connected to the pairs of adjacent terminals 1, 1 -2, 2' and 3, 3'4, 4 of the network and the high-pass filters being respectively' connected to the'other pairs of terminals 1, 1 4, 4' and 2, 2-3, 3, the phases being inverse along the'two we have:

paths comprised between two pairs of opposite terminals,-

characterized in this, that in order to ensure, at each pair of access terminals, the same strictly constant-effective impedance as a function of the frequency of the voltage applied to said terminals, the low-pass filter and the high-pass filter respectively have mutual admittances:

and admittances on the terminals y' and y" obtained by developing y' and y" along theiripolar portions and by taking as residues the 'moduIi-oftheresidues-of the mutual admittance's, the function-q having the form:

2 q2=-/P +1% LAP) and M (P designating polynominals P? of degree n (with P= 'Q, 9 being the ratio of the'applied frequency to the cut-off frequency) and 'of constant term equal to 1, determined by means 'of 1' parameters 0' 0 a all less than 1, by the condition that (1 -1 .7 has double zeros in P for the values corresponding to a the function of :1 having the form:

q RHAP PvP +1sm 1'( R (P3) and Sm (P )j designating polynominal's in P of de'greelm and ml, of which 'theterms'of highest degree "have 1 as coefiicient, and ofwhich the 2 parameters are chosen so that (1 -1 has double zeros in P for the n zeros of M (P and for the infinite frequency.

References Cited in the file of this patent UNITED STATES PATENTS 2,115,138 Darlington Apr. 26, 1938 (US. corresponding to German Patent 673,336)

FOREIGN PATENTS 673,336 Germany Mar. 24, 1939

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