, y - -
VOLUME X IX
JULY, 1940
NUMBER 3h 2 S / t o
THE BELL SYSTEM
TECHNICAL JOURNAL
DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS
341
C rosstalk B etw een Coaxial Conductors in Cable
— R. P . Booth and T. M . O darenko 358
C om p ressed P o w d ere d M o ly b d en u m P erm a llo y for H igh Quality Inductance Coils— V. E. Legg and F. J. G iven . 385
H igh Accuracy H eterodyne O scillators— T. S lo n czew sk i . . 407
R elations B etw een Attenuation and P h ase in Feedback Ampli- fier D esign— H . W. B o d e ... 421
Analysis of the Ionosphere— K arl K . D a r r o w ...455
Abstracts of Technical P a p e r s ...489
Contributors to th is I s s u e ... 492
AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK
50c p e r Copy $1.50 p e r Year
Crosstalk in and Open
B ased on Short-Circuited
Published, qu arterly by the
A m erican Telephone and T elegraph Com pany 195 B roadw ay, N e w York, N . Y.
E D IT O R S
R. W. King J. O. Perrine
F. B. Jewett A. B. Clark S. Bracken
EDITORIAL BO ARD H. P. Charlesworth
O. E. Buckley M. J. Kelly
W. Wilson
W. H. Harrison O. B. Blackwell G. Ireland
SUBSCRIPTIONS
Subscriptions are accepted at SI.50 per year. Single copies are fifty cents each.
The foreign postage is 35 cents per year or 9 cents per copy.
Copyright, 1940
American Telephone and Telegraph Company
PRINTED IN U. S. A.
The Bell System Technical Journal
Vol. XIX July, 1940 N o. 3
Crosstalk in Coaxial Cables—Analysis Based on Short-Circuited and Open Tertiaries
By K. E. GOULD
T h e p ro b le m c o n sid e r ed h er ein is t h a t o f e s tim a tin g , from m e a s
u r e m e n ts o n sh o r t le n g th s o f co a x ia l ca b le, th e c r o s sta lk t o be e x p e c te d in lo n g le n g th s o f th e sa m e c a b le. T h e m e th o d d e v e l
o p e d , w h ic h is p a r tic u la r ly a p p lic a b le t o c a se s in w h ich th e effe c t o f te r tia r y c ir c u its on th e c ro s sta lk is large, is b a sed o n m e a su r e m e n ts of c r o s sta lk in a sh o r t le n g th , w ith th e te r tia r ie s first sh o rt- c ir cu ite d a n d th e n o p e n . T h e a p p lic a tio n of th is m e th o d t o th e c a b le d escrib ed in th e c o m p a n io n p a p er b y M essrs. B o o th and O d a ren k o g a v e c r o s sta lk v a lu e s in g o o d a g r e e m e n t w ith th e ir ex p e r im e n ta l r esu lts.
In t r o d u c t i o n
T ? O R a n u m b er of years th e problem of cro sstalk su m m atio n in long open-w ire lines or cables h as been stu d ied by m easuring crosstalk, in p hase an d m a g n itu d e , in sh o rt lengths. T h e cro sstalk w ithin a sh o rt len g th , betw een tw o c ircu its te rm in a te d in th e ir ch a ia c teristic im pedances, w ould be m easured w ith all im p o rta n t te rtia ry circuits also a p p ro x im a te ly te rm in a te d . T h e n th e cro sstalk betw een tw o circ u its in ad jo in in g sh o rt lengths would be m easured w ith th e te r tia ry circ u its te rm in a te d . F rom these coefficient m easu rem en ts th e cro ssta lk in a long length could be estim a te d b y a process of in teg ratio n .
T h e ap p licatio n of th is general m eth o d to cro sstalk in th e usual ty p e s of coaxial cable would require g re a t accu racy in th e coefficient m easu rem en ts, because in ’onger lengths th e desired cro ssta lk v alue dep en d s on th e difference betw een tw o n e arly equal q u a n titie s in v o lv ing th e coefficients. In the following analysis th e co m p u ta tio n of th e cro ssta lk for long lengths of coaxial cable is based on cro sstalk m easure
m en ts, in p hase and m ag n itu d e, betw een tw o coaxial circuits in a single sh o rt length w ith th e te rtia ry circuits first open an d th en short- circuited, no cro sstalk m easu rem en ts w ith te rm in a te d te rtia ry circuits being involved.
341
T h is m eth o d of an aly sis, w hen ap p lied to th e tw in coaxial cab le described in th e co m p an io n p a p e r b y M essrs. B o o th a n d O d aren k o , g av e re su lts in good a g re e m e n t w ith th e m easu red c ro sstalk .
In th is an aly sis it w as assu m ed t h a t all th e te r tia r y c irc u its could be com bined a n d considered a s a single c ircu it. A lth o u g h no evidence h a s been found t h a t w ith th e ty p e s of s tru c tu re stu d ie d so far, b e tte r ac c u ra cy w ould re su lt from th e f u rth e r refin em en t of co n sid erin g tw o or m ore d issim ilar te r tia r y circ u its w ith co upling b etw een th e m , th e re is one case of p ra c tic a l im p o rta n c e w hich c a n n o t be h a n d le d w ith th e sin g le -te rtia ry an aly sis. T h is case is t h a t of th e in te ra c tio n cro ss
ta lk ( th a t is, th e cro ssta lk b y w ay of a te r tia r y c irc u it) betw een tw o ad jo in in g len g th s of coaxial cable, w hen, a t th e ju n c tio n , p a r t of th e te r tia r y co n d u c to rs are sh o rt-c irc u ite d to th e o u te r coaxial co n d u c to rs while th e rem ain in g te r tia r y co n d u c to rs c o n tin u e th ro u g h w ith no d isc o n tin u ity . T h is p roblem m ig h t be of im p o rta n c e w here, a t a re p e a te r, th e o u te r coaxial c o n d u c to rs a n d th e s h e a th a re b o n d ed t o geth er, b u t p a p e r-in su la te d p airs in th e sam e s h e a th p ro v id e an u n in te rru p te d te r tia r y c ircu it. T h e n ear-en d c ro ssta lk u n d e r such co n d i
tions m ig h t also differ sig n ifican tly from th e v alu es in d ic a te d b y th e sin g le -te rtia ry analysis.
T h e tw o -te rtia ry an aly sis is to o long to be given here in d e ta il, an d hence h as been o u tlin ed o n ly to such a n e x te n t as to in d ic a te th e d e riv a tio n of th e form ulas for in te ra c tio n c ro ssta lk w hen one of th e te rtia rie s is sh o rt c ircu ited a n d th e o th e r te rm in a te d in its c h a ra c te r
istic im pedance. T h e fo rm u la for n e a r-en d c ro ssta lk u n d e r th is c o n d i
tion is given w ith o u t d eriv a tio n .
I-— Id e n t i c a l Co a x i a l Li n e s Sy m m e t r i c a l l y Pl a c e d w i t h Re s p e c t t o a Si n g l e Te r t i a r y
T h e first case we shall consider h erein is t h a t of a n y n u m b e r of id en tical coaxial lines w ith th e o u te r coaxial c o n d u c to rs in c o n tin u o u s electrical c o n ta c t an d sy m m e tric a lly p laced w ith re sp e c t to a single te r tia r y c irc u it, such as t h a t w hich m ig h t be p ro v id ed b y a s h e a th s u r
ro u n d in g th e coaxial lines a n d in su la te d from th e o u te r coaxial c o n d u c to rs, o r b y a su rro u n d in g lay er of p a p e r-in su la te d p airs. T h ro u g h o u t we shall assum e t h a t th e re actio n of th e in d u ced c u rre n ts u p o n th e d istu rb in g line is negligible.
