(2)

R957 Philips Res. Repts, 32, 16-43,1977

THEORY OF THE PROXIMITY EFFECTIN MULTIWIRE CABLES

by V. BELEVITCH

AbstractThe paper deals with the computation of the impedance per unit lengthof a side or phantom circuit in a system of parallel cylindrical wires,with or without screen, taking into account the eddy-current losses inall conductors. Owing to various methodological innovations, it bringsmany new results and offers shorter and much more transparent proofsof several known results. It is therefore written so as to present a unifiedand self-contained analysis of the subject.

PART I

1_ Introduction

In a multiwire cable, an alternating current exp Grot) in one conductorgenerates a magnetic field, hence eddy currents, in all conductors (includingitself), thus causing an increase of the impedance per unit length with frequency,called proximity effect (skin effect), respectively.

The theory of the skin effect is well-known. With the notation

k = (-jro,ua)t (1)

(0' = conductivity, ,u = permeability), where the square root with negative realpart is taken, the impedance per unit length of a wire of radius a is the ratioof Bessel functions

where1

Ro=--7taa2

(3)

is the d.c. resistance. At low frequency, the first two terms of the Taylor ex-pansion

Part II, containing secs 6 to 10 and the appendices, will appear in the next issue.

THEORY OF THE PROXIMITY EFFECT IN MULTIWIRE CABLES 17

zJo(z) Z2 Z4--=1---- ...2J1(z) 8 192

(4)

givejwft

ZSk =Ro +--8"

(5)

hence a contribution of ft/8" per wire to the d.c. inductance, whereas the thirdterm of (4) yields an increase in w2 of the resistance. In terms of the normalizedfrequency

_ Gftwa2_ ( radius )2

Q----. 2 skin-depth

(6)

the relative increase isRSk Q2

-=1+-.s; 48(7)

At high frequency, the first terms of the asymptotic expansion

zJo(z) -jz 1--=-+-2J1(z) 2 4

(8)

giveRSk Qt 1-=-+-s; 2 4

(9)

whereas 'the inductance tends to zero proportionally to I/Qt.The theory of the skin effect only takes into account the current in one con-

ductor, assuming that the return conductor, and all other idle conductors, areinfinitely distant. With that approximation, the impedance per unit length ofa pair of wires of radii al and a2 whose centres are separated by D is

Z = ZSkl + ZSk2 + jwL, (10)

where ZSkl is (2) with a replaced by al (also in the value (3) of Ro) with i = 1,2,and where L is the inductance per unit length for thin wires, i.e. *)

# D2L=-log--

2" al a2(11)

hence, in particular,

L = !:._ log (D/a)

"(12)

*) Natural logarithms are used throughout and noted log.

18 v. BELEVITCH

for the symmetric pair with al = a2 = a. The above theory is always correctat d.c., where there is no proximity effect, so that the d.c. inductance is (11)completed by a contribution of #/8TC per wire, hence

#Lo = - [log (D/a) +·!l

TC(13)

for the symmetric pair. Since, however, the magnetic lines of force at d.c. crossthe wire boundaries, it is not correct to interpret the term (12) of (13) asrepresenting the external inductance. The question is further clarified inappendix A (of the forthcoming Part 1I).

As soon as the go and return conductors are not infinitely distant, or whenother conductors are present in the neighbourhood, (10) must be replaced by

Z = Zskl + Zsk2 + jwL + Zprox, (14)

which defines Zprox by difference with (10). The essential purpose of this paperis to furnish means for evaluating the term Zprox of (14) taking into accountnot only the interaction between the go and return conductors, but also theinfluence of all other idle conductors and of the screen. All conductors areassumed parallel; effects due to twisting will be analysed in a following paper.

Even in the simplest case of the unscreened symmetric pair, no closed-formexpression is available for Zprox. The problem has been treated by Mie 1),Carson 2), Butterworth 3) and Arnold 4). It will be shown in this paper thatthe result is the infinite series

(15)

n=l

where the An are the solutions of the infinite linear system

I AnAnT. Am - -- = 1 (n = 1, 2, ... )nm <52n

(16)

m=1

with the notations(m+n-I)!

Tnm=------(n-I)! (m-I)!

(17)

(18)

and<5 = al D (19)

THEORY OF THE PROXIMITY EFFECT IN MULTlWIRE CABLES 19

(a = radius of conductors, D = distance between their axes). The equivalentof system (16), in various forms and notations, occurs in the papers quotedabove, which mainly differ from each other in the way series expansions areobtained for the solution. In the notation we have adopted for (16), the matrixis symmetric and its off-diagonal elements are real and constant, an advantagefor numerical work.The main objective of this paper is to generalize (15) and (16) to multiwire

cables. For unscreened cables this has been achieved by Lorenz 5.6) but ourderivation is considerably simpler as justified in the next sections.All authors deduce the impedance from the complex power evaluated by

integrating the flux of the Poynting vector. In a previous paper 7) we haveshown (on one example) that integration can be avoided if the problem isformulated from the beginning in terms of scalar and vector potentials : thispermits to deduce the voltage drops per unit length directly from the boundaryconditions. Our methodology is described in sec. 2 starting from Carson's quasi-stationary approach 8).The general equations for multiwire cables with screen are established in

