PH I LI PS TECHNICAL REVIEW------------ VOLUME 40,1,982, No. 4

Computer-aided research on multiwire telephone cables

r. Veldhuis

Informed forecasts of the future of telecommunications include the emergence of an 'Inte-grated Services Digital Network' (ISDN). The idea of such an all-embracing network, inwhich text, sound and image would be transmitted by purely digital methods, may still seemrather visionary, at least on a world-Wide scale, but it powerfully infiuences current thinkingon existing telecommunication systems. The adaptation of these systems for the most eco-nomic yet properly engineered transition to such a 'network of the future' is a challengingtopic of Philips research. The article below deals with theoretical work that is particularlyrelevant to this research. The main question was the extent to which the existing telephonesystem, with its conventional copper cable, can be adapted for the digital transmissionof speech and the provision of a wide variety of new' data services. The analytical methodpresented here, based on an advanced theoretical study of telephone cables with twisted wires,clearly illustrates that simulation by software provides a most promising research method forthe study of large systems.

The copper cable and digital communication

The conventional copper cable used in existing tele-phone networks is once again the subject of a great dealof technological and scientific research. This tends tobe overshadowed somewhat by such a revolutionaryinnovation as the' glass-fibre cable, whose introduc-tion has of course attracted a great deal of attention.As is often the case, however, this new developmentdoes not signify the immediate demise of the existingtechnology [1], nor even that all the research on itis complete. Much practical and theoretical workremains to be done.

Research on the copper cable has two objectives:making better telephone cables and making better useof existing telephone networks. The first objective isof major importance to the cable manufacturer. In hisview, a telephone cable of high quality should give nomore than 1 to 2 dB attenuation per kilometre of cable(at frequencies up to 100 kHz) in each of the groupedtransmission channels. 'High quality' also implies thatinterference between the signals travelling along theIr J. Veldhuis is with Philips Research Laboratories, Eindhoven.

individual transmission channels is negligible. Themanufacturer therefore seeks to construct his cablesso as to minimize signalleakage between neighbouringchannels ('crosstalk'). In addition to low attenuationin the direction of transmission, his aim is to achievehigh isolation between the transmission channels (morethan 40 dB at frequencies up to 100 kHz in cables upto 5 kilometres long).The second objective, making better use of-existing

cables, is of particular interest to those who wish totransmit signals other than analog speech signals onthe existing networks. A good example of such ex-tended use is the transmission of a wide variety ofdata with the aid. of digital signals. Indeed, this is_[1) The Netherlands Postal, Telegraph and Telephone service,

which is responsible for the Dutch telephone system, requiresfor reasons of operational reliability that subscriber connec-tions should be energized from the telephone exchanges. Theglass-fibre cable does not at the moment appear to be suitablefor this power-supply function. Research is however beingdone in this field (R. C. Miller and R. B. Lawry, Bell Syst.tech. J. 58, 1735, 1979). It is not yet entirely clear how theglass-fibre cable will ultimately be used, along with the coppercable, in local telephone networks.

86 J. VELDHUIS Philips tech, Rev. 40, No. 4

already happening, though only on a small scale. Indi-vidual users who have an appropriate terminal canrent telephone lines to transmit their own data to acentral computer, and receive data from the com-puter. In future all the ordinary subscribers may beoffered digital communication facilities through thenational telephone network.A consequence of the new applications will be that

the frequencies of the electrical signals in the existinglocal network will extend over a larger bandwidththan has so far been necessary. This will become mostapparent when eventually the analog speech signals ofsubscribers are also converted into digital signals.Such a situation will require transmission channelswith a bandwidth of the order of 100 kHz, some 20times the bandwidth required for analog speech sig-nals alone (4 kHz).A disadvantage of such a larger bandwidth is that it

entails greatly increased crosstalk between the trans-mission channels, owing to the higher frequenciesin the signal. The attenuation and phase shift alsodeteriorate rapidly as the frequency increases. Con-sequently, the signals with a broad frequency spec-trum will be much more seriously mutilated than thepresent analog speech signals, with their bandwidth ofonly 4 kHz.The only reallimiting factors in the plans to provide

telephone subscribers with digital communication andnew data-communication services are the existing net-work and its cables. It is not easy to obtain an overallpicture of the limitations introduced into a cable net-work because of crosstalk and attenuation. Making acomplete series of tests for every cable in the networkwould be an almost impossible task.A simpler and much cheaper method of investigat-

ing the effect of cable limitations is to use a computerto simulate the telephone network as a system, withthe emphasis on transmission and crosstalk. A simula-tion program of this type is based on a formal modelof the cable taken as an elementary structural unit.Network calculations will then give a good simulationof the transmission behaviour of the cable and theinterference due to crosstalk. Worst-case situations inthe cable network can then be determined.