Follow ing a n o m e n c la tu re analo g o u s to t h a t of th e Schelkunoff- O d aren k o p a p e r,1 we will d e sig n ate b y Z u th e m u tu a l im p e d an ce p e r u n it le n g th betw een a n y tw o coaxial lines in th e presence of th e o th e r coaxial lines b u t in th e ab sen ce of a n y o th e r co n d u c to rs. T h e m u tu a l
1 B ell System Technical J o u rn a l, April, 1937.
C R O S S T A L K I N C O A X I A L C A B L E S 343
im pedance per u n it length betw een a n y coaxial line (in th e presence of th e o th e r coaxial lines) a n d th e te r tia r y c ircu it consisting of all of th e coaxial o u te r c o n d u cto rs w ith re tu rn b y w ay of th e sh e a th or o th e r te rtia ry conductors, we will d esignate b y Z i3.
If we consider th e cro sstalk betw een tw o coaxial lines of len g th , I, such th a t th e coaxial lines an d th e te rtia ry are electrically sh o rt, each coaxial line being te rm in a te d in its ch ara c teristic im pedance Z an d th e te rtia ry open a t each end, th e cro sstalk (near-end an d far-end being identical for such a length) is given by
Z \jl 2 Z '
If, now, we consider a case sim ilar except th a t th e te rtia ry is sh o rt- circuited a t each end, th e cro sstalk is th e above te rm plus th e effect of th e te rtia ry c u rre n t / 3, w hich, for u n it c u rre n t in th e d istu rb in g coaxial line, is given by
w here Z 33 is th e series im pedance of th e te rtia ry circu it per u n it length.
(
Z2
I \— *3 ) in th e dis- 2 Z Z33 /
tu rb e d coaxial line and th e to ta l crosstalk will be Z u l Z2i3l
~2Z ~ 2Z Z33 '
Z Z2
If we d esignate ~ b y X an d „ *3 by £, th en , for an electrically
¿ Z Z12Z 33
sh o rt length, X will rep resen t th e cro ssta lk per u n it len g th betw een tw o coaxial lines w ith th e te r tia r y open, an d X (1 — £) th e cro sstalk per u n it length w ith th e te r tia r y sh o rt-circu ited . In th e form ulas developed below these q u a n titie s will be found to be of fu n d am en tal im portance.
Tertiary Terminated in its Characteristic Impedance Far-End Crosstalk
F rom th e S chelkunoff-O darenko paper, th e sum of th e d ire c t far-end cro sstalk (eq. 19) a n d th e in d irect far-end cro sstalk (eq. 40) for an y length u n d e r these conditions gives th e to ta l far-end c ro sstalk F t, as
Z u l Z\ 3 r 2731
m
2Z 4 Z Z 3 1_ 732 — 72 (73 - 7 ) 2 ( 7 3 + T)2 J ’ { )
w here
Z 3 = c h a ra c te ris tic im p ed an ce of te r tia r y circu it,
7, 73 = p ro p a g a tio n c o n s ta n ts of coaxial lines a n d te r tia r y circ u it resp ectiv ely .
T h is m a y be re a rra n g ed a n d w ritte n (since Z 373 = Z 33) F t = Z n l Z \ z l Z 213
2 Z 2 Z Z 33 2 Z Z 33
X r h2 73
l 732 - 7 2 2
1 — g—(r¡—yU 1 — g - ( r 8+TU
= X [ / ( ! - £ ) - /£
732 - 7
(73 - 7 )2 _ i £73 ,2 "T- 9
X 1
— g—(73-7)*
+
(73 + 7 )2
1 — g—(rs+T)1 (73 + 7 ) 2
(2a)
(73 - 7 )2 (2b)
T h is form ula h as been found to be ap p licab le, w ith good ac c u ra cy , to th e ty p e s of coaxial cable w hich h av e been stu d ie d so far. T h e q u a n titie s X a n d X (1 — £) are d e te rm in e d from cro ssta lk m e a su re m e n ts on a sh o rt len g th , a n d th e p ro p a g a tio n c o n s ta n ts are of course re a d ily d e te rm in e d .
N ea r-E n d Crosstalk
A sim ilar ap p ro a c h to th e p ro b lem of th e n ear-en d c ro ssta lk N t w ith th e te r tia r y te rm in a te d in its c h a ra c te ristic im pedance, using e q u a tio n s (10) an d (32) of th e S chelkunoff-O darenko p a p er, gives
1 — £ £ 7
N , = X (1 - e~2yl)
2y 2 ( 7 s 2 - 7 2)
£7 2(732
> j L _ ( i + e- i y i _ 2e-<Tg+ir>») j (3) H ere, as in th e case of e q u a tio n (2b) above, th e c ro ssta lk m a y be co m p u te d rea d ily from cro ssta lk a n d im pedance m e a su re m e n ts on a s h o rt sam ple.
Interaction Crosstalk
Far-End Far-End and F ar-End N ea r-E n d
W e will consider th e in te ra c tio n c ro ssta lk betw een tw o ad jo in in g sections of len g th s I a n d I', resp ectiv ely , th e te r tia r y being c o n n e cted th ro u g h a t th e ju n c tio n , w ith no d isc o n tin u ity . T h e te r tia r y c u rre n t is(l) a t th e far end of a section of len g th I, for u n it sen d in g -en d c u rre n t, w ith th e te r tia r y te rm in a te d in its c h a ra c te ristic im p ed an ce, is re ad ily fo rm u la te d as
. . . . Z u e - t l -
C R O S S T A L K I N C O A X I A L C A B L E S 345
In th e adjoining section, w ith th e te rtia ry te rm in a te d in its c h a r
ac te ristic im pedance, th e te r tia r y c u rre n t i z i y ) will be given by w here y is th e d ista n c e m easured from th e ju n c tio n of th e tw o sections.
T h is te rtia ry c u rre n t fs(y) will produce a far-end c u rre n t in th e d istu rb e d coaxial of
Z213 e~y' - 4 Z Z 3 73 — 7
- e~y>1 f '' - 7 J o
e- y ,i/ e- y C '- ii) d y .
T h e equal-level far en d -far end in te ra c tio n cro sstalk FF, being this far-end c u rre n t divided by e~yil+l' \ m a y be o b tain e d as
F F = d - (5)
T h e near-end c u rre n t in th e d istu rb e d coaxial d u e to th e c u rre n t isiy) is given by
Z 13 e n1 - e ~ y f '
e~y^e~yydy.
- 7 Jo 4Z Z3 73
F rom th is th e equal-level far en d -n ear end in teractio n cro sstalk F N , being th is near-end c u rre n t divided b y e~yl, m a y be o b tain ed as
F N = 2- ^ y l \ 2) (1 - (6)
N ear-End N ear-End
T h e near-end te rtia ry c u rre n t in th e section of length I is sim ilarly form u lated as
*»(°) = § g l ~ e~ 'y3+y)‘ - 2Z 3 73 + 7 (7) T h e near-end near-end in te ractio n c ro sstalk N N is readily o b tain ed , in a fashion sim ilar to th a t outlined above for th e far-end near-end in te r
action crosstalk, as
N N = (! - ^ (T‘+7)I)(1 - e-<y+y»')- (8) Tertiary Short-Circuited
T h e general case of th e cro sstalk betw een coaxial lines of length I w ith th e te rtia ry sh o rt-circ u ite d a t each end m ay be a tta c k e d as follows.
A t a n y p o in t a t a d ista n ce x from th e sending end, th e v o ltag e g ra d ie n t along th e o u te r surface of th e o u te r coaxial co nductors, for u n it sending- en d cu rre n t, will be Z u e~rx. E ach differential elem ent, Z u e ~ r xd x , of th is v o ltag e d ro p will produce a c u rre n t in th e te rtia ry circ u it de-
te rm in e d b y th e im pedances, Z ' a n d Z " , of th e te r tia r y as seen in th e tw o d irectio n s from th is p o in t, th e se im p ed an ces being
a n d
Z ' = Z3 ta n h y 3x Z " = Z3 ta n h y 3(l — x)
(9) (10) re sp ectiv ely to w a rd a n d a w a y from th e sending end.
A t a n y o th e r p o in t a t a d ista n c e y from th e sen d in g en d , th e te r tia r y c u rre n t d u e to th e v o lta g e Z u e~yxd x will be given, for y > x, by
*s60 _ Z l:ie~yxd x cosh y 3(l — y)
~ Z ' + m cosh y 3(l - x) ( 1 1 ) F ro m th is th e tra n sfe r a d m itta n c e A ( x , y) betw een th e se tw o p o in ts is o b ta in e d as
A (x, y) = 1 cosh 73 (/ — y)
Z3 ta n h y 3x cosh y 3(l — x) + sinh y 3{l — x) Sim ilarly, for y < x, th is tra n sfe r a d m itta n c e is o b ta in e d as
A ( Xty) = cosh y & ________
Z3 sin h y 3x + cosh y 3x ta n h y 3(l — x) T h e te r tia r y c u rre n t, i3(x), is given b y
*«(*) = f J o
Z 13e yvA ( x , y ) d y
(12)
(13)
(14) from w hich we o b ta in
it(x ) = Z 13
2 Z3 sinh 731
X
1 c o s h y 3{l — x) + e~yx s i n h y 3l 73 + 7 — e r yX c o s h 7 3X
1 c o s h y 3(l — x) — e~yx s i n h y 3l 7 3 - 7 — e r yl c o s h y 3x
(15)
Far-End Crosstalk
T h e in d ire c t far-en d c ro ssta lk F / d u e to th is te r tia r y c u r re n t (eq. 15) is given b y
'/ = er' r Z ^ e-r('~*>dx
Jo 2 Z
13
2Z Z33
\ _ h l _ L 7 s 2 - 7 2
2 7 3 7 2 cosh 731 — cosh y I (7 32 - 72 ) 2 sinh 731
(16a)
(16b)
C R O S S T A L K I N C O A X I A L C A B L E S 347
Z I
If th is is com bined w ith th e d ire c t far-end cro sstalk , an d th e term s rearran g ed as in th e case of eq u a tio n (2b), th e to ta l far-end cro sstalk Fe is o b ta in e d as
= * [ / ( ! - Ö ~ l k
Y372 cosh 73/ — cosh 71
(7 a 2 ~ 7 2) 2 sinh 731 (17)
I t will be n o ted t h a t eq u a tio n s (2b) a n d (17) differ only in th e term s which are n o t p ro p o rtio n al to th e length a n d w hich th u s are of d e creasing im p o rtan ce as th e length becom es g re at.
N ea r-E n d Crosstalk
T h e in d irect near-end cro ssta lk N , due to th e te rtia ry c u rre n t iz{x) is given by
J 0 (18)
B y su b s titu tin g f3(x) from eq u a tio n (15) herein, a n d com bining th e re su lt w ith th e d irect n ear-end crosstalk,
Z „ 1 - e -^ i 2 Z 27
we o b tain th e to ta l near-end cro sstalk N s, which m ay be w ritten in th e form
N . = X
1 - e ~ i y l I ^ ^ , ? t 2(7 3 2 + 7 2) 27
?7372 (7 32 - 7 2) !
(7 32 - 7 2) :
(1 + e~2yl) cosh 731 — 2e~yl sinh 731
■ (19)
I I— I d e n t i c a l C o a x i a l L i n e s S y m m e t r i c a l l y P l a c e d w i t h R e s p e c t t o E a c h o f T w o D i s s i m i l a r T e r t i a r i e s
W e will now consider th e case of a n y n u m b er of identical coaxial lines w ith th e o u te r co n d u cto rs in co n tin u o u s electrical c o n ta c t an d sy m m etrica lly placed w ith resp ect to each of tw o dissim ilar te rtia rie s w ith coupling betw een them .
In an u n p u b lish ed m em o ran d u m by J. R io rd an , th e general form s are developed for th e c u rre n ts an d voltages in tw o parallel circuits h av in g u n iform ly d is trib u te d self an d m u tu a l im pedances an d a d m itta n c e s, w hen these circu its are su b jected to im pressed axial fields.
T h e se c u rre n ts ( I1 a n d I 2) a n d v o lta g es ( V x an d V 2) (th e su b sc rip ts ap p ly in g , of course, to th e resp e ctiv e te rtia rie s) are given b y th e coefficient a rra y ,
(fll + P J e - v i * (bi + Qi ) e^x (a, + P 2)e~y>* (b2 + (?*)«*•*
/
1 1
-1
V2
- Vih Vi - Vi
1
-1
V ! K i K i — V iK
2
— V iK2
v
2
— T]2K l — V2K 1 k2
k2
w here a 2, bi , a2 a n d b2 are c o n s ta n ts to be d e te rm in e d from th e b o u n d a ry co n d itio n s, an d
e l = 2x , ( i - „ „ ) / + r " h ) d x '< 2 0 )
Q , -2m + (21)
/ i an d f2 being th e im pressed fields along c ircu its 1 a n d 2 resp ectiv ely . If we consider th e tw o te r tia r y c irc u its as con sistin g of (1) th e o u te r coaxial co n d u c to rs in parallel w ith re tu rn b y te r tia r y p a th 1 a n d (2) te r tia r y p a th 1 — te r tia r y p a th 2, o n ly te r tia r y c irc u it 1 will be su b je c te d to an im pressed field. T h u s we will h a v e f2 = 0 a n d f i = Z u e ~ yx (for u n it sending-end c u rre n t in th e d is tu rb in g coaxial line), w here Z 13 is th e m u tu a l im p ed an ce, p e r u n it le n g th , b etw een a coaxial (in th e presence of th e o th e r p aralleling coaxials) a n d te r tia r y 1, a n d y is th e p ro p ag atio n c o n s ta n t, p e r u n it len g th , for th e coaxial circu it. T h e o th e r q u a n titie s in th is a r r a y are c irc u it p a ra m e te rs given as follows in te rm s of th e series im p ed an ces Z n , Z22 a n d Z\ 2 p er u n it len g th a n d a d m itta n c e s a n , a22 a n d ai2 p er u n it le n g th (su b sc rip ts 11 an d 22 for self im p edance or self a d m itta n c e of c ircu its 1 a n d 2 resp ectiv ely , a n d 12 for m u tu a ls ):
7 i 2
2 = a n Z n + a22Z22 -T 2«12Z 12 ± (( o ijZ h — a22Z22) 2
+ 4 ( a u Z i2 + a \2Z22) ( a \2Z \ \ a22Z i 2) ) :l23, (22) 7 l2 — Ö11Z11 — d \ 2Z 12
011Z12 + ö i 2Z22 (23)
722 d \2Z\ 2 — d22Z22
d l2Z \ l + d 22Z 12 (24)
-V Z \ \ + V iZ\ 2 7x
K j — >
7 l Û11 ~ ’720X2 (25)
js Z22 + V2Z12 72
iv 2 ' •
72] d22 — Vla U (26)
C R O S S T A L K I N C O A X I A L C A B L E S
F ro m eq u a tio n s (20) an d (21) above, we h av e P , — --- --- --- --- g(Yi—t)*
2X x(l — viVi)(yi — y)
Ox - — —--- g-(ri+Y>*
^ 2 X x( l - VlVt)(yi + 7 )
p . = _________Zl3V2 ■ _______- g(Ta-r)*
2 X 2(1 — 57m ) (7 2 — 7 )
0. = _______~ - Zl37?2_______ e-(7a+7)x.
2X2(1 — 171772) (7 2 + 7 ) If we d esig n ate
Z13__________, .
X x ( l - 7 7 1*7 2) ( 7 i 2 - 7 2) Y ^ an d
__________^ 1 3 7 7 2 , .
x 2( l - 77X77.) (7 2 2 - 7 2) y ^ we hav e
Zx = aie- '>'i;c — b\ey‘x + ri2a2e~y^x
— 7i2b2ey*x + (^x7x + \p 2V2y2) e - yx, (31) Z2 = via ie~ y'x ~ vibiV7iX + a2e~y‘x
— b2ey‘x + (1/777 x7i + 4/2J i)e ~ yx, (32) Fx = Kia.ie~y ix + K \b ie y'x — 171 K2a2e~y*x
— rnK2b2ey*x + ( i i K xy — i2K2rny)e~yx, (33) F2 = — r]2K ia ie ~ y'x — V2K1b1ey^x + K2a2e~y*x
+ K2b2ey*x — (ipiKiT]2y — \p2K2y)e~ yz. (34) Before proceeding w ith th e ap p licatio n of these re su lts to specific c ro sstalk problem s, we will estab lish ce rta in relatio n s w hich, as in th e sin g le -te rtia ry analysis, will be fu n d am en ta l in re la tin g cro sstalk m easu rem e n ts on sh o rt len g th s of cable to th e c ro sstalk to be expected in a longer length.
L e t us consider th e c ro ssta lk as m easu red on a sh o rt len g th u n d e r th e following tw o c o n d itio n s: (1) b o th te rtia rie s open a n d (2) te rtia ry 1 sh o rt-c ircu ited a t each end a n d te r tia r y 2 open. W e will desig n ate th e cro sstalk u n d e r condition (1) b y X I a n d u n d e r condition (2) by X l ( 1 — £). U n d er co n d itio n (2) th e te r tia r y c u rre n t (Zx) for u n it
£
c u rre n t in th e energized coaxial is given b y a n d th e in d irect cross-
¿11 2p 1
ta lk c u rre n t in th e d istu rb e d coaxial is th u s 13- , so t h a t we hav e LZjZj11
(27)
(28)
(29)
(30) 349
Interaction Crosstalk with One Tertiary Short-Circuited Far-End
F o r th e sake of sim p licity , a n d w ith no considerable loss of a p p lic a b ility , we will p o s tu la te th e re stric tio n t h a t eyA a n d eyA a re large co m p ared w ith eyl, w here I is th e len g th of th e section in w hich we are fo rm u la tin g th e te r tia r y c u rre n ts.
R eferring to e q u a tio n s (31) to (34), u n d e r th e abo v e re stric tio n s th e te rm s in volving e~y'x a n d e~y*x are negligible in th e region n e a r x = I a n d th u s in th is region
11 = — biey'x — r\2b2ey* + \_{pi — 2i) + V i( p i — 22) Je~yx, (36) 12 — — vib iey'x — b2ey*x + £vi(Pi — 2i) + (pi — 22) Je~yx, (37) Vi = K i h1e ycx - 771 K2b2ey*x
+ \ _ K i ( p i + q i ) — r \ \ K 2( p 2 + qi)~\e~yx, (3 8 ) V2 = — n2K \ b i e y'x + K2b2ey*x
+ C — m K i ( p i + 2i) + K 2(p 2 + qi)^\e~yx, (39) w here 2
p l = 2 ^ ( 1 - v Z ) ( y1 - 7) ’ (4 0 )
q i = 2 X i ( l - v i m ) ( y i + 7) ’ (4 1 )
, ________ E13272________
p2 ~ 2 K 2(l - , i „ ) (72 - 7 ) ’ (42)
_ _________________~ ^13??2________________ - ¡ s
Q2 2 K 2(1 - , ! , 2) ( 7 2 + 7 ) ’ ^ j
If, now, x is m easu re d from th e fa r end, a n d th e following s u b s titu tio n s a re m ad e as a m a tte r of co n v en ien ce: 3
ai — bie(yi+y)l, (44)
a 2 = b2e<-y*+y)l, (45)
e q u a tio n s (36) to (39), m u ltip lie d b y eyl so t h a t th e c u rre n ts a n d v o ltag es are given for u n it received c u rre n t in th e energized coaxial, becom e
I i — — aie~yix — ri2a 2e~y‘x + [_(pi — qi) + r\2( p2 — q2)~]eyx, (46) I i = — v i d i e - y^x — a 2e - y*x + [ , i ( £ i — qi) + ( p 2 — q2) ~]eyx, (47) 2 T he term s in volvin g e~yx here are identical w ith the corresponding term s in equations (31) to (34) except for the change in nom enclature, which in each case has been chosen so th at the a ’s and b’s will be given b y sim ple functions of the parameters employed (p ’s and q’s here: p ’s in the previous equations).
3 a\ and a 2 here have no relation to a\ and a 2 in equations (31) to (34).
Vi = K\a\er~<'x — y i K 2d2e~yie
+ L K i(p i + <?i) — r\iK2{p2 + qi)~\e'ix, (48) V2 = — i\2Kid\e~y^x + K2d2e~y^x
+ [ — V ïK iip i + qî) + K2(p2 + q2) ~\eyx. (49)
In th e section, of len g th a d ja c e n t to th e far end of th e energized section, th e im pressed fields are zero an d th u s (u n d er th e condition th a t
ey A' a n d ey A' are large co m p ared w ith u n ity ), using prim es to in d icate c u rre n ts an d voltages in th is region, w ith th e d ista n c e x ' ta k e n positive from x — I,
11 = a / e “ T i*' + t/2 a2e - y ^ ’, (50) 12 = ijiai'e-T»*' + a i ' e - * * ’, (51) V i’ = K i d i ' e - i * ' - v i K2a2'e-i>x\ (52) V 2' = — y 2K i d i c~ 7 iX -j- K 2d 2 6 ~ . (5 3 ) W ith te r tia r y 1 sh o rt-c irc u ite d , th e b o u n d a ry conditions to be satisfied are t h a t a t x = x ' = 0, V\ — V\ = 0 an d I2 = I 2 . F rom th ese b o u n d a ry co n d itio n s, we o b ta in
g l - - (, , + g l ) + ^ ( y . + w , (54)
- {p! + 3,) + Kk T + m 'K ? ’ (5S>
_ , V iK 2(yipi + p 2) /e ^
01 - - - K i + ^ K r ’ (56)
_ t _ K l i v i p l + pi)
a > - K i + • (5 7 )
T h e equal-level far-end far-end in te ra c tio n cro sstalk F F S is given b y 4
F F S = ^ § e v 1'
f
h ' e - y w - ^ d x ' . (58) LZj j oW ith I i as given b y e q u a tio n s (50), (56) an d (57), u n d er th e re stric tio n s we h a v e placed on y il' a n d y2V, we hav e
C R O S S T A L K I N C O A X I A L C A B L E S 351
F F . = Z2
4 Z (1 — y i y 2) ( K i + y i 2K 2)
X vi _j_ Vi _ i r l
i £ i (7i — 7) - ^2(72 — 7) J L 7 i — 7 7 2 — 7
T m K2
J L
7 i - (5 9 )4 As pointed out in the Schelkunoff-Odarenko paper in the section on mutual impedance, since a coaxial circuit is involved, the current distribution external to this circuit does not affect the mutual impedance, and hence the current I I contributes nothing to the crosstalk.
or, w ith th e use of e q u a tio n (35), X t-Z n
F F S =
2(1 — 771772) (-KT1 + r ,J K 2)
X r ÜL
U i ( 7 i - + V2
7 ) ^ 2 ( 7 2 - 7 ) _
riiK2 V2X 1 _ 7 i 7 72 — 7
( 6 0 )
T h e equal-level far-en d n ea r-e n d in te ra c tio n c ro ssta lk F N , is given b y
F N , = 1
ï f !
I ' e~ yx'd x '' (61)a n d u n d e r th e re stric tio n s we h a v e placed on y il' a n d y2V , we h a v e
F N , =
X
4Z(1 — 771772) ( K i + Vi2K 2)
[ ^ 1(7 1 - 7 ) + K 2( y 22- y ) ] [ X Ç Z n
V iK2 t]2K i 7 i T 7 72 + 7
2 (1 — 771172) {K i + t]i2K 2)
X r * _
L ^ i ( 7 i - + V2
7 ) K 2 (72 — 7 ) .
V iK2 _j_ t]2K 7 i + 7 72 +
7 ]
(62a)
(62b)
N ea r-E n d
U n d e r th e abo v e re stric tio n t h a t ey »*, ey ' l’t ey A a n d ey *1’ a re large c o m p ared w ith eyl, th e c u rre n ts a n d v o ltag e s in th e d is tu rb in g section n e a r x = 0 a re given b y e q u a tio n s (31) to (34) w ith th e ¿»-terms o m itte d , a n d th e c u rre n ts a n d v o ltag e s in th e d is tu rb e d section a d ja c e n t to th e sen d in g en d b y e q u a tio n s (50) to (53).
T h e b o u n d a ry co n d itio n s to be satisfied a re t h a t a t x = x ' — 0, V i = V\ = 0 a n d 12 = — I2 ■ F ro m th ese b o u n d a ry c o n d itio n s we o b ta in
a i = — ( p i + 31) riiK2(q2 + 17131) K \ + rii2K2
- ( * +
A i i ? r A 2 / _ V \K2(q2 + 77igi)
fll _ K ! + r,i2K2 ’ CL 2 —K i ( q2 + ?7igi)
K i + 17 12 K 2
(63)
(64)
(65)
(66)
C R O S S T A L K I N C O A X I A L C A B L E S 353
N N .
T h e near-end near-end in te ra c tio n cro ssta lk N N , is given by
= ^ r W e 2 Z Jo 11'e~yxdx
Z h z
X
4Z(1 — + Tli2K 2)
Vi , V2
_ K 1 (7 1 + 7 ) K 2(72 + 7 ) . ________ X $ Z n _________
2(1 — V1V2) { K \ + 7] \2K 2)
7]\K2 r)2K \ 7 i + 7 72 + 7
X »11 | V2 Vi K 2 1
1 K i ( 7 1 + 7 ) K 2( y 2 + 7 ) 7 1 + 7 T 2 7 ^
(67a)
(67b)
(67c) N ear-E nd Crosstalk with One Tertiary Short-Circuited
A lth o u g h th e d e riv a tio n of th e form ula for n ear-en d c ro ssta lk N s w ith one te r tia r y sh o rt-circ u ite d is to o long to be included here, it seem s ad v isab le to give th is form ula w ith o u t d e riv a tio n . U n d er th e above m en tio n ed re stric tio n t h a t ey d an d ey *1 are large co m p ared w ith eyl,
N s = N t + N N - N N S, (68) w here
N t = n ear-end crosstalk, te rtia rie s term in a te d ,
N N = n ear-end n ear-end in te ra c tio n cro sstalk betw een tw o ad jo in in g lengths, te rtia rie s w ith no d isc o n tin u ity , N N a = near-end n ear-end in te ra c tio n cro sstalk betw een tw o
ad jo in in g lengths w ith te rtia ry 1 sh o rt-circu ited a t th e ju n ctio n .
T h e first tw o te rm s ( N t an d N N ) m ay, w ith th e ty p es of cable stu d ied so far, be d eterm in ed w ith sa tisfa c to ry accu ra cy from th e single- te r tia r y analysis. In such a case, th e form ulas given herein are suffi
cien t for co m p u tin g th e near-end cro sstalk w ith one te rtia ry sh o rt- circu ited .
I l l —C o m p a r i s o n o f C o m p u t e d C r o s s t a l k w i t h M e a s u r e d V a l u e s
W ith 72-ft. a n d 145-ft. sam ples of th e tw in coaxial cable described in th e com panion p a p e r b y M essrs. B o o th an d O darenko, cro sstalk an d im pedance m e asu rem en ts were m ad e in th e la b o ra to ry , a t frequencies from 50 kc to 300 kc, th e sh e a th an d q u ad s in parallel being considered as p ro v id in g a single te rtia ry , t h a t is, as being connected to g e th e r a t sh o rt in terv als.
T h e far-e n d c ro ssta lk for a len g th of 5 m iles w as c o m p u te d from th ese la b o ra to ry m e a su re m e n ts a n d in F ig. 1 th e re su lts are c o m p ared w ith m e a su re m e n ts on th is len g th , th e c ro ssta lk in e ith e r case being p ra c tic a lly th e sam e w h e th e r th e te r tia r y w as te rm in a te d o r s h o rt-c irc u ite d .
95
100
105
110
120
125
130
135
140
145
\ \
A -C O M P U T Ed' 1 FAR-END,TERTIARY TERMINATED B - M E A S U R E D j OR SHORT CIRCUITED
C ( A N D o ) — COMPUTED \ NEAR-END,TERTIARY T E R - D — MEASURED FROM WEST ENDJMINATED
N X N
A B
\ \ N \
\ \ / \ / V"
/ c \
1 /
\ - / i\¡
\V
%\ '----\
\\
( ----\
\
\
1■
•
!
(
50 60 70 80 90 100 150 200
FREQUENCY IN KILOCYCLES PER SECOND
250 30Q
Fig. 1— Far-end crosstalk and near-end crosstalk for 5-m ile length.
In F igs. 1 a n d 2, th e co m p u te d n e a r-en d c ro ssta lk for a le n g th of 5 m iles is co m p ared w ith re p re s e n ta tiv e m e a su re m e n ts on th e abo v e m en tio n e d tw in coaxial cable. F ig u re 1 show s th is co m p ariso n w ith th e te r tia r y te rm in a te d a n d F ig. 2 w ith th e te r tia r y sh o rt-c irc u ite d .