sec. 3, and the unscreened case is obtained by making the screen radius infinite:this makes the presentation slightly heavier but shortens the paper. An addi-tional point where our analysis differs from Lorenz's is that all Fourier series(containing sines and cosines in the angular coordinates) are written as complexexponentials, which halves the number of equations and simplifies considerablytheir appearance. Since the imaginary unit j already occurs in the complexfrequency jw, a distinct Imaginary unit i is used to denote rotation in physicalplane, but the product ij is left undefined: the algebra is a direct product oftwo separate complex algebras. In the whole analysis the expressions real,imaginary, complex conjugate <*) refer to i alone; it is only at the very endthat the impedance (which is real in i) is separated with respect to j into aresistance and a reactance.The equations of sec. 3 are specialized for the pair in sec. 4 and for the quad

(both for side and phantom circuits) in sec. 5. As regards the screen, we onlytreat the case where it is perfectly conducting (its thickness is then irrelevant)because the general case is considerably heavier, whereas the effect of an im-perfectly conducting screen is analysed approximately by a perturbation techni-que (in sec. 8). In addition to the symmetric pair we also treat the case of one-way induction from a thin to a thick wire without screen: the correspondingexplicit expression

Zprox = jw,u "" Jn+1(ka) (ajD)2n,27t l__; nJn_1(ka)

n=l

(20)

where a is the radius of the thick wire, coincides with the one deduced by

20 V. BELEVITCH

Dwight 9) from Manneback's solution 10) for the current density. Expression(20) is of interest because the two-way induction effects in a symmetric pairmust produce losses that are larger than twice the one-way effect. For the samereason we also analyse the quad formed by one thin pair and one thick pair.The rest of the paper is devoted to approximate solutions of the equations.

For the pair and the quad without screen, the impedance only depends on twodimensionless parameters: Q of (6) and 15 of (19). The following expansions aretherefore investigated:(a) expansion in powers of 152 valid for all jQ,(b) expansion in powers of jQ valid for all 15,(c) asymptotic expansion in jQ valid for all ö.Expansion (a) is obtained in sec. 6, even for the screened case. This generalizes

(and yields a simpler proof of) the Mie expansions for the unscreened pair. Insec. 7. the first term (in Q2) of expansion (b) is obtained: it is a transeendentalfunction of èJ2 without closed-form expression, so that further continuation ishopeless.At high frequency, the first term in the asymptotic expansion of Zprox gives

a negative inductance, to be called -L', because the total inductance

Loo =L-L' (21)

resulting from (14), where ZSk does not contribute, must be smaller than itsprincipal value L for thin wires, since thick wires form a larger obstacle to theflux. In particular, the well-known value

Zprox = -jooL' + A (jQ)t + B, (23)

f1,Loo= - arccosh (D/2a)

7t(22)

for the symmetric pair is asymptotic to Lof (12) for D » a but is 0 for D = 2a(the wires with infinitesimal insulation touch each other and there is no pathfor the flux). By equation (21), 7tL'/t-t thus increases from 0 for 15 = 0 tolog 2 = 0.693 ... for 15 = t.

In sec. 8 we prove that the expansion corresponding to case (c) is always ofthe form

where the coefficientsA and B are real functions ofthe parameters characterizingthe cable geometry, just like L'. Moreover, A and B can be deduced from L'by partial differentiation with respect to the radius (or radii) of the conductors.When L' is known in closed form, as for the symmetric pair, so are A and B,and this justifies and completes the results of Arnold 4). Else A and B can bededuced from a tabulation of L' by' numerical differentiation.The high-frequency inductance Loo is independent of the conductivity (J and

THEORY OF THE PROXIMITY EFFECT IN MULTIWIRE CABLES 21

remains the same for perfect conductors: (J = ex) or w = 00 have the sameeffect in (6). The computation of Loo is thus the solution of a problem ofpotential theory, and (22) is indeed simply obtainable by conformal repre-sentation. For the quad and the screened pair, even the potential problem isnot elementary and is solved numerically in sec. 9. This, however, raises anumber of interesting points which will be first described on the example ofthe unscreened pair where the explicit solution is known:For large Ikal the asymptotic value of (18) is

Än =n. (24)Setting

vnAn=--

n(25)

in (16) simplified by (24) and noting that

00

I vm I(m+n-I) 1To - = vm = -1nm m n-I (1 _ v)n

(26)

m=l m=l

it appears that all equations are satisfied if

1 v

I-v (F(27)

i.e. for1± (1 - 4(P)t

v± =------2

(28)

For °< c5 ~ t, i.e. as long as the wires do not overlap, both roots (28) arereal and have moduli < 1, so that convergence is ensured in all sums. Math-ematically, one can thus accept both signs in (28) and take in (25) a linearcombination of both solutions, provided their coefficients have unit sum. Thus

1An= - - [a v+ n + (1 - a) v., n]

n(29)

is a solution, with a arbitrary. By contrast, when the problem is solved byconformal representation, one obtains as unique physical solution the particularcase of (29) corresponding to a = 0, and that solution, inserted into (15),yields,Zprox = -jwL', where L' is the correct value

P,L' = - [(logD/a) - arccosh (D/2a)]

tt(30)

22 V. BELEVITCH

resulting from (12), (21) and (22). The physical solution thus selects the smallestroot v_ in (28), which tends' to zero for 15 = 0, so that there is no proximityeffect for an infinitely large separation of the go and return conductors; bycontrast, v+ tends to 1 for 15 = 0 and (26) diverges.