To keep the simulation flexible the software shouldbe organized in modules, in such a way that the indi-vidual blocks corresponding to parts of the network,e.g. cables, switchgear, transmitters, etc., can easilybe included ..The treatment in this article is confined to the cable

block; a substantial part of the software for this blockhas now been completed. The calculations for thisblock are made at three hierarchical levels, each withits own cable model. At each level the program takes

particular input data, and employs the model to arriveat a result at the appropriate level. These results inturn form the block of input data at the adjacentlevel. This division into levels simplifies the cable cal-culations, which is important for efficiently debuggingthe programs in the initial phase.

In addition to these three levels, the schematic ar-rangement adopted has a fourth level. At this level thecomplete telephone network can be simulated by acomplicated product of various transfer matriceswhose characteristic data come from models of thethree main parts of the network (cables, switchgear,transmitterslreceivers). Improvement and expansionof the communication facilities will eventually be at-tained at the fourth level. For the work described inthis article the fourth level only acts as the environ-ment for the cable-simulation program. This simula-tion program has been completed at the third level.

In the sections that follow, the geometry of themultiwire telephone cables used in the Netherlandswill first be described, and a number of interactionsbetween the conductors will be discussed. Attentionwill then be turned to the three levels in the model cal-culations, in particular to finding a cable model andderiving from it expressions for the primary cableparameters at the first level. These form the basis forthe network calculations, which eventually result insignal-transfer matrices. The final section deals withthe software developed and the values of the primaryand secondary cable parameters calculated with thissoftware. A detailed example of one program moduleis included, mainly to demonstrate the functionalstructure of the program and to show that certaingeneral quality criteria for software are satisfied.The article concludes. with a comparison betweenvalues that we have calculated and the results ofcable calculations and measurements carried out else-where.

Geometry and interference

Conductor configuration

The geometry of a multiwire telephone cable ismainly determined by the distribution of the wiresover the cross-section of the cable. Fig. 1 shows thecross-section of a commonly used type of telephonecable in the Netherlands. The cable contains a largenumber, of conductor wires (there may be severalhundred). The wires are distributed in groups of four,called 'quads', in concentric layers. There can be asmany as six or seven layers, including the central partof the cable. From the inside outwards the number ofquads increases by six per layer. In each quad thefour wires are twisted together at a particular pitch,

Philips tech. Rev. 40, No. 4 TELEPHONE CABLE~ 87

tions between the pitches are properly chosen the along or crosstalk between the individual circuits just

called the group pitch. The wires do not therefore runparallel to the axis of the cable but are wound concen-trically in the form of interwoven helices. In their turnthe quads are wound as helices in the individuallayerswith a particular pitch per layer, called the layer pitch.The layer pitches differ from one another, and thegroup pitches also differ for adjacent quads in thesame layer.

This relatively complicated structure with groupand layer pitches has the advantage that if the rela-

crosstalk effects between pairs of wires can be greatlyreduced. This is true whether the crosstalk is due toinductive or capacitive coupling. Because the wiresare twisted, the structure is compact and strong - an-other advantage of some importance.

The aim is to select the wires for the required trans-mission channels in such a way as to achieve optimumtransmission with minimum crosstalk in the cable.

Many telephone subscribers in the Netherlands areconnected to the system by a single quad. The tele-phone set is connected to the network by one of thetwo side circuits (fig. 2). The remaining connectionsare held in reserve.

In this article, calculating the transmission or cross-talk in a cable is taken to refer to the transmission

described. The electrical quantities such as potential,current and also the cable parameters in the rest ofthis article always refer to a particular circuit, unless itis explicitly stated that the quantity relates to a par-ticular wire.

