THE CAPACITANCE OF CIRCUITS IN ACABLE WITH TWISTED … Bound... · THE CAPACITANCE OF CIRCUITS IN ACABLE WITH TWISTED QUADS ... based on more solid foundations and brings anumber

R969 Philips Res. Repts 32, 297-321, 1977

THE CAPACITANCE OF CIRCUITS IN ACABLEWITH TWISTED QUADS

by V. BELEVITCH *), R. R. WILSON **) and G. C. GROENENDAAL ***)

AbstractApproximate theoretical formulae are established for the electrostaticcapacitance per unit length of a side or phantom circuit in a star quadcable, taking into account the proximity effect and the longitudinalcharge separation produced by twisting, in the six surrounding quads.The theory essentially follows the original approach of Martin 2) but isbased on more solid foundations and brings a number of important cor-rections to Martin's results.

1. Introduetion

In a cable with parallel wires, the electrostatic capacitance C per unit lengthof any circuit and its h.f. inductance Loo are related by

27t1>c=-h '

(1)

P,Loo =-h,

27t(2)

where h is a dimensionless quantity depending only on the cable geometry. Ina star quad cable, the geometrical parameters are the wire radius a, the dis-tance 2b between the axes of opposite wires in a quad and the distance Dbetween the centres of two adjacent quads. There are thus two dimensionlessratios

aw=-

b '(3)

bs=-.

D(4)

It is convenient to write the expression of h as a function of wand s in theform

h =g-p-t, (5)

*) MBLE Research Laboratory, Brussels, Belgium.**) NKF Kabel, Delft, The Netherlands.

***) Philips Research Laboratories, Eindhoven, The Netherlands.

298 V. BELEVITCH. R. R. WILSON AND G. C. GROENENDAAL

where g is the value for thin wires, i.e.

2gs = 21n~,

w(6)

1gF =ln-

w(7)

for a side (S) or a phantom (F) circuit, respectively. In (5), which will be usedwith subscripts S or F everywhere, P is the proximity effect within the centralquad and t the effect due to the six surrounding quads forming a regularhexagon.The expansions

Ps = t w2 + ~J w4 + l~w6 + . . . , (8)

(9)

have been obtained by Belevitch 1) in terms of the notation

w6=-.

2(10)

In section 2 and appendix A, the effect of the six surrounding quads is evaluatedas the sum of 24 independent effects per wire (which is correct to order w2)

due to the field of the central circuit, but the dipolar (side) or quadripolar(phantom) component of the field at large distance is computed by solving thefull Belevitch equations for the central quad to order w4• The resulting fieldsdiffer from the ones of a thin dipole or quadripole by the field factors

(11)

(12)

and the resulting proximity corrections are

(13)

(14)

At this stage our results differ from the ones of Martin 2) because he onlyuses the first terms (in w2) of (8) and (9) and because he replaces Is and IFby 1 in (13) and (14).

THE CAPACITANCE OF CIRCUITS IN A CABLE WITH TWISTED QUADS 299

In a cable with twisted conductors, the inductance (2) is only very slightlyaltered by the spiral effect whereas (1) is replaced by

2nl>C=_· h'=h-u

h' ' ,

where the term u originates from the interaction of the central quad with thecage formed by all other circuits. The rest of this paper is devoted to theevaluation of u; we now state the results and point the differences with respectto Martin.For the side circuit at the centre we use the notations

One then has

where hA is defined by (5) with subscripts A. For the phantom at the centre,the notations are

tB = 5 w2 S2 (1 + 13265 S2)

and PB = PA. One then has

In Martin's formulae, all h's in the denominators of (19) and (22) are approx-imated by the corresponding g's. In (19), Is 4 is erroneously replaced by IS2 inthe second term, and the last two terms are completely overlooked. Moreover,only the first two terms are kept in (11) and IS2 is further replaced by theapproximation 1 - 3w2; a similar remark holds for (12). Finally, our ex-pressions (19) and (22) are evaluated up to terms in s\ while Martin has aterm in S6 which is only one among many overlooked terms of the sameorder.

(15)

(16)

(17)

(18)

(20)

(21)

(22)


2. The proximity effect

At large distance (ro »b) the electrostatic potentialof the line of :fig.1 withunit charges per unit length is

b(/Jo = --- cos eo

7tero(23)

and is independent of the wire radius a for thin conductors. For a thick sidecircuit in a quad, it is proved in appendix A that (23) is replaced by

(24)

where j', is (11).We now consider a second quad (:fig.2) for which it is convenient to take

temporarily a different diameter (c :;6 b). If the side circuit 1-3 of :fig. 2 isexcited, the second quad lies in the :fieldcorresponding to the potential (24).In the :first approximation (where the effects of all wires are additive), the

p

+oI.

Fig. 1. Two parallel wires.

5

2o o

7Ijl--------------~

Xo 03 4

Fig. 2. Configuration of two different quads.


correction due to the proximity effect of wire 5 alone is *)

If one replaces

1 exp (-Ï7p) [ c . b . J-1- = 1+ - exp (Iy) - - exp (1(3)DS1 D D D

by its Taylor expansion in powers of bl D and t[D up to the third order, anduses the same expression with b changed into -b for 11Ds3, (25) is, correctlyup to the fourth order,

4a2 b2 fS2 (4C 4c2 6c2 2b2)1 - - cos y + - +- cos 2y +- cos 2(3 .

D4 D D2 D2 D2

The odd term in c disappears by adding the effect of wire 7 and the term incos 2y disappears by adding the effect of the other pair 6-8 for which y isy + Te12. Finally, since (3 + 1jJ is fixed in fig. 2 whereas 1jJ increases anti-clock-wise by Tel3 in the hexagon, (3 decreases by Te13, and the term in cos 2(3 dis-appears upon summation over the six quads. By making c = b in the result,one obtains (13). It is meaningless to improve the accuracy to higher powersof s because the next terms involve the angle y which depends on the individualtwisting pitch of each of the six quads.At large distance the electrostatic potentialof a thin quad in phantom mode

is of the quadripolar type, proportional to (blro)2 cos 280• The correctionfactor (12) for a thick quad is derived in appendix A. The proximity correctionfor wire 5 is

fF2 a21 1 1 1 1 12

•

-4- DS1 + DS3 - DS2 - DS4

(6c 9c2 12c2 )= 1 - - cos y + - + -- cos 2y .D D2 D2

It is correct up to the sixth order and has been obtained by using the Taylorexpansions of the inverse distances up to the fourth order. The odd term in cdisappears by adding the effect of wire 7, the term in cos 2y disappears byadding the effect of the second pair and the total for six quads is (14) aftermaking c =b.

*) Except for the factor fS2, this correction on Loc" hence on h of (2), results from eqs (177)and (179) of ref. 1 when the Bessel functions are replaced by their asymptotic values.The factor fS2 is justified by the fact that the correction is proportional to the square ofthe field.

(25)


When the wires of a quad touch each other, i.e. for w = I/V2, all capac-itances must become infinite, independently of the distance of the surroundingquads. One must thus have separately g - P = 0 and t = 0 in (5) and thelatter must be produced by the disappearance of the field, i.e. by / = O. Infigures 3 and 4 we show gs - Ps and/s2 as functions of w with truncations atvarious orders in (8) and (11). In figure 4, Martin's approximationjj," = I-3w2

is also indicated. The results are very similar for the phantom mode.

3. The longitudinal charge separation

For a system of n parallel conductors, there are n - 1 potential differencesVI and n - 1 independent charges ql per unit length (the sum of the chargesis zero). Consequently the electrostatic equations are of the form

n-lVI = L s., qj

)=1

For thin wires, the influence coefficients Klj are immediately deduced from thelogarithmic potential expressions; for thick wires corrections due to the fieldfactors and to the proximity effects apply. For non-parallel wires the sameelectrostatic relations are approximately valid if the variations of the mutual

(i = 1, ... , n -1). (26)

6

4

2(0)

o

Fig. 3. Curve (0): gs of eq. (6). Curves (1) to (3); gs - Ps with the first 1, 2 or 3 terms of eq. (8).


""'," ,,,,,,,,

\\\\\,\B\\,\

.5

o .2 .4

Fig. 4. Curve A: first two terms ofeq. (11). Curve B: eq. (11). Curve M: Martin's approx-imation.

distances between the wires along the axial coordinate z are small over a dis-tance of the order of the transverse dimensions, and this will always be assumedin this paper. Since the potentialof a perfect conductor is constant in d.c.,whereas the coefficients Klj in (26) are functions of z, so will be the charges q).If the system (26) is inverted to give

n-l

ql = L Cl) V),)=1

where the ci) are the Maxwell capacitances per unit length, one can integrate(27) over the length L of the cable to yield

n-l

QI = L c., VJ>)=1

where

Lc., = J Cl) dz.o

If some circuit, corresponding to subscript 1 in (28), is' connected to a d.c.source, whereas all other circuits are idle, one has

Qz = Q3 = ... =0

(27)

(28)

(29)

(30)

(31)

304 v. BELEVITCH. R. R. WILS ON AND G. C. GROENENDAAL

in (28), which becomes, in matrix notation,

[QI] [Cll Cu C13 ] [VI]o Cu C22 C23 V2o = C31 Cn C33 . . . V3.· .· .· ... .

(32)

By solving (32) one determines all VI as proportional to Ql and, in particular,the ratio QI/VI is the total capacitance of the excited circuit.In a system of parallel conductors the above computations are unnecessary

since the whole problem is independent of z, so that (31) forces

(33)

and (26) gives immediately VI = Kll ql' so that I/Kll is the capacitance perunit length of circuit 1.By contrast, for non-parallel conductors, the solutions VIof (32) determine a posteriori the ql by (27), so that (33) does not hold ingeneral. This phenomenon has been called longitudinal charge separation(longitudinale Ladungstrennung) by Martin.In a cable of twisted pairs or quads the mutual capacitances Cu (i =1= 1)

between an excited symmetric circuit and all other circuits (sides, phantoms,superphantoms ... ) are periodic functions of z which can be expanded inFourier series. Moreover, if the twisting has been realized so as to cancel thesystematic capacitive cross-talk, the Fourier series have no constant terms, andthe corresponding integrals (30) vanish when the length of the cable is an exactmultiple of the twisting pitch. Consequently one has

Cl2 = Cl3 = ... = C21 = C31 = ... = 0

in (32), hence

(34)

The theory based on (27) to (32) then becomes unnecessary, since (26) simplifiesto

rVI] rKll s., Kl3 ... ] [ql]K21 K22 K23 .. . q2o K31 K32 K33 .. . q3.· .· .

- • _o • • •

(35)

By solving (35) one determines all ql as proportional to VI'4. The seven quads

The seven quads contain 28 wires, hence 27 independent circuits. It is con-venient to take as circuits the 14 side circuits (2 per quad), the 7 phantoms(1 per quad) and the 6 superphantoms formed by the 6 peripheral quads with


the central quad. In (35) we set

With a unit charge on the excited circuit (of subscript 1), (35) partitioned intosubmatrices of order 1 and 26 is

where x T is the row-vector

The equation (37) separates into

and elimination of q yields

where Hl1 is the coefficient hs or hF of the excited circuit, so that one has

1l=XTX-1X

by comparison with (15).The computation of (42) involves the solution of the linear system (40) of

order 26, which will be obtained approximately as an expansion in powers of s,correct to order S4, because the algebra becomes too heavy for justifying a higheraccuracy. Since all elements of x are at least of order s, it is sufficientto computeall HIJ to order S3.

The diagonal elements HII are simply hs (hF) for side (phantom) circuits.For the superphantom circuits, the field corrections are much more complicatedbecause the six circuits (having the central quad in common) are stronglycoupled to each other, and the proximity effect resulting from the superpositionof the six fields can only be computed as a perturbation after having obtainedthe approximate solution for thin wires. Moreover, this computation is madeeasier if each superphantom is treated as the superposition of two asymmetrie(homopolar) circuits, with concentric returns at infinite distance cancelling eachother. In the following we thus compute the mutual coefficients HIJ betweenside, phantom and asymmetrie circuits, and the self-coefficient HII for asym-metric circuits, for thin wires and to order S3. .

For thin wires, all coefficients are linear combinations of logarithms of dis-

(36)

(37)

(38)

(39)

(40)

(41)

(42)

306"

v. BELEVITCH, R. R. WILS ON AND G. C. GROENENDAAL

tances. For a typical distance, such as D1s of fig. 2, we set

In ID1s1 = In D + Re In (1 + z), (43)with

c bz = - exp (iy) - - exp (i{J),

D D(44)

and replace the logarithm by its Taylor expansion up to Z3. The distancesbetween the various wires of fig. 2 are deduced from (43-44) by replacing b by-b or ± ib, and similarly for c, and the algebra is made easier if the real partis taken at the very end. The results are given in table I for the influence coeffi-

TABLE I

interaction wire numberingHIJj i

4bcSS 1-3 5-7 - - cos (y + (J)- - D2

2b b3SA 1-3 5678 - cos {J + t - cos 3{J- -- D D3

4bc2SF 1-3 57-68 - cos (2y + (J)- - D3

b2FA 13-24 5678 -cos 2fJ- -- D2

AA 1324 5678 -lnD--

cients between circuits belonging to different quads (the influence between dif-ferent circuits of the same quad is zero by symmetry), and the wires having thepositive polarity are underlined. At the order S3, the interaction FF is O.Finally,an asymmetrie circuit consists of a wire of radius a with two adjacent wiresat distance b V2 and one opposite wire at distance 2b, each carrying a chargei; consequently its self-coefficient is

HAA = -!: (In a + 2ln b V2+ In 2b) = -In d (45)

hence

(46)


If one makes c = b in the results of table I, it appears that the strongestinteraction is AA (of order SO), then SA (order s), then SS and FA (order S2)

and finally SF (order S3). The strongest interaction submatrix is thus the onebetween the seven asymmetrie circuits. If the central quad is labeled 0 and thesix surrounding quads are numbered anticlockwise from 1 to 6, the corre-sponding subrnatrix is a bordered symmetric circulant of the form

where the first row and column correspond to the central quad. The value.

al = -In d

results from (45) and the values

a2 = -In D; a3 = -In D V3; a4 = -In 2D

result from the expressions for AA in table I and from the mutual distancesin the hexagon. The submatrix ofthe interactions between the 6 superphantomsis deduced from (47) by subtracting row and column 1 from all other rows andcolumns. The result is a symmetric circulant of first row

with

Db, = 2In-;

d

Db2 = In-;

d

5. Phantom circuit at the centre

For a phantom excitation in the central quad, it results from table I thatthe dominant interaction is FA of order S2 giving an effect of order S4 in (42)and there are no other effects of that order. Since there is no interaction betweenthe phantom in the central quad and its own asymmetrie circuit, the interactionsbetween the central phantom and the six superphantoms are simply given bythe coefficients FA of table I, with f3 respectively replaced by (3- rr:J3,f3 - 2rr:J3, ... as established in sec. 2. Consequently the vector xT of (42) is

xT =S2 [cos 2(3, cos 2 (f3- : ), cos2 (f3- 2;), ... , cos 2 (f3- 5; ) ] (52)

and X is the circulant defined by (50) and (51). The linear system (40) thusfalls to order 6.

(47)

(48)

(49)

(50)

(51)

308 V. BELEVITCH, R. R. WILSON AND G. C. GROENENDAAL

Equation (40) is of the type defined by (BI) and (B2) of appendix B withX=B, x=-Rey,

YI = _S2 exp (2i{3)

and the value (B5) of À. Consequently, the solution is

q = Re z.

(53)

(54)

By (B4) one has

(55)

where gB is (B6), which reduces to (20) by (51) and (46). By (B3), (53), (54)and (55) one thus has

S2 [ (k - 1) 7tJqk = - - cos 2 {3- .

gB 3(56)

Now, by (40), (42) is equivalent to

u=-xTq. (57)

By (52) and (56), this is6

u = S4 "\"' cos- 2 [{3_ (k - 1) 7tJ = 3s4

•

gB ~ 3 gBk= I

(58)

The result (58) is for thin wires, and it remains to show that it is changedinto (22) for thick wires. The coefficient FA of table I results from the quadri-polar field of the phantom in the central quad and is affected by the fieldfactor IF of (12) as shown in appendix A. The factor IF thus affects (52), hencealso the solution vector q of (40) and thus generates IF2 in (22).The change of gB of (58) into hB = gB - PB - tB of (22) is due to the prox-

imity effect in the six superphantoms. The term PB originates from the effectwithin each quad whereas tB results from their interactions. The first term canbe computed as well for an isolated asymmetrie mode with return at infinity;this is done in appendix C and yields PA of (17). The correction PA must besubtracted from the diagonal elements al of (17), hence also from the diagonalelements bI of (50). Consequently (B6) is also decreased by PA and so is gB of(55) which is thus replaced by gB - PB with PB = PA'At large distance the field of an asymmetrie mode is the one of a single unit

charge at the centre of the quad, without any field correction. The proximityeffect due to wire 5 in that field of quad 1234 is

(59)


where

1 1 1 1E=-+-+-+-

DSl DS2 DS3 DS4

4 [C c2

]= - exp (-Ï1p) 1 _ - exp (iy) + - exp (2iy)D D D2

(proportional to the field) is correct to order c2/D2. Since the true field is asuperposition of six asymmetrie modes, the complex fields (60) must be com-bined vectorially before taking the modulus in (59). Also (60) assumes a unitcharge in one asymmetrie mode, whereas the real charges are (56). Now, if(58), which is of the form u = Hu2/H22 were produced by a coupling Huto a single asymmetrie circuit of self-coefficient H22 = ga, the charge in thatcircuit would be defined by u = Hu q2' hence q2 = Hu/H22' SinceHu = S2V3by comparison with (58), one has

instead of 1. Coherently, the normalized charges of the six peripheric asym-metric circuits, playing the role of a unit charge in a single equivalent asym-metric circuit, are (56) divided by (61), i.e.

qk = __1_ cos 2 [p __ (k_-_l)_7tJV3 3'

and the asymmetrie charge on the central quad is zero, since the sum of (62)is zero.The field in wire 1 of the central quad produced by an asymmetrie mode of

unit charge in the peripheral quad 1 is deduced from (60) by permuting thetwo quads of fig. 1, i.e. by replacing p, y and 1pby y _ re, P _ 7t and 1p- rr,respectively. This gives

4El = _ - exp (-i1p) [1 + s exp (iP) + S2 exp (2iP)]. (63)

D

The field Ek produced in the same wire 1 of the central quad by a unit charge inthe kth peripheral quad is deduced from (63) by changing 1pinto 1p+ (k -1) 7t/3and pinto P _ (k _ 1) 7t/3. The total field in wire 1 of the central quad is, by(62) and the values of Ek resulting from (63),

6 12sE = LEk qk = -- exp [-i (P + 1p)].

k=l D V3

The resulting value of (59) is 3a2s2/D2 and must be multiplied by 4 to accountfor the four wires of the central quad. The proximity effect due to the central

(60)

(61)

(62)

310 v. BELEVITCH. R. R. WILSON AND G. C. GROENENDAAL

quad is thus(64)

We next compute the field in one wire (5 of fig. 2) of the peripheral quadno. 1 produced by the asymmetrie charges (62) in the five other peripheralquads. From figure 5 it appears that the lines joining vertex 1 of a regularhexagon to all other vertices form a star with angles 1t/6, so that the successiveangles, playing the role of 'Ij)in (60), between quad I and the remaining quadsare

41t 51t 81t'Ij)+-, 'Ij)+-, ... , 'Ij)+-.

6 6 6

Since the angle y + 'Ij)of the excited quad 1 with respect to coordinates remainsconstant, the y's of (60) must be replaced by y - 41t/6, Y - 51t/6, .... Finallythe distances are D, D V3 and 2D. The total field in one wire is anyway ofthe form

4E = - - exp (-i'Ij) [A + Bs exp (iy) + Cs2 exp (2iy)]

D

similar to (60). To order S2, one has

16 {IEI2 = D2

AA· + s [A·B exp (iy) + AB· exp (-iy)]

+ S2 [BB· + A·C exp (2iy) + AC· exp (-2iY)]}.

Fig. 5. Numbering of the seven quads.

(65)


If the same expression with s changed into -s is added for the effect in wire 3,the terms in s disappear. If the result with y changed into y + 7tJ2 is addedfor the effects in wires 2 and 4, the terms in exp (± 2iy) disappear. By (59),the proximity effect due to the 4 wires of quad 1 is thus

where C does not appear. It is therefore permitted to neglect from the startthe term in S2 of (60). With that approximation the total field in one wire is

E = - _4_exp (-Ï1p) {exp (-4i7tJ6) [1-s exp i(y-47tJ6)] cos 2({3-7tJ3)D V3exp (-5i7tJ6) [S ]+ 1- - exp iCy- 57tJ6) cos 2({3- 27tJ3)V3 V3exp (-6i7tJ6) [s ]+ 2 1- 2 exp iCy- 67tJ6) cos 2({3- 37tJ3)

exp (-7i7tJ6) [S ]+ 1- - exp iCy- 77tJ6) cos 2({3- 47tJ3)V3 V3+ exp (-8i7tJ6) [1-s exp iCy- 87tJ6)]cos 2({3- 57tf3)}

= - _4_ exp (-i1fJ){tcos 2{3-i sin 2{3-s exp (iy) [172 cos 2{3+isin2{3J}., D V3

By (65), the proximity effect due to quad 1 is

a2

[ S2 ]- 5 - 3 cos 4{3+ - (193 - 95 cos 4(3) .6D2 36 (66)

The terms in cos 4{3disappear when (66) is summed for 6 quads and theresult is

By adding (64) one obtains (21).

6. Side circuit at the centre. Effects of order S2

It results from table. I that the dominant interaction is SA of order s givingan effect of order S2 in (42) and this effect alone is computed in the presentsection; the more complicated effects of order S4 are obtained in sec. 7. The


linear system (40) is again of order 6, with

xT = 2s [COS fJ, cos (fJ- :). cos (fJ- 2;). ... cos (fJ- 5;)] (67)

and X is the circulant defined by (50) and (51). Equation (40) is of the type(BI) of appendix B with x = -Re y,

Yl = -2s exp (ifJ)

and the value (B7) of À.. By (B4), one has

(68)

(69)

where s: is (B8), which reduces to (16) by (51) and (46). By (B3), (54), (68)and (69) the solution is

2s [ (k-I) 7tJqk = - - cos fJ - .

gA 3(70)

By (57) one has6

U = 4s2

'" cos" [fJ _ (k - 1) 7tJ = I2s2 .s; i...J . 3 s:

k=l

(71)

The result (71) is for thin wires. For thick wires, the coefficient SA of table Iis affected by the field factor of (11). The factor fS2 thus affects (71) andappears in the first term of (19). The additional change of gA of (71) intohA = gA - PA - tA of the first term of (19) is due to the proximity effect inthe six superphantoms. The term PA originates from the effect within eachquad and is (17). The term fA is the sum of the effects of the central quadand of the six peripheral quads, and will now be computed. From (71) and theequations preceding (61) it results that one has Hu = 2s V3 so that theequation replacing (61) is

2s V3q2 =--.

gA

Consequently, the normalized charges of the six asymmetrie circuits are (70)devided by (72), i.e.

(72)

1 [ (k-I) 7tJqk = - ï73 cos fJ- 3 .

By analogy with (63) and the next equations, the total field in wire 1 of the

(73)


central quad, due to the asymmetrie charges (73) in the six peripheral quads, is

12E = - -- exp [- i({J + 7p)].

D V3By (59), the proximity effect due to the 4 wires of the central quad is thus

(74)

By a computation similar to the one preceding (66), the field in one wire ofthe peripheral quad 1 due to the asymmetrie charges (73) in the other fiveperipheral quads is

4E =--- exp (-i7p) [t cos {J - 2i sin {J + s exp (iy) (~ ~ cos {J - 2i sin (J)]

D V3The analogue of (66), i.e. the proximity effect due to the 4 wires of quad 1 is

a2

[ S2 ]- 17 - 15 cos 2{J + - (697 - 455 cos 2{J) ,6D2 36

so that the effect for six quads is

17 w2 S2 (1 + ~à S2).

By adding (74) one obtains (18).

(75)

7. Effects of order S4

As mentioned in sec. 4, in order to obtain (42) to order s\ it is necessary totake into account terms up to order S3 in the vector (38). We thus set

x = S Xl + S2 X2 + S3 X3,q = S ql + S2 q2 + S3 q3'X = Xo + SXl + S2 X2 + S3 X3

in (40). Separating the various powers of s, one obtains

Xl + Xo ql = 0, (76)

(77)

X3 + X2 ql + Xl q2 + Xo q3 = °and X3 is irrelevant. Similarly, (57) becomes

u = _S2 [xJ ql + s(xJ q2 + xi ql) + S2 (xJ q3 + xi q2 + xj ql)]'

(78)

(79)

In principle, all vectors and matrices in the above equations are of order 26


but, since the SF interaction of table I is of order S3, its effect in (42) is onlyof order S6 and is neglected in (79). We can thus forget the 7 phantom circuitsand reduce all vectors and matrices to order 19.We further partition all vectorsand matrices into symmetric (subscripts S, 13 elements) and superphantom(subscript A, 6 elements). From table I it results that one has

(80)

since SA-interactions are odd in S while SS-interactions are even. The matrix Xoof order SO is

_ [hs 113 0JXo - o B(81)

where 113 is the unit-matrix of order 13 and B the circulant of (30). Thematrix Xl only results from the SA-interactions and is thus of the form

(82)

Finally X2 only results from SS-interactions and is of the form

X2 =[: ~lIf (81) to (83) are inserted into (76) to (78) and if all vectors ql are also par-

titioned into qlS, qlA, it immediately results that one has

(83)

(84)

so that the vectors ql are of the same type as (80). The remaining equationsreduce to

(85)

(86)

(87)

and N does not appear in these equations, so that (83) becomes irrelevant.Also, owing to (80) and (84), the term of order S3 disappears in (79) whichreduces to

(88)

Set

(89)

THE CAPACITANCEOF CIRCUITSIN ACABLEWITHTWISTEDQUADS 315

The solution of (86) isy + X2S

q2S =----hs

Premultiplying (87) by qTA and eliminating qTA by the transposes of (85) andof (89) one obtains

(90)

By (90) this gives

This, and (90) transform (88) into

_ 2 T 4 (2 T x~s X2S + 2yT X2S + yT y)U - -s x q - s q x - .IA IA IA 3A hs

(91)

Equation (85) only involves terms of order s and corresponds to the approx-imation of sec. 5. Consequently the elements of its solution vector qlA are (70)divided by s, and the term in S2 of (91) is (71). Since q2S and q3A have dis-appeared from (91) it becomes unnecessary to solve (86) and (87), and it onlyremains to evaluate the four scalar products appearing in the s4-term of (91).The vector X3A originates from the contribution in S3 of the SA-interaction

of table I, and its elements are proportional to cos 3 [,8 - (k - 1) "Tt/3].Owingto (70) one has

(92)

The elements of the vector x2s originating from the SS-interactions appearin the form -4 cos (y + ,8) in table I for the interaction 13-57. For the inter-action 13-68 one must change Y into Y + "Tt/2and this yields 4 sin (y + ,8).The angles playing the role of y in fig. 2 may be different for each peripheralquad and will be called YI (i = 1,2, ... , 6). Also, both ,8 and YI are decreasedanticlockwise by "Tt/3.The vector X2S has 13 elements; the first is the zerointeraction between the side circuits of the central quad while the next 12 onesare the interactions between the excited central side circuit and the 6 couplesof peripheral side circuits. The first zero element does not contribute to thescalar products of (91) and will be omitted. The shortened 12-vector is

xT = 4 [-cos (Yl + ,8),sin (YI+ ,8),2S

-cos (Y2 +,8- 2; ). sin(Y2 +,8- 2; ), ... ] (93)


and one has6

XT X = 16L (cos" + sin") = 96.25 25

1

(94)

The submatrix M of (82), of order 6X 13 results from the interaction, limitedto order s, between the 6 superphantoms and the 13 side circuits other than theexcited one. We temporarily replace the 6 superphantoms by the 7 asymmetriecircuits and also consider the 14th excited side circuit, thus enlarging M toorder 7 X 14. With the numbering of fig. 5 for the quads the enlarged matrixis (95) as will now be justified *). For convenience, the couples of side circuitsare distinguished by subscripts a and b.The first column of (95) is the vector XIA of (67) preceded by a zero, and the

second column is deduced by changing fJ into fJ + n/2. The first row of (95)is deduced from the first two columns by changing fJ into Yl - n, whichcorresponds to the interchange of the two quads of fig. 2, as explained before(63). Column la is deduced from column 0a, by adopting the angles and thedistances discussed before (65), and column l, is similarly deduced fromcolumn Ob. Finally, with the exception of row 0, the next columns are deducedby cyclic permutations from the set la, lb. Only 2a is indicated in (95) as anillustration.The submatrix M of (82) is deduced from (95) by subtracting row ° from

all other rows and then dropping the first row and column. We will not executethe subtraction because M only appears in the combination (89) where thismakes no difference, because the elements (70) divided by sof qlA sum up tozero, so that the subtraction of a constant from every column of M has a zeroeffect. Consequently we may work as if M were the submatrix separated bydotted lines in (95). The resulting vector y of (89) has 13 elements. The firstone resulting from (70) and column Ob of (95) is zero.The next one, resulting from column la of (95) is

- :A[2COS(Yl- 4;)cos(fJ- ;)+ :3 COS(YI- 5;)cos(fJ- 2;)

+COS(YI- 6;)cos(fJ- 3;)+ :3 COS(YI- 7;)cos(fJ- 4;)

+ 2 cos (YI - 8;) cos (fJ - 5;)] = -_!_ [5 cos (YI-fJ)-3 cos (YI+ fJ)].s; (96)

Similarly, the element resulting from column l, is

1- [5 sin (YI- fJ) - 3 sin (YI + fJ)]·gA

*) Eq. (95) is on page 317.

(97)

Equation (95)

Oa Ob la lb 2a I~0 I 0 0 -2 COS1'l 2 sin 1'1 -2 COS1'2................................................................................................................................................................................................................................. £>

I 2 cos f3 -2 sin f3 2 cos (1'2 _ 8;) 181 0 0 ,>'z0

t>10"%j

2 cos (p- ;) -2 sin (p- ;) 2 cos ( r 1_ 4;) 2sin(1'l- :n) 02 I - 0 :<I0c::ïCIl

2 cos (f3 _ 2;) -2 sin (f3 _ 2;) 2 ( 5n) - :3 sin (1'1 _ 5;) 2 cos (1'2 _ 4;) Z3 I j73 cos 1'1- 6 >~tzj

r-t>1

2 cos (f3 _ 3;) -2 sin (f3 _ 3;) cos (1'1 _ 6;) sin (1'1 _ 6;) 2 ( 5n) ~4 I - )/3cos 1'2-6 ~~

2 cos (f3 _ 4;) -2 sin (f3 _ 4;) 2 ( 7n) - ~3sin (1'1- 7;) cos (1'2 _ 6;) CIl

It;!5 I V3cos 1'1-6 tl

o~I;)

2 cos (f3 _ 5;) -2sin(f3- 5;) 2 cos (1'1 _ 8;) 2 sin (1'1 _ 8;) 2 ( 7n) til6 I - j73cos 1'2- 6

Iw......I....:)

318 V. BELEVITCH. R. R. WILS ON AND G. C. GROENENDAAL

The next pairs of elements of y are deduced from (96) and (97) by replacing Pand Yl by P - 7t/3, P - 27tJ3, ... and Y2 - 7t/3, Y3 - 27t/3, ... respectively.In the scalar product yT y of (91), the first zero element of y brings no

contribution, the next two elements (96) and (97) give a contribution

1- (34 - 30 cos 2P)gA2

(98)

and the term in cos 2p of (98) cancels in the sum of six contributions, to yield

(99)

Finally, it results from (93) and the element values of y discussed after (96)and (97), that the last scalar product of (91) is

yT x2s = - ;: . (100)

The expressions (94), (99) and (100) are for thin wires and must be modified forthe proximity effect and the field corrections. Expression (94) originates froman SS-interaction which disappears if the wires of the potentiating quad toucheach other, and this introduces in x2s the field factor is of (11). But the inter-action also separately disappears if the wires of the potentiated quad toucheach other, and this introduces an additional factor, which is also is byreciprocity. Consequently x2s is multiplied by is2 and (94) by is 4.

In the theory of sec. 5, the vector qlA (there called q) was affected by thefield factor is originating from an SA-interaction, as mentioned after (71).The same factor affects all elements of (95) which are also SA-interactions.By (89), y is thus multiplied by is2, and (99) by is 4. Also (100) is multiplied byis 4 because both X2Sand y are multiplied by is2. Finally, the proximityeffectchanges gA of (99) and (100) into hA = gA - PA - tA as in sec. 5. After thesemodifications (94) + (99) + 2 X (100) becomes

96is4 - 144 fs4/hA + 204is4/hA2

and the corresponding change in (91) generates the last terms of (19).

April1977

Appendix A

We use the notations of ref. 1 (where, in particular, D is 2b of the presentpaper) and refer to eq. (84) of ref. las eq. (1-84). From that equation withoutscreen (Apo = 0), it results that the dipolar potential at large distance isifJO•1 exp (iOo) with

(Al)


For the paraIlel mode (1-112) one has, by (1-116) and with the coordinatesof fig. 1-4,

4

L = 2[a (Al + Al ') - D/V2].s=l

For the second paraIlel mode, the 90° rotation multiplies (AI) by -i; nextthe rotation of coordinates from fig. 1-4 to fig. 1-5 multiplies all results byexp (-3i7t/4) and the reduction from (1-114) to 11 = 1, 13 = -1 halves theresult, which is

t (1 - i) exp (-3i7t/4) if>O.1= -if>O.1 cos 7t/4

hence, by (AI) and (A2),

,u V2- [a (Al + Al') -D!V2].27tro

Now, the coefficient Al used in (A3) is the one of the paraIlel mode beforethe transformation (1-120). This transformation, and (1-133), change (A3) into

,uD- - [1 + 2(Al + Cl)]'

27tro

In (A4), Al and Cl are the solutions of the equations (1-135) at h.f.. Con-sidering the first two equations and working to order 0\ one obtains

Al = _02 (I + 02); Cl = -202 (I + 02)

and the bracket of (A4) becomes (11) by (10).For the phantom mode, the qüadripolar potential at large distance is

C/JO•2 exp (2i ()o) with,u 4 , , 2

C/JO•2= - --2 L (a2 A2• + a Dos AIs - t Is Dos)27tro se r

by (1-84). By (1-111), (1-116) and (1-120), one has4

L = iD2 (-4B2 + 2Bl + t),s=l

hence

where BI and B2 are the solutions of (1-126) at h.f.. Considering the first twoequation~ and working to order 0\ one obtains

BI = _02 (1 + 302); B2 = 04/2

(A2)

(A3)

(A4)

(A5)


and these values inserted in (AS) transform the factor between parenthesesinto (12).

Appendix B

We consider the linear system

Bz=y (BI)

where B is a circulant of order 6 of first row (50) and where y is a vector ofelements

Yk =YI Àk-l (k = 1, ... ,6). (B2)

By setting

(B3)

all equations contained in (BI) reduce to

(bI + À b2 + À2 b3 + À3 b4 + À4 b, + À5 b2) Zl = Yl (B4)

multiplied by successive powers of À. In particular, for

J. = exp (-27tij3), (B5)

the coefficient of z I in (B4) is

27t 47t 67tbI + 2b2 cos - + 2b3 cos - + b4 cos - = bI - b2 - b3 + b4• (B6)

3 3 3

For

À = exp (-7tij3), (B7)

the coefficient is

Appendix C

For the asymmetrie mode with infinitely distant screen the equations (1-143)at h.f. become 3)

00 xnL Knlllx'"-- =Yn (n = 1,2, ... ,00)111=1 (J2n

(Cl)

with

[(n-m) 7t ]

K = -T, 21+(III+n)/2 cos + 1nm nm 4' (C2)

-L' = _!:__ (9(52 - ¥ 154+ 198156).81t"

By (2), the proximity correctionp of(5) is -21t"//-l times (CS) and this gives (17).

CS)


Yn = -[ 21+n/2 cos n41t"+ 1] (C3)

and the proximity effect on the h.f. inductance resulting from (1-153) is adecrease of

-L' = _!:__ ~ a: 152111 (C4)81t" 111=1

where the h.f. values resulting from (1-155) are

Q1 = Y12 = 9,Q2 = t Y/ + Kll .f1

2 = - 8-1,Q3 = t Y32 + K12Y1 Y2 + K11

2 Y12 = 198

so that (C4) becomes, to order 156,

Acknowledgement

The authors are grateful to O.R. Bresser (NKF Kabel, Delft) for an impor-tant correction.

REFERENCES1) v. Belevitch, Philips Res. Repts 32,16-43, 1977 and 32,96-117, 1977.2) H. E. Martin, Arch. Elektr. Übertr. 18, 293-308, 1964.3) G. C. Groenendaal, R. R. Wilsonand V. Belevitch, Philips Res. Repts, 32, no. 5,

1977.

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