We now examine the situation for Ö > t, i.e. when the wires overlap.Physically the problem is meaningful, although trivial, since the solution mustbe Loo = 0, so that one must have L = L', hence (IS) must yield

cc

L An = log (D/a).n=l

(31)

Now, for 15 > t, the roots (28) are complex conjugate. For physical reasons(29) must be real, and this forces CJ. = -}.With that choice, (31) is satisfied.

In conclusion, the infinite linear system (16) is always mathematically indeter-mined, and the unique physical solution is selected by imposing the condition

An -+ 0 for 15 -+ 0, all n (32)

for 15 ~ t, and by the reality condition for 15 ~ t. Note that, for 15 = t, onehas v+ = v., and (29) becomes independent of CJ..

For the screened pair or quad, the infinite linear system has also a closed-form solution when the wires overlap, or when the screen crosses the wires andit will be proved analytically that the result is Loo = 0 in all cases. Since theproof involves non-trivial algebra, it yields a strong argument for the correct-ness of the general equations. For non-overlapping wires Loo must be computednumerically by truncating the infinite linear system in the unknowns An atincreasing finite orders n. From a general theorem of Pincherlé 11) it is knownthat the process converges to the so-called distinguished solution iff the infinitesystem possesses such a solution. A distinguished solution is one for whichthe sequence IAnl grows more slowly with increasing n than the similar sequenceof any other solution. For the case of the pair, the solution selected by (32) isthe distinguished solution since it imposes CJ. = 0 in (29) and thus chooses thesolution involving only the smal/est root v; of (28). For the cases where noanalytical solution is available, numerical results indeed converge for non-over-lapping wires, and the solution satisfies (32), thus proving empirically that thedistinguished solution is the physical solution. Mathematically, this can onlybe proved in the extreme case 15 = 0 where (32) is the distinguished solutionby definition. One can then argue by continuity that the distinguished andphysical solutions stay identic'al for small 15:

Finally, the matrix relations generalizing (15) and (16) are given an equivalentcircuit interpretation in sec. 10.

2. Foundations

Inside the conductors, the displacement current is always negligible with

THEORY OF THE PROXIMITY EFFECT IN MULTlWIRE CABLES 23

respect to the conduction current, so that the quasi-stationary equation

curIH= a E (33)

holds. The vector-potential A and the scalar potential V are introduced by

u H = curiA (34)

(35)E = -jwA - grad V

with the quasi-stationary condition

div A = O. (36)

Elimination of A between (33) and (34) yields, making use of (35) and (36)and (1),

ÖA + k2 A = ua grad V. (37)

Outside the conductors, the rigorous equation replacing (33) is

curl H = jcos E (s = dielectric constant)

and the Lorentz condition replacing (36) is

div A + êp,jW V = 0

(38)

(39)

but (34) and (35) remain. The elimination of H between (38) and (34) nowyields, making use of (39) and (35),

ÖA + êp,W2 A = O. (40)

In the TEM mode with propagation constant y in the z-direction, thez-dependence of A is exp (-yz), and (40) becomes

b2A ()2A- + -- + (y2 + êp,W2) A = O.bx2 by2

(41)

For perfect conductors, it is well-known that one has

(42)

so that (41) reduces to the Laplace equation in x, y. For non-perfect con-ductors this is no longer true and the solution of (41) will be a function of thetransversal coordinates x, y, times (y2 + êp,W2)t. If D is the largest transversaldimension of the system of wires, the largest value of the dimensionless quan-tities occurring in the boundary conditions will be

(43)

Whenever the modulus of (43) is much smaller than 1, it can be neglected inthe boundary conditions. Since the resulting approximate boundary conditions

24 V. BELEVITCH

coincide with the ones obtained by making y2 + Bf-lW2 = 0 in (41), it is thenlegitimate to replace (41) by the Laplace equation in the xy-plane.For good conductors, the value of y for the TEM mode is close to (42), hence

(44)

where a is the attenuation per unit length (Neper/meter), for the correctionon the phase (f3 - f3o) is ofthe second order in (43). Neglecting a2, the modulusof (43) becomes' D (2af3o)t. Since f30 is 2rt/À, where À is the wave length, (43)is negligible if

[(4rtaD) (D/À)]t « 1. (45)

In all cable applications, both D/À and «I) are much smaller than 1, so that(45) holds.Suppose now that the quasi-stationary approximation (displacement current

neglected) is accepted from the beginning, also in the dielectric. Equation (38)is then replaced by curl H = 0 and eq. (39) by eq. (36). Consequently (40)is replaced by

6.A = O. (46)

Since, however, propagation also disappears, all fields (and the vector-potential)become independent of z, and (46) reduces to the Laplace equation in x, y.In conclusion, the rigorous theory with the approximation (45) is equivalentto the quasi-stationary theory as regards the x, j-dependence of the fields,i.e. as regards the transverse (penetration) problem. In the quasi-stationarytheory, the current in each wire is constant and only flows in the z-direction,just as in d.c., so that the only non-zero component of A is Az and the equa-tions (37) and (46) reduce to

bV6.Az + k2 Az = /-la - inside,

bz(47)

outside, (48)

where 6., from now on, denotes the (x, y) Laplace operator.Assuming f-le = f-li> the boundary conditions are the continuity of H, and

Hno hence of bAz/bn and bAz/bs by (34), and of

bVE; =-jwAz--

bz(49)

resulting from (35). Since, inside a conductor, ö V/()Z takes the same constantvalue as on its boundary, the continuity of Az is also required by (49), and thecontinuity of ()Az/bs is an automatic consequence of the latter. The only

THEORY OF THE PROXIMITY EFFECT IN MULTIWIRE CABLES 25

boundary conditions are thus the continuity of Az and of oAz/on. (50)Finally, the current I in any wire in the positive z-direction is related to theintegral of oAz/on on its boundary, because (34) gives

1 f oAz1= 1Hsds =-- -dsfl on

(51)

for the orientation of the coordinates depicted in fig. 1.In conclusion, the solution of the quasi-stationary equations only furnishes

the constant values of the voltage drops on each wire, as linear combinationsof the currents (51) in the various wires, thus defining an impedance matrixper unit length, but is incapable to yield the secondary line parameters, sincepropagation has been suppressed. Propagation is restored, in first approxima-tion, by computing the current leak per unit length as it results from (39) whichreduces to

oAz- =-sfljwVoz (52)

and yields

ol = sjw ,c 0V dsOZ jon

(53)

by (51). Now the right-hand side of (53) is simply -jwQ, where Q is the chargeper unit length on the wire boundary, and this is related to Vby the electrostaticcapacitance matrix. Consequently, the transversal penetration problem is com-pletely decoupled from the longitudinal propagation problem because differentorders of magnitude are involved: the transversal problem is solved in the quasi-stationary approximation and is only used to compute the impedance matrix Zper unit length; the longitudinal problem is then solved by elementary linetheory by combining Z with the electrostatic capacitance matrix per unit length,so as to determine the secondary parameters (characteristic impedances andpropagation constant).

n

z

Fig.l

26 V. BELEVITCH

Additional simplifications are introduced in the quasi-stationary theory byadopting the notations

cp = Az outside, (54)

1 <>V'IjJ = Az +:- - inside.

JW <>ZEliminating Az between (47) and (55), and remembering that ö V/'öz is constant

(55)

inside, one obtains

/j.'IjJ + k2 'IjJ = 0 inside, (56)whereas (48) is

D.cp = 0 outside. (57)

The continuity of Az in (54) and (55) imposes the constancy of the voltagedrop U per unit length

'öVU =-- =jw (cp-'ljJ)

'öz(58)

whereas the continuity of <>Az/'ön becomes simply

'ön 'ön(59)-=-.

The problem thus amounts to solving (56) inside and (57) outside, with theboundary conditions (58) and (59).If a conductor is perfect one has Ikl = 00 inside it and (56) forces 'IjJ = 0,

so that (58) reduces toU = jwcp. (60)

On the other hand (59) does no longer hold since there is a surface-currentdensity creating a discontinuity of Hs. In fact the value of 'ö'IjJ/'ön can be usedto evaluate the current density.

3. General equations

We consider a number N of parallel wires and call r.,(Js the polar coor-dinates relative to the centre of wire s (s = 1,2, ... , N). The radius as andconductivity O's may be different for each wire and the parameter correspondingto (1) is called kso The screen is supposed to be a cylinder of internal (external)radius ao(b) around the origin of the polar coordinates ro,(Jo and has con-ductivity 0'0' In order to avoid minor complications, we assume that the screenhas the same permeability as the wires and the dielectric and that it carries nooverall current (Io = 0). The general solution 'ljJs of (56) inside wire s is

THEORY OF THE PROXIMITY EFFECT IN MULTIWIRE CABLES 27

(61)

where the coefficients Bns are complex in i, and where all expressions have tobe ultimately replaced by their real parts with respect to i. The first coefficientin (61) is proportional to the current Is in wire s so as to satisfy (51). In thedielectric, the general solution of (57) is

s=l . m=l

m=l

where the series in negative powers of 's accounts for the allowed singularityinside wire s, whereas the logarithmic terms and the series in positive powersof '0 allow for singularities at infinity. Also, the coefficients Is have been chosenso as to satisfy (51), and (62) must be ultimately replaced by its real part.

In order to write the boundary conditions on 's = as> one must express 't/)tfor all t =1= s in terms of '.,()s so as to obtain a Fourier series of the form

CPs= %,0 +I CPs,n exp (in()s)n=l

(63)

(whose real part must be taken) for (62) referred to the s-coordinates. Thecomplex distance from centre of wire s to centre of wire t is called Dst (fig. 2)and one has

(64)and, of course

(65)

p

5Fig.2

28 V. BELEVITCH

The Taylor expansion of the logarithm of (64) gives

Wt + log r, = log Dst + i7t - '\' ~ (~)n exp (inOs). (66)~n o;n=1

By (26) with v = rs exp (Ws)/D:t the conjugate negative m-th power of (64) is

exp (imOt) _ '-m [rs exp (-Ws) ]- m _ '-nt [ r, exp (-Ws)]- m _--m---Dst D' -1 -Dts 1- D' -

rt st st

1 [ ITnm ( r, )n ]= --om 1+ - -. exp (-inOs)Dts n Dst

(67)

n=1

whereas an ordinary binomial expansion based on (64) for t = 0 yieldsm

rom exp (imOo) = DosmI(:)(;:sJ exp (inOs) (68)

n=O

but the upper summation limit m can be replaced by 00, since

(:) = 0 for n > m.

When inserting (66) in (62), the imaginary terms Wt and i7t can be omittedsince they do not contribute to the real part. When (67) is introduced into(62) it contributes terms in exp (-in Os)which are not kept explicitly in (63);the implicit conjugate term in A:'Jt of (62), however, contributes to the explicitpart of (63) through the conjugate of (67). One thus obtains

CPs.O=- ~ {IsIOgrs+ I[ItIOgDst+

fJ, { ( as)n 1I[(r, )nCPs.n=-- Ans - +- -It - +27t rs n Dst (70)

m m

m m

F' --.-,'

THEORY OF THE PROXIMITY EFFECT IN MULTIWlRE CABLES 29

In these expressions, and in the following, all summations on n or m are for1,2, ... , 00, unless stated otherwise, whereas the summations in t are from 1to N, excluding s. Finally, since

(:) = 0 for m < n,

the last term of (70) can be rewritten as00 00

m=n m=OWith (71), (7.0) is of the form

(72)

where Cns is an abbreviation for

m

00

(73)

m=O

The boundary condition (58) on rs = as applied to the terms in n = 0 of (61)and (69) yields, using Jo' = -J1>

jw,u { Is Jo(ks as)U, =--- Is log as+ +

27t' ks as J1 (k,as)

m m

For n =1= 0, (58) is equivalent to cp = 'Ijl since U, is constant, hence independentof es> and this yields

_,u_ B Jn(ks a.) = _!_ (A + C )kns 'k) ns ns •2rcas s Jn ( s as 2rc

(75)

The boundary condition (59) for n = 0 is automatically satisfied, owing to (51).For n =1= 0 it yields

(76)

30 v. BELEVITCH

Elimination of Ens between (75) and (76) gives

Ä.nCns =--Ans

n(77)

owing to the identity

1+ nJnCka)f[kaJn'(ka)]

1- n Jn(ka)f[ka Jn'(ka)]

Jn_l(ka)

Jn+1(ka)

resulting from the recurrence relations for Bessel functions. With a notationsimilar to (18) with k; as, elimination of Cns between (73) and (77) yields thelinear system

co

m=l

eo

+ n(as)n ""' An+m 0 (n +m) (Dos)m = "\' Ir (!:_)n.ao ~ . n ao U o;

m=O

(78)

At this stage one can already obtain the final equations for the cable withoutscreen by making ao = 00 in (74) and (78). One thus gets

m

jw,u { Is Jo(ks as) L[ L ·(ar )mJ}U,=--- [s log as+ + [rlogDsr + Amr - .27t k, as J1(ks as) Drs

m (80)

In order to express the boundary conditions on the screen we expand (62)into a Fourier series in 00, using (64) and (69) with t = 0 rewritten in the form

rs exp (iBs)= ro exp (Wo) - Dos·

lts m-th negative power is, by a binomial expansion,

exp (-imOs) exp (-imBo) [Dos J-m---- = 1 - - exp (-Wo) =

~m ~m ~<Xl

= exp (-imOo) ""' (m + n-l) (Dos)n exp (-inOo) , (81)rom ~ m-I r«

n=O

THEORY OF THE PROXIMITY EFFECT IN MULTIWIRE CABLES 31

and its logarithm is

log's + iBs = log z, + iBo- "\' ~ (DOs)n exp (-inOo). (82)Ln '0·n

Inserting the conjugates of (81) and (82) into (62) and changing a dummyindex for later comfort, one obtains

(83)

s m n=O

Expansion (83) is of the form

with

(/Jo= (/Jo,o + L: (/Jo,p exp (ipOo)p=l

fJ,(/Jo,o=--(L:ls)log,o =0

27t s

since there is no overall current in the screen. For p =1= 0, the terms exp (ipOo)originate from combinations with m + n = p in the last term of (83) wherethe double sum in m,n thus reduces to a simple sum in n from 0 to p - 1and one obtains

sp-l

n=O

Which is of the form

(85)

32 V. BELEVITCH

where Cpo is an abbreviation for

p-l_L Is (D~s)P LL ( p-l ) (as)p (D~s)nCpo _- - - + Ap-n.s - -. (86)p ao p-n-l ao as

s n=O

Inside the screen the solution of (56) is

# w ."Po= - - L [Ep Jp(ko ro) + Fp Yp(ko ro)] exp (lP()o).

27t o=1

In the outer space the solution of (57) is

eo

# L (b)P .f{J = -- Gp - exp (lp()o).27t ro

p=l

In both expressions, Ep, Fp and Gp are undetermined coefficients. The termsin p = 0 are missing due to 10 = 0 and the corresponding boundary condi-tions are automatically satisfied. For p =F 0, the boundary conditions onro = bare

Ep Jp(ko b) + Fp Yp(ko b) = Gp,

ko [Ep J/(ko b) + Fp Y/(ko b)] = -p Gp/b.

(87)

(88)

The boundary conditions on ro = ao are

Ep Jp(ko ao) + Fp Yp(ko ao) = Apo + Cpo, (89)

ko [Ep J/(ko ao) + r, Y/(ko ao)] =!__ (Apo _ Cpo). (90)ao

Elimination of Ep, Fp and G; between (87) and (90) yields the relation

(91)where

p-l

Jp-1(ko ao) Yp-1(ko b) - Jp-1(ko b) Yp-1(ko ao)~= . ~Jp-1(ko b) Yp+1(ko ao) - Jp+1(ko ao) Yp-1(ko b)

Finally, elimination of Cpo between (86) and (91) yields

Apo =wp L[~ (::sY - I Ap._n.s(pP~~I)(::y-n(::s)"1(93)n=O

The final set of equations is (93), (78) and (74), but Apo can be eliminated

THEORY OF THE PROXIMITY EFFECT IN MULTIWIRE CABLES 33

from (74) and (78) since (93) is explicit in it. The elimination is rather heavyand replaces the double sum of (78) by a triple sum. For the case of a perfectlyconducting screen, however, (92) gives Wp = 1 for all p, and the triple sumcan be reduced analytically to a double sum, as shown in appendix B (of theforthcoming Part 11).With the notation

mln(n,m)

I (m+n-r-l)! ( U )m+n-rFnm(u) = nm --

(n-r)! (m-r)!r! 1 - u(94)

r=O,(78) and (74) become respectively

co

m=l

co

q m=l

= ~ It (.!2_)n _ ~ (a 2 as~~q D )n Iq, (95)~ o; ~ 0 Oq Os

q

co

q m=l

co

(96)

In the final equations (95) and (96) the summation in q is for 1, 2, ... , N,including s.Note that, if the x-axis of fig. 2 is rotated anticlockwise by an angle q;, all

distances are multiplied by exp (-iq;) so that (95) is not invariant. It is, how-ever, easily checked that if, in addition, all' Ans are replaced by Ans exp intp,hence A:t by A:t exp (-imq;), the equations (95) turn into themselves aftermultiplication by exp (-inq;). The same transformations leave (96) invariant,except for a constant imaginary term produced by the logarithms. This has no

34 v. BELEVITCH

effect since only the real part of (96) with respect to i is physically meaningful.In conclusion, a rotation cp of the coordinate system multiplies all An. byexp (incp).

4. The pair

We first consider the screened symmetric pair of fig. 3 with al = üz = a,0"1 = 0"2 = 0", 11 = 1, 12= -1, D12 = D, D2l = -D with D > 0, andDOl = -Dj2; D02 = Dj2. The problem is symmetric with respect to thex-axis of fig. 3 and has odd symmetry with respect to the y-axis. The x-sym-metry means that the problem is invariant when '1 exp (iOl) and '2 exp (i02)are changed into their conjugates, so that all coefficients Anl,An2 are real in i.The y-symmetry means that the solutions must be changed into their negativeswhen '1 exp (Wl) is changed into '2 exp [i(n-- (2)], and conversely, and thisforces

(97)

Fig.3

THEORY OF THE PROXIMITY EFFECT IN MULTIWIRE CABLES 35

Owing to (97), the two sets of equations (95) with s = 1,2 become identical,and it is sufficient to consider the first set, which becomes (with Anl written An)

m m

(98)

Also the impedance per unit length is simply Z = 2Ul, hence, by (96),

jw,u { Jo(ka) a 1 _ D2/4ao2Z=--- +log--log -

7t ka Jl(ka) D 1+ D2/4ao2

(99)

m m

We setu = (D/2ao)2 (lOO)

and in:troduce the abbreviations

(101)

s, (u) = ( 2U) n_(~)n.I-u I+u

Finally, replacing An by An/~n, (98) simplifies to

(102)

(103)

m

and, by comparison with (14) and (2), (99) separates into

,u (D 1+ U)L = - log - -log -- ,7t a I-u

(104)

jw,uZprox = -- L Am(l + gm)'

7t m(l05)

In particular in the absence of screen (ao = 00, hence u = 0 and Gnm = 0,gn = 0), (103) and (105) reduce to (16), (12) and (15). Note that the screenalways decreases L.

We next treat the unsymmetric pair with al =1= a2, al =1= a2 but only without

36 V. BELEVITCH

screen, so as to obtain the Manneback pair as a particular case. We stilI take/1 = 1,12= -1. The equations (79) and (80) become

(106)

m

(107)

m

(108)

m

U2= - jw,u [-lOg a2 _ JO(k2 a2) +log (-D) + '\:' A;"l (a1)m].27t k2 a2 J1(k2 a2) L..,; D

m (109)

The impedance per unit length is deduced from (108) and (109) by Z = U1-U2

where term ire can be disregarded. By comparison with (I4), one obtains (11)and

(110)

m

which must be restricted to its real part in i. For the Manneback pair witha2 = 0, (I07) yields An2 = 0 and the solution of (106) is

An1=-_I (a1)n.Ä.n1 D

Substitution in (110) yields (20) where a is the radius al of the thick wire.

5. The quad

The numbering of the wires and their positions in the coordinate system areshown in fig. 4. We will treat the phantom mode with

11 =13 = 1; 12=h =-1

and the parallel mode with

11= 14= 1; 12= 13=-1.

(111)

(112)

If all fields and currents of the second mode are turned clockwise by 90°, whilethe wire numbering is unchanged, one obtains the situation

11= 12= 1; 13= h=-1.

The superposition of (112) and (113) gives

(113)

THEORY OF THE PROXIMITY EFFEcr IN MUhTIWIRE CABLES 37

y

12

~ x

03 4

Fig.4

11 = 2; 13 =-2; 12 = 14 = 0 (114)

and corresponds to the excitation of pair 1-3 alone (side mode). Since theproblems (112) and (l13) only differ by notation, it is sufficient to consider(111) and (112). These will be treated simultaneously with the conventions

(115)

where the upper signs refer to the phantom mode and the lower signs to theparallel mode.The parallel mode is symmetric with respect to the x-axis, whereas the

phantom mode has odd symmetry. This forces

Both modes have odd symmetry with respect to the y-axis, and this forces

An2 = -r-r A:I; An3 = _(_)n A:4.

As a consequence, all Ans are expressed in terms of AnI = An by

An2 = _(_)n A:; An3 = ±(_)n An; An4 = =F A:. (116)

and (116) checks with (115) for n = 0 and Aos = Is.Both for the parallel mode and for the phantom mode, the total voltage

drop per unit length is twice the voltage drop on any wire carrying the positiveunit current, for instance 2UI• Since the total current in one direction is 2 inboth cases, the impedance is always

(117)

For the side mode (114), the voltage drop is UI - U3 and results from thesuperposition of UI - U3 of the parallel mode (112) with the voltage UI - U3

38 V. BELEVITCH

of (113), which is U4 - U2 of (112), due to the 90° rotation. Now, in the mode(112), UI - U3 = U4 - U2 = 2UI so that the voltage drop in the side modeis 4UI• Since the total current in (114) is 2 as in (112), the impedance of theside mode is twice the impedance of the parallel mode.Owing to (116) and (117) it is sufficient to deal with the equations (95) and

(96) with s = 1. They become

m

_ [ 1 F. (DOl D~2) ± _1 F. (DOl D~4)J A' } =(_D')m nm 2 D'm nm 2 m02 ao 04 ao

[1 (1 1) ( D' )n ( D' )nn 01 02=a --n-± -n---n- - 2' + 2' =+D12 Dl3 D14 ao - DOl DOl ao - D02 DOl

(118)

jw,u {a Dl3 JO(ka)z=--- Iog-±log-+---21t D12 Dl4 ka JI(ka)

m m

m

(119)

m

We haveD21 = D/V2; D41 = iD!V2; D31 = D exp (i1t/4);

Dos = (D/2) exp (i1t/4) exp [i (s - 1) 1t/2]

and the remaining distances are deduced by (65). We also set

THEORY OF THE PROXIMITY EFFECT IN MULTIWIRE CABLES 39

(_I)n

An .. T B; exp (-in7t/4). (120)

With the notations (19), and (100), the equations (118) times _(_)n exp (in7t/4)become

ÀBTnm {[exp (n-m) i7t/4± exp (m-n) i7t/4] 2(m+nl/2 Bm =F B:.} -~ +(i2n

m

m

= 2n/2 [exp (in7t/4)± exp (-in7t/4)] =+1 +

[( U )n (_U)n ( iu )n (-iU )nJ+(_2)n -- ± -- =F -- - --I - U 1+ U 1- iu 1+ iu

(121)

whereas (l19) separates into

ft ( D I-u I+U)L=- log-V =+logV2+log--±log-- ,

27t a 2 1+ iu 1- iu(122)

Zprox = - j;: [2.:{2m/2 s; [-exp (-im7t/4) =F exp (im7t/4)]± B;}-m . (123)

For the phantom mode, all coefficients in (121) are real, so that the equationsand their conjugates force Bm to be real. With the notations

(124)

_ (-2U)n (2U)n (-2iU)n (2iU )nhn(u)- -- + -- - --. - --. ,1 - u 1+ u 1- lU 1+ JU

(125)

defining real functions, (121) becomes

'" {T. [2(m+nl/2+1 cos (m-n)7t -IJ + H. } B _ An B =~ nm 4 nm m (i2n nm

n7t= 21+n/2 cos--l + h4 n

(126)

40 v. BELEVITCH

whereas (122) and (123) give

L= ~(IOg D -log 1+ U

2

),27t 2a 1 - u2

Zprox = j;: I (21+mI2 cos :7t - 1 + hm) e;m

(127)

(128)

For the parallel mode, (121) becomes, using (101) and (102),

I {Tnm[ _i2(m+n)/2+1 sin (m~n)7t s; + B: ] + s; Gnm(u) + B: Gnm(iU)}-

m

Àn ntx- - Bn = i 21+n/2 sin - + 1 + gn(u) + gn(iu).

62n 4

In (122) and (123) only the real parts in i are relevant. This gives

L = _!__ (lOg D _ log 1+ U)27t a 1 - u

which is one half of (104), and

Zprox =- j;: I[i21+mI2 sin :7t Bm-B:-Bmgm(u) + B:gm(-iu) ].(131)

m

Since (101) and (102) are odd real functions of U, we set

Gnm(iu) = irnmCu); gn(iu) = iYn(u)thus defining real functions rand y. With

and(m-n)7ts.; = Tnm2(n+m)/2+1 sin--4--·

the real and imaginary parts of (129) separate into

(129)

(130)

(132)

(133)

(134)

m

whereas the real part of (131) gives

Bns = !5n Ans exp (-3mti/4). (137)

THEORY OF THE PROXIMITY EFFECT IN MULTIWIRE CABLES 41

m

The final equations (135) and (136) of the parallel mode also hold for theside mode. Moreover, the coefficients An and C; have then a simple physicalinterpretation, which will now be established. The coefficients are deducedfrom the original coefficientsAn of (118) and (119) by the transformations (133)and (120), and the original coefficients determine the various space-harmonicsof the potential on the boundary of the four wires by (116), where the lowersigns are used for the parallel mode. In accordance with the last paragraph ofsec. 3, the transformation (120) which replaces An by

B; = !5n An exp (-3ni7t/4)

corresponds to a rotation by -37t/4 of the coordinates or, equivalently, to arotation by 37t/4 of the cable in the original coordinate system, changing fig.4into fig. 5. Consequently all coefficients Ans are changed into

For the second parallel mode (113), the 90° rotation multiplies all coefficientsby exp (-ni7t/2) and changes the subscript s into (s - 1) (mod 4). The coeffi-cients of the side mode are thus

Bns' = Bns + (_i)n Bn,s-l(mod 4)' (138)

Eliminating Bns and Ans from (137) and (138) and (116), and using (133), oneobtains

4

3

2

Fig,5

42 V.BELEVITCH

Bn1' = _(-)n Bn3' = 2An !5n,

Bn2' = (_I)n Bn4' = 2(_i)n+1 C; !5n.(139)

This shows that the coefficients An and C; occurring in (135) and (136) areproportional to the various space-harmonics of the potentials in pairs 1-3 and2-4 respectively, for excitation in pair 1-3 alone, and the symmetries in (139)reflect the symmetry with respect to the x-axis and the odd symmetry withrespect to the y-axis of fig. 5.

We now consider the more general quad of fig. 6 where the mutual distancesbetween the wire centres are the same as in fig. 5 but where the two pairs havedifferent radii and conductivities: a1,G1 for pair 1-3 and a2,G2 for pair 2-4. Itis proved in appendix C (of the forthcoming Part II) that the impedance ofthe side mode in pair 1-3 is still twice (136) where Am and Cmare solutions of aset of equations, called (a) and (b) and deduced from (135) by changing An/!52n

into An1/!5/n and An2/o22n, respectively, with 151 = a1/D, 152 = a2/D.For a2 = 0, eq. (b) multiplied by !52ngive C; = 0 and eq. (a) and the double

of (136) reduce to (103) and (l05) for pair 1-3 alone. For al = 0, eq. (a)multiplied by !51ngive An = 0 and eq. (b) and the double of (136) reduce to

L A2C me(G T.) C n n - 21+n!2' +nm- nm m---- - S1O- Yn,

!5/n 4(140)

m

_ Jwft '\' ( 1+m!2 . n7C )Zprox - ---;- i.....J 2 sm""4 + Ym c; (141)

m

These relations are of interest since (141) characterizes the eddy-current losses inpair 2-4 produced by the side mode in pair 1-3, as measured by an impedance

3 X

2

Fig.6

THEORY OF THE PROXIMITY EFFECT IN MULTIWIRE CABLES 43

increase in pair 1-3, in the case where there is no back reaction from pair 2-4on the current distribution in pair 13, because the latter is thin.Finally, although the functions (124), (125) and (132) are real in u, i occurs

in their algebraic definitions, which is unsuitable for numerical work. Thedifficulty is circumvented by the elementary trigonometrie transformation

( iU)n (iU)n { U}n--. + --.- = 2 tCOS (n arccot u).1 + lU I-lU (1+ UZ)

(142)

M BLE Research Laboratory Brussels, November 1976

REFERENCES1) G. Mie, Ann. Phys, 2, 201-249, 1900.2) J. R. Carson, PhiI. Mag. 41, 607-633, 1921.3) S. Butterworth, PhiI. Trans. Roy. Soc. 222A, 57-100, 1921.4) A. H. M. Arnold, J. lEE 88, 49-58, 1935.5) R. W. Lorenz, Die Frequenzabhängigkeit der Leitungsbelagsmatrizen von zylindrischen

verlustbehafteten Leitersystemen. Diss. Darmstadt, 1971.6) R. W. Lorenz, Frequenz 25, 227-234, 1971.7) V. Belevitch, Philips tech. Rev. 32, 221-231, 1971.8) J. R. Carson, Bell. Syst. tech. J. 7, 11-25, 1928.9) H. B. Dwight, Journ. AIEE 42,961-970, 1923.

10) C. Manneback, Journ. Math. Phys. (MrT) 1, 123-146, 1921.11) S. Pincherlé, Acta Math. 16, 341-363, 1892.12) H. Kaden, Wirbelströme und Schirmung in der Nachrichtentechnik, 2-te Aufl., Springer,

Berlin, 1959, p. 33.