!]Fig. 1. A telephone cable with quad geometry (made by NKF Kabel B.V., Waddinxveen, TheNetherlands). A quad (Q) consists of four conductors. The central part of the cable has threequads; the next layer (L I) has 9 quads, then there is a layer with 15 quads (L2), a layer with 21quads (L 3), and so on. The entire configuration is contained inside a plastic-encapsulated screen-ing shield of aluminium strip. The type shown here is widely used in the Netherlands for local net-works as the cable between exchanges and subscriber groups. (The cable sample has been madeavailable by the Creative Services and Publicity Division of NKF Kabel B.V.)

Wires within a quad, and also quads themselves,can be used for making a transmission channel (intelephone cables often called a circuit). It is possibleto use a wire for more than one transmission channelat the same time (a familiar example is given infig. 2).In this case one quad and a conducting shield formfour transmission channels, which can operate simul-taneously and independently of each other.

In cross-section the wires of a quad form the corners of a square.Two circuits, the side circuits, each consist of two conductors dia-gonally opposite each other. Another circuit, the phantom circuit,is obtained by taking each side pair as a single conductor. Thefourth circuit, the asymmetrical circuit, is produced by taking theshield as one conductor and the full quad as the other.

The case of fig. 2 is based on the general proposi-tion that a cable with N conductors inside a conduct-ing shield can provide a totalof N independent trans-mission channels [21. The transmission channels in atelephone cable are not chosen arbitrarily, of course.

The electrical quantities in the circuits and the correspondingquantities for the wires are linearly related in a way that can bedescribed by a matrix equation. In the case of fig. 2, for example,there are four circuir potentials (VsI, VS2, VF, VA), referring to thetwo side circuits, the phantom circuit, and the asymmetrical circuitin that order. The matrix equation is:

o -I

o

where VI, V2, Va and V4 are the potentials of the individual wires(defined with respect to the sheath). The same matrix applies to therelation between the charges on the circuits and wires. The fourwires and the shield are shown as perfect conductors in fig. 2; thisassumption is made when the capacitances of circuits (and alsobetween circuits) are calculated, as ratios of charges and potentials.On the other hand, finite conductivity is assumed when dealing withthe currents and voltages in the longitudinal direction of a cable;this assumption enables the resistances in the longitudinal direction

[2] W. Klein, Die Theorie des Nebensprechens auf Leitungen,Springer, Berlin 1955.

88 J. VELDHUIS Philips tech. Rev. 40, No. 4

to be derived in addition to the self-inductances and the mutualinductances. This dual approach to the cable calculations wil! beexplained in more detail later.

Interactive effects

Although the electrical phenomena in multiwiretelephone cables are described by Maxwell's classicalequations, it is not so easy to obtain a clear picture ofall the possible interactive effects inside and betweenthe conductors. The principal effects are listed in

reduced) most strongly on the side of the cross-sectionthat lies closest to an adjacent conductor. It is relatedto the classical skin effect, in which each conductor isaffected by its own magnetic field rather than the fieldfrom adjacent conductors. Consequently, for the skineffect, the current density in the cross-section is onlyradially dependent, with a maximum at the outside ofthe conductor. If the smallest spacing between the con-ductors in a cable is at least five times the radius of theconductor, the proximity effect is negligible in com-parison with the skin effect. In the determination of

signal in- 1---- cable ----I

TN' t. - f I ~ !V -..- ~+ Vî VS1..

I ~ t J

~- I,T (2) 1 ---

~i T~ VF ~r-

IT ~ Ç01 T ~.., .. 1 -- ~ t• -

t IVS2. I -- 3. ~

T 10 . 1II t-I

._}\

T

signal out --

r~

T

Fig. 2. Simple example of commonly used conductors in a multiwire telephone cable. The cablecontains four wires (numbered), which are screened by a conducting shield (sh). Four independentcircuits are formed: the two side circuits (blue), the phantom circuit (green), and the asymmetricalcircuit (red). Each conductor is used for three signal currents. The input signals are supplied viathe primary windings (not shown) of four input tr ansforrners T. The secondary windings of thesetransformers are connected by terminals to the conductors and the shield. In a similar way thesignals at the end of the cable are delivered to the outside world via terminals and four outputtransformers T (whose secondary windings are not shown). Only one signal is effective on a par-ticular transmission channel; any other signals cancel each other out by balancing. VSI, VS2 circuitpotentialof side circuit. VF circuit potentialof phantom circuit. VA circuit potentialof asym-metrical circuit. VI, ... , V4 conductor potential (defined with respect to sh).