C R O S S T A L K I N C O A X I A L C A B L E S 355
T h e assu m p tio n of u n ifo rm ity of th e coaxial lines, as reg ard s tra n sfe r im pedances, a n d of th e te rtia ry circu its as reg ard s tran sm issio n c h a r
acteristics, is a m ore serious restrictio n in th e c o m p u ta tio n of n ear-end cro sstalk th a n in th e case of far-end cro sstalk . E v en for long len g th s of cable, th e n ear-end c ro sstalk is d e te rm in e d a lm o st e n tire ly b y th e c ro sstalk b eh av io r of a rela tiv e ly sh o rt le n g th of th e cable n ear th e sending end, w hereas th e av erag e c ro ssta lk c h a ra c teristic s d e te rm in e
Fig. 2—-Near-end crosstalk for 5-mile length, tertiary short-circuited.
th e far-end cro sstalk for a long length of cable. T h u s, from m easu re m en ts on re p resen tativ e sh o rt lengths, th e far-end cro sstalk for a long length m a y generally be co m p u ted m ore a c c u ra tely th a n can th e n e a r
end crosstalk.
Sim ilarly, th e various ty p es of in te ra c tio n cro sstalk depend largely upon th e cro sstalk b eh av io r of re lativ ely sh o rt lengths of th e cable near th e ju n ctio n . In F ig. 3, th e far-end far-end in teractio n cro sstalk has been chosen as an illu stratio n of th e correlation w hich has been o b tain ed betw een co m p u ted in te ra c tio n cro sstalk for th e above m en
tio n e d tw in coaxial cable a n d th e m easu red in te ra c tio n c ro ssta lk T h e cu rv es in F ig. 3 are for eq u al len g th s e ith e r of 3000 ft. o r 12,000 ft.
In th e case of th e m e asu red valu es, th e ju n c tio n p o in t of th e tw o sec
tio n s w as n o t th e sam e for th e se tw o len g th s. A lth o u g h th e a g re e m e n t betw een c a lc u la ted a n d m easu re d v a lu es is o n ly fair, th e sp re a d in th e e x p e rim e n tal re su lts for th ese tw o cases, w hich for u n ifo rm cable w ould be slig h t, is a b o u t th e sam e as th e sp re a d betw een c a lc u la te d a n d m easu red values.
100
105
110 if) UJ5 Q Z
*_l
< 120 ii)if) OCL
U 125
130
135
5 0 6 0 7 0 8 0 9 0 100 150 2 0 0 2 5 0 3 0 0
FREQUENCY IN KILOCYCLES PER SECOND
Fig. 3— Far-end far-end interaction crosstalk betw een tw o equal lengths.
T h e tw o -te rtia ry form ulas h a v e so far been a p p lied o n ly to one ty p e of cable w ith fo u r coaxial lines a n d a lay er of p a p e r-in s u la te d p a irs.
T h e longest len g th of th is ty p e of cable on w hich c ro ssta lk m e a su re m e n ts h a v e been m ad e is 1900 ft. T h e v a rio u s ty p e s of in te ra c tio n c ro ssta lk w ith one te r tia r y sh o rt-c irc u ite d , as co m p u te d from th e fo rm u las given ab o v e, agree ro u g h ly w ith th e m ea su re d in te ra c tio n c ro ssta lk u n d e r th is sam e c o n d itio n . H ow ever, th e re stric tio n t h a t th e te r tia r y circ u its in volved are e le c tric ally long, as p o s tu la te d in d eriv in g th e in te ra c tio n c ro ssta lk fo rm u las for th is case, is n o t sa tisfie d ,
C R O S S T A L K I N C O A X I A L C A B L E S 357
a n d com parisons of th e calcu lated an d m easu red values are n o t v ery significant.
I t m ay be rem ark ed th a t th e ap p lic a tio n of th e s in g le -te rtia ry a n a ly sis to all cases in w hich th e tw o te rtia rie s w ere tre a te d alike (eith er te rm in a te d or sh o rt-circ u ite d ) gave v e ry s a tisfa c to ry a g reem en t b e
tw een c o m p u ted an d m easu red c ro sstalk for th is 1900 ft. length of 4-coaxial cable.
By R. P. BOOTH and T. M. ODARENKO
The available literature on crosstalk between coaxial conductors in contact makes it clear th a t the presence of any other conducting m aterial in continuous or frequent contact with the coaxial outer conductors simply reduces the coupling per unit length w ithout altering the law of crosstalk sum m ation with length.
W hen the conducting m aterial is insulated from th e coaxials, as in the case of quads and sheath in coaxial cables, the situation is more complicated. Instead of simply reducing the coupling per unit length the quads and sheath, with the outer conductors for a return, provide a tertiary circuit in which interaction crosstalk can take place between elem entary line sections. The sum m ation with length for this type of crosstalk is quite different from th a t between two coaxials in contact and therefore the combined sum m ation is obviously more involved.
Tests on sections of a five-mile length of coaxial cable were made a t Princeton, New Jersey, in the latter p art of 1937 and early in 1938 in order to obtain experimental verification of the m anner in which the quads and sheath affect crosstalk sum m ation with length.
I t is shown th a t the crosstalk component due to the presence of the sheath and quads opposes the component which is present between two coaxials in free space so th a t the resultant crosstalk is con
siderably lower than would be computed ignoring the tertiary effects.
In t r o d u c t i o n
In sp ite of th e g eo m etrical a n d electrical s y m m e try of th e coaxial c irc u it a n d th e excellent shielding p ro p e rtie s of th e o u te r c o n d u c to r, a p a r t of th e elec tro m a g n etic en erg y escapes from th e c irc u it th ro u g h th e o u te r c o n d u c to r a n d se ts u p a n e le c tro m a g n etic field in th e sp ace a ro u n d it. A n y circu it, be it even a n o th e r coaxial p lac ed in th is field will ab so rb a p a r t of th e e n erg y s to re d in th e field a n d d e liv e r it to th e te rm in a ls of th e c irc u it in th e form of a n u n w a n te d o r in te rfe rin g c u rre n t— th e cro ssta lk c u rre n t. T h e m a g n itu d e of th is c ro ssta lk c u rre n t d ep e n d s on a v a rie ty of fa c to rs, such a s th e p h y sical c h a r a c te r istics of th e co n d u c to rs a n d of th e in te rv e n in g space, th e freq u en cy a n d th e len g th of th e c ircu it.
E xpressions for tw o im p o rta n t cases o f c ro ssta lk b etw een tw o coaxial circ u its in free space, n am ely , th e so-called “ d i r e c t ” c ro ssta lk w ith th e o u te r c o n d u c to rs in c o n tin u o u s c o n ta c t a n d th e “ in d ir e c t” c ro ssta lk w ith th e o u te r c o n d u c to rs in su la te d from each o th e r, w ere d e te rm in e d
358
C R O S S T A L K B E T W E E N C O A X I A L CON DUCT ORS 359 a n d discussed in a prev io u sly published p a p e r.1 I t w as show n th ere t h a t th e d ire c t far-en d cro ssta lk is d ire c tly p ro p o rtio n al to I an d th e d ire c t n ear-en d c ro ssta lk is p ro p o rtio n al to
1 - (T2yl 2y ’
w here I is th e length an d y is th e p ro p ag atio n c o n s ta n t of e ith e r coaxial u n it. T h e in d irect cro sstalk was shown to be a m ore com plicated function of th e length.
T h e p re se n t p a p e r ex ten d s th is earlier w ork to include th e case w here th e coaxials are enclosed in a com m on sh e a th or, in th e general case, paralleled by a n y co n d u ctin g m a te ria l sy m m etrically d isp o sed.2 W hen th is co n d u ctin g m ateria l is in tro d u ced in th e neighborhood of tw o coaxials in c o n ta c t th e conditions for cro sstalk p ro d u ctio n are n a tu ra lly changed from those existing in free space. If th e m aterial is uniform ly d is trib u te d along th e coaxials a n d is in co n tin u o u s or fre q u e n t c o n ta c t w ith th e o u te r co n d u cto rs th e su m m atio n of crosstalk w ith len g th is th e sam e as before b u t th e m a g n itu d e is reduced. T his reduction is d u e to th e fa c t t h a t p a rt of th e c u rre n t form erly flowing on th e d istu rb e d o u te r co n d u cto r now flows on th e new co nducting m aterial in stead , th u s reducing th e d ire c t cro sstalk coupling per u n it length.
In m ost cables, th e coaxial o u te r c o n d u cto rs are in c o n ta c t b u t th e o th e r con d u ctin g m a terial (sh eath an d quads) is in su lated from th e o u te r c o nductors. T h e q u ad s m u st obviously be in su lated for norm al use a n d th e sh e a th is k e p t in su late d except a t th e ends of a re p e a te r section in o rd er to p e rm it th e use of in su latin g jo in ts for electrolysis p re v en tio n w here req u ired . T h is m ate rial th u s p ro vides an e x tra tran sm issio n circu it, or te r tia r y circuit, in w hich te r tia r y c u rre n ts can be p ro p a g a te d u p a n d dow n th e line. In such a case th e resu ltin g c ro sstalk in a n y len g th consists of b o th th e d ire c t cro ssta lk betw een th e c o n ta c tin g coaxials a n d th e in d irect cro sstalk v ia th e o u te r c o n d u cto r-sh eath a n d q u ad te rtia ry circuit. T h e gen
eral form ulas given in th e S chelkunoff-O darenko p ap e r ap p ly for these com ponents. Since th e tw o com ponents follow different laws regarding su m m atio n w ith length th e re s u lta n t su m m atio n is q u ite com plicated except for v e ry sh o rt or v e ry long lengths.
T h e s tu d y of th e te rtia ry effects on c ro ssta lk su m m atio n is th e m ain c o n trib u tio n of th is p ap er to cro sstalk th eo ry . E m p h asis will be placed on th e d ev elo p m en t of a sim ple physical p ictu re w hich will help one to
1 Schelkunoff-Odarenko paper in B ell S ys. Tech. Jour., April, 1937.
2 In the interim between our tests and this publication a paper by H. Kaden concerning this general subject was published in the Europäischer Fernsprechdienst, no. 50, October, 1938, pp. 366-373.
visualize clearly th e influence of th e te r tia r y c irc u its in th e su m m a tio n process. T o pro d u ce such a p ic tu re a c e rta in a m o u n t of review of th e general c ro ssta lk p ro b lem will be n ecessary. T h is is u n d e rta k e n in P a r t I of th is p a p e r.
P a r t II is d e v o te d m a in ly to th e p re se n ta tio n of te s t d a ta ta k e n in N o v em b er a n d D ecem ber, 1937, J a n u a r y a n d F e b ru a ry , 1938 on sec
tio n s of a five-m ile len g th of a tw in coaxial cable n e a r P rin c e to n . T h ese d a ta confirm a n d g ra p h ically illu s tra te c e rta in re la tio n sh ip s dev elo p ed in P a r t I. In a d d itio n th e y p ro v id e in fo rm a tio n on th e te n d e n c y of te r tia r y c irc u its to c o m p licate th e effectiveness of tr a n s p o sitio n s a n d show how in te ra c tio n c ro sstalk ta k e s place a ro u n d re p e a te rs v ia th e te r tia r y c ircu its.
P A R T I— T H E O R Y
In a n y series of c ro ssta lk te s ts on sh o rt len g th s of p aired o r q u a d d e d cable w here th e p ro b lem of com bining a n u m b e r of such le n g th s is co ncerned it has gen erally been th e p ra c tic e to te rm in a te b o th th e te s t c ircu its a n d im p o rta n t te r tia r y circ u its in c h a ra c te ris tic im p ed an ce.
U n d er such a co n d itio n th e norm al influence of all circ u its in th e p ro d u c tio n of cro ssta lk w ith in each s h o rt section is p ro v id ed for a n d th e su m m a tio n process, in clu d in g in te ra c tio n betw een successive sections, can be stu d ie d u n d e r a c tu a l line co n d itio n s. T h is is a g en eral m e th o d ap p licab le to a n y ty p e of coupling a n d w as a d o p te d for th e P rin c e to n in v e stig a tio n . T h e effect of d isc o n tin u itie s such as sh o rt-c irc u ite d te rtia rie s a t th e ex trem e ends of a re p e a te r sectio n can be re a d ily h a n d led m a th e m a tic a lly as co rre ctio n te rm s d u e to “ en d effe c t.”
T o sim plify th e p re se n ta tio n of th e fa c to rs in v o lv ed , th e discussion in th is section will be confined m ain ly to th e case of far-e n d cro ssta lk . In a tw in coaxial cable w here th e tran sm issio n in th e tw o u n its is in o p p o site d irectio n s th e re a c tu a lly ex ists no far-en d c ro ssta lk p ro b lem since o n ly ta lk e r echo, a n e a r-en d c ro ssta lk p h en o m en o n , is in v o lv e d.3 In m u lti-u n it cable, how ever, th e re will be fa r-e n d c ro ssta lk betw een d ifferen t sy stem s. Since th is ty p e of c ro s s ta lk te n d s to in crease d ire c tly w ith th e n u m b e r of re p e a te r sectio n s it is im p o r ta n t to u n d e rs ta n d its n a tu re th o ro u g h ly . M o reo v er, in a s tu d y of fu n d a m en ta ls it is possible to av o id c e rta in co m p lic a tio n s n o t esse n tia l to an u n d e rsta n d in g of th e p ro b lem b y in v e stig a tin g far-en d r a th e r th a n n ea r-e n d cro sstalk .
T o p re se n t a clear p ic tu re of th e p h y sical m ean in g of som e of th e fo rth co m in g m a th e m a tic a l expressions th e ir d e riv a tio n s will be a p -
3 T his statem ent m ay not hold if the repeater im pedances fail to m atch the line im pedance since in that case the far-end crosstalk can be reflected and appear as near-end crosstalk.
C R O S S T A L K B E T W E E N C O A X I A L CON D UC T OR S 361 proached in as e le m e n ta ry a fashion as possible. In o rd er to do this we shall s ta r t w ith th e sim ple a rra n g e m e n t of tw o coaxial co n d u cto rs in free space, a case a lre a d y covered in previous p ap ers. T o th e c ro sstalk eq u a tio n s covering th is case will th en be ad d ed term s to allow for th e effects of q u ad s a n d sh e a th . In all t h a t follows in P a r t I th e q u ad s a n d sh e a th will be considered as one u n it referred to as th e
“ s h e a th .” T h is is a good ap p ro x im a tio n as will be show n in P a r t II.
T h e conception of tw o in d e p en d en t cro sstalk c o m p o n en ts— a d ire c t or tra n sv e rse co m p o n en t betw een coaxials in c o n ta c t an d an in d irect o r in te ra c tio n c o m p o n en t v ia th e sh e a th te rtia ry circu it— is n o t neces
sa ry for th e solution of th e problem . I t is preserved here, how ever, as offering a fam iliar an d m uch sim pler ap p ro a c h to a clear u n d e rsta n d in g of th e processes involved in cro sstalk su m m atio n w ith length.
