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Патент USA US2115138

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