Table I. Some of these effects increase strongly withfrequency. An example is the proximity effect in theconductors - a current-concentration phenomenonthat affects the resistances and self-inductances ofconductors as well as the mutual inductances betweenconductors; the effect increases strongly with fre-quency [41.

This effect, highly undesirable at high frequenciesin telephone cables, is an asymmetry in the currentdistribution over the cross-section of a cable con-ductor, caused by the magnetic field of the currentsin adjacent conductors. The essential feature is thatthe current density in the conductor is increased (or

capacitances in telephone cables, which is essentiallyan electrostatic-field problem, a second proximityeffect has to be taken into account. This is an asym-metry in the charge distribution (compare with thecurrent distribution above) on a conductor as' a con-sequence of charges in adjacent conductors. To avoidconfusion this effect is referred to in Table I as the'electrostatic proximity effect' with the symbol Pq. Allthese effects ultimately contribute to an increase in theattenuation and phase shift in the conductors, meas-ured per metre of cable. The proximity effect Pandthe skin effect S increase with the signal frequency,but Pq is independent of frequency.

~---------------------------------------------------------Philips tech. Rev. 40, No. 4 TELEPHONE CABLES 89

H. E. Martin's cage effect [3], mentioned in Table I,also increases the attenuation per metre; more pre-cisely, it increases the effective capacitance per metreof a circuit. The effect arises because each circuit in atelephone cable consisting of twisted wires is effectivelyenclosed by a Faraday cage. This cage - not to beconfused with the cable shield - is formed by thegroup of other circuits most closely surrounding thecircuit under consideration. All these cages behavelike conducting shields at the same constant potential.If the wires forming a cage are twisted together cor-

separation'. If the cage encloses two circuits insteadof one - this could happen if non-ideal twisting ofthe wires effectively makes one ofthe adjacent circuitsnot part of the cage - then the induction charge willalso appreciably increase the capacitive couplingbetween both of the enclosed circuits. V. Belevitchhas recognized this extension of the effect, and hasgeneralized Martin's theory and at the same time suc-ceeded in finding an explanation for the occurrence ofcertain non-ideal twisting pitches [5], an effect knownin practice but not really understood previously.

Table I. Influence of skin effect, shield, proximity effects and cage effect on the primary cableparameters in a telephone cable with the geometry given in fig. 1. S skin effect. P proximity effect(current). Pq electrostatic proximity effect (charge). K cage effect as described by Martin [3].

A separate column indicates which of the primary cable parameters are frequency-dependent andwhich are not.

Primary cable parameter of circuit m,or between circuits m and n(defined per metre)

Change due to frequencydependent

S Shield P Pq KResistance Rm Yes Yes Yes No No Yes

Self-inductance Lm Yes Yes Yes No No Yes

[a] No Yes [cl No Yes Yes Yes

Capacitance Cm [a] No Yes [c) No Yes Yes No

[b] Yes Yes Yes

Shunt conductance Gm

[b] Yes Yes Yes

Mutual resistance Rmn Yes No No

[a) No Yes [c) No Yes Yes No

Mutual inductanceLmn Yes No No

[a) No Yes [c) No Yes Yes

Capacitance Cmn

Conductance Gmn Yes

[a) Conductance and capacitance are related by the fixed relation G =wCtan ó, where ca (= 2nf)is the angular frequency and ó is the loss angle of the material between the conductors.

[b) Rmn + jwLmn = Um IIn, the ratio of the induced voltage per metre Um to the inducing currentIn. Owing to the screening effect of the shield, and also because of the proximity effect, thisratio also has a real part, Rmn, here called mutual resistance (per metre).

[c) The change brought about by the screening action of the cable shield only occurs in so far asthe circuit is not screened by the cage effect K.

rectly, their spatiallocation is such that the voltages atthe connection terminals of all the circuits belongingto a particular cage do not change when the centralcircuit in the cage is energized. (The surroundingcircuits and the central circuit are then decoupled elec-trostatically, making capacitive crosstalk impossible.)Local static-induction charges arise on the cage, andthese oppose the field-strength of the central field,which produced them. The corresponding voltagedrop between the two conductors of the central circuitin turn produces an increase in the effective capaci-tance, for constant charge. This increase may amountto 10 to 150/0.Also, because of the twisting of thewires - with a periodic variation of the distancebetween central circuit and cage - the induction in thelongitudinal direction of the cable will be alternatelystrong and weak, resulting in 'longitudinal charge

With all these effects influencing the behaviour ofcables, it is not surprising that it is so difficult to cal-culate the crosstalk between pairs of conductors prop-erly. The amplitude of the crosstalk signals dependscloselyon the method adopted for twisting the wiresin the cable. For these reasons the simple cable modelsnow in use are often no more than rough approxima-tions to the actual cable. The effects that occur in thefrequency range already important today cannot becorrectly described by these models, and the difficultyis only aggravated at even higher frequencies.