Fa r- En d Cr o s s t a l k
C onsider first an ele m e n ta ry section, dl, of a long single coaxial in free space as in d icated in S ketch (a) of Fig. 1. If th e c u rre n t a t th is p o in t in th e ce n te r co n d u cto r is I \ th e c u rre n t in th e o u te r co n d u cto r is p ra c tic a lly — I x since th e re is no o th e r re tu rn p a th (except th ro u g h th e1 air dielectric w hich offers a high im pedance especially a t th e lower b ro a d -b an d frequencies considered here). U sing Schelkunoff’s nom en cla tu re we m ay s ta te th a t an open-circuit v o ltag e equal to ex = I i Z aSdl is developed on th e o u te r surface of th e o u te r coaxial conductor. T h e te rm Z aS rep re se n ts th e surface tra n sfe r im pedance (m u tu al im pedance) p er u n it len g th betw een th e in n er a n d o u te r surfaces of th e o u te r coaxial co n d u cto r.
N ow suppose t h a t we place a n o th e r long coaxial parallel to th e first one an d , for g en erality , in su late d from it as show n by S k etch (b) of Fig. 1. T h e o p en -circu it v o ltag e ex on length dl of th e first coaxial o u te r co n d u c to r will now cause c u rre n t to flow in th e in term ed iate circ u it com posed of th e tw o o u te r co nductors. T h e p a ra m e te rs of this circu it are y3 an d Z3 as show n on th e sk etch . In re tu rn in g on th e second coaxial o u te r c o n d u cto r th is c u rre n t causes c ro sstalk in to th e second coaxial circu it.
I t is co n v en ien t a t th is p o in t to replace th e original im pressed voltage ei b y th e se t of e m f’s show n in S ketch (c) of Fig. 1. T h e insertion of equal a n d opp o site v o ltag es ei/2 on th e o u te r surface of th e d istu rb ed coaxial o u te r co n d u c to r does n o t change conditions b u t enables us to consider ce rta in effects se p a ra te ly . T h e first effect to be considered is t h a t d u e to th e p a ir of equal a n d opp o site v o ltag es ei/2 in th e loop com posed of th e tw o coaxial o u te r c o n d u cto rs. T hese voltages com bine to form a “ b a la n c e d ” v o ltag e e\ w hich te n d s to d riv e c u rre n t
a ro u n d th e b alan c ed circ u it com posed of th e tw o o u te r co n d u c to rs.
F o r th e p re se n t we shall n o t consider th e v o lta g es Ci/2 w hich a re in th e sam e d ire c tio n in th e o u te r co n d u cto rs.
T h e c u rr e n t in th e “ b a la n c e d ” in te rm e d ia te c irc u it of c h a ra c te ris tic im p ed an ce Z3 a n d p ro p a g a tio n c o n s ta n t 73 d u e to th e b a la n c e d v o lta g e ei in th e e le m e n ta ry len g th dl is i3 = ei/2Z 3. T h is c u rre n t flowing along th e o u te r coaxial co n d u c to r of th e d is tu rb e d c irc u it p ro d u ces a v o ltag e e-i = i3Z apdl on th e in n e r surface of th is o u te r c o n d u c to r a n d th is v o ltag e in tu rn causes a c u rre n t i ia in th e d is tu rb e d coaxial c ircu it
- d l -
Z zz i
COAXIAL I
-e! = I| za p d i
ta)
r 3>z 3 I
COAXIAL I
ei I y3>z3
COAXIAL 2
(b)
JZ
SHEATH
Fig. 1.— Coaxial crosstalk schem atics.
equal to e2/2Z , w here Z is th e coaxial c h a ra c te ris tic im p e d a n c e.4 In a long line o th e r e le m e n ta ry le n g th s of th e d is tu rb e d coaxial are also affected b y i 3 because of its p ro p a g a tio n along th e in te rm e d ia te c ircu it.
(T h is cro ssta lk b y w a y of a te r tia r y c irc u it from one le n g th in to a n o th e r is know n as in d ire c t or “ in te ra c tio n c r o s s ta lk ” a n d b ecau se of its presence th e su m m a tio n of cro ssta lk w ith le n g th is n o t a sim ple fu n c tio n of le n g th even for sy ste m a tic co upling such as occurs w ith coaxials.) T h is is a cro ssta lk case for w hich th e g en eral so lu tio n is a lre a d y
4 T he subscript “ a ” in i 2a relates this current to the so-called “ mode a ” current used by Carson and H oyt in their paper entitled “ Propagation of Periodic Currents Over a System of Parallel W ires,” B ell S y s. Tech. J o u r., July, 1927.
C R O S S T A L K B E T W E E N C O A X I A L C ONDUC T ORS 363 av ailab le. W hen th e effects are in te g ra te d it is found t h a t th e far-end c ro ssta lk is q u ite a co m p licated function of length an d of th e te r tia r y a n d coaxial p ro p ag atio n an d im p edance c h a ra c te ristic s.6 H ow ever, if th e coaxial u n its a re in a c tu a l c o n ta c t as in th e case of th e coaxial cable to be considered here, th e form ula for th e far-end c ro sstalk F3 expressed as a c u rre n t ra tio is q u ite sim ple, nam ely,
F3 = 2ZZ73 ’ l '
w here Z33 = Z3y3 is th e series im pedance per u n it len g th of th e circu it com posed of one coaxial o u te r co n d u c to r w ith re tu rn on th e o th er.
T h u s, for th is co m p o n en t, th e far-en d c ro sstalk is d ire c tly p ro p o rtio n al to len g th . T h is sim ple re latio n re su lts from th e fa c t t h a t th e in te r
m ed iate circu it, being co n tin u o u sly sh o rted , h as such high a tte n u a tio n t h a t no in te ra c tio n cro ssta lk betw een ele m e n ta ry len g th s can exist.
W e shall now consider th e c ro sstalk c o n trib u tio n due to th e longi
tu d in a l v o ltag e ei/ 2 ac tin g along b o th coaxial o u te r co n d u c to rs in p arallel. Suppose t h a t a sh e a th is placed sy m m etrically aro u n d th e tw o coaxials b u t in su lated from th e m as show n in S k etch (d) of Fig. 1.
T h e lo n g itu d in al v o ltag e sends a c u rre n t aro u n d th e circu it com posed of th e tw o parallel o u te r c o n d u cto rs w ith sh e a th re tu rn equal to n = ei/(2 )(2 Z 4), w here Z4 is th e c h a ra c te ristic im pedance of this circu it. H alf of th is lo n g itu d in al c u rre n t flows on th e d istu rb e d coaxial o u te r co n d u c to r in opposition to th e balan ced c u rre n t i3 flowing there.
Follow ing previous p rocedure it can be show n t h a t in th e ele m e n ta ry length a c ro ssta lk c u rre n t izc = iJZcpdl/iZ will flow in th e d istu rb e d coaxial c irc u it.6 O th e r e le m e n ta ry len g th s are also affected b y u th u s producing in te ra c tio n cro sstalk . W hen th e effects are in te g ra te d over a len g th I th e far-end cro sstalk for th is c o m p o n en t is found to be as follows:
Fa = —
z a?
r2i
/ 2 ( y 42 + y 2) _ € ~ ( r » ~ r ) i _ e-(.yt+ y)i
\ ( y42 — y 2) 2 (74 — y ) 2 ( y4 + y ) 2 (2) w here y4 is th e p ro p ag a tio n c o n s ta n t of th e sh e a th -o u te r co n d u cto r circu it. If th e sh e a th is in a c tu a l c o n ta c t w ith th e coaxial u n its th e
5 See equation (40) in the Schelkunoff-Odarenko paper in B ell S ys. Tech. Jour., April, 1937.
6 T he subscript “ c ” in i 2c relates this current to the “ mode c ” current used by Carson and H oyt in their paper of July, 1927.