[3) H.-E. Martin, Die Berechnung der Übertragungseigenschaftensymmetrischer Leitungen unter Berücksichtigung des Verdral-lungseffektes, Arch. elektro Übertr. 18,293-308, 1964.

[4] • See for example P. Grivet, The physics oftransmission lines athigh and very high frequencies, Vol. I, Academic Press,London 1970.

[6) V. Belevitch, On the theory of cross-talk between twisted pairs,Philips Res. Repts 32,365-372, 1977.

90 J. VELDHUIS

The models

The simulation program on which we are workingattempts to take into account as far as possible all theinteractive effects between the conductors. A greatdeal of attention therefore had to be paid to the con-sequences of the complicated interweaving of the con-ductors. This was made possible by the theoreticalwork published in recent years, particularly by Bele-vitch, working with G. C. Groenendaal and R. R.Wilson [5]-[8]. Their treatment gives particular atten-tion to the two proximity effects and the cage effect.

INPUT SYSTEM

Philips tech. Rev. 40, No. 4

It provides a better understanding of the frequencydependence of the attenuation and the phase shift,which determines the transmission behaviour.

The approach can also provide a better descriptionof the inductive and capacitive couplings, which deter-mine the crosstalk between the pairs of conductors.

The block structure

Fig. 3 shows the block diagram of the cable-simula-tion system we have designed, giving the levels atwhich calculations can be carried out on the cables.

MODELDESIRED

CALCULATION OUTPUT

TELEPHONE CABLE==:> geometry

material

PHYSICS MODEL

Moxwell's equations

R,L,C, G___ primary cable para- ..-

meters I cableIcross-sec.

DISTRIBUTED

~ NETWORK ELEMENTS f--R,L, C, G network theory

2N-PORT,j.. m= 1, 2, ,2N'rmr, n=1,2, ,2N

f----- secondary cable para- f--meters I cable

I section

SECTION-MATRIXELEMENTS

cp m=1,2, ,2Nmn n=1,2,. ,2N

CASCADE NETWORK___ 2N-ports

matrix multiplication

CABLING TELEPHONE NETWORK~ TRANSFER-MA TRIX f--

ELEMENTS ?!i-~~/~C~/~; - - ~:- ::r ,...< "« f--I 1---" _____y./~ _/,I MODEL I ~ ~

L J i-~ .F-;:9~1\ --L.~---------, -_ Jr--'I TRANSMITTERS/ I I

I RECEIVERS r--'I MODEL IL _j

1> m=1,2, ,2Nmn n = 1, 2, ,2N

f-- system parameters f---I cable

COMMUNICA TI ONIMPROVEMENT & EXPANSION

e. g. digitaltransmission

reduction oftransmissionlosses andcrosstalk

.-ig. 3. Block diagram of computer-simulation software for telephone cables, The first three levelsrepresent the calculation of the elements of the transfer matrix - the system parameters <IJ mn forsignal transfer and hence the final cable model. These parameters, combined with the models forthe switching elements and transmitters/receivers (not dealt with here), will form a programsimulating the entire telephone network, shown at the fourth and lowest level. A program of thistype gives an efficient analysis of the transmission attenuation and crosstalk in existing telephonesystems, R resistance per metre. L inductance per metre, C capacitance per metre. G conductanceper metre, A 2N-port is a network consisting of N circuits, each with two connection terminals(a port) at the input and the output. The parameters CPmn are defined in the text.

:..Philips tech. Rev. 40, No. 4 TELEPHONE CABLES 91

At the upper level the calculation starts with dimen-sions and materials, the 'simplest' data for establish-ing the geometry and the electrical properties of a tele-phone cable. These data are substituted in Maxwell'sfield equations, which serve as a model for calculatingquantities such as the resistance per metre, the self-inductance per metre, etc., i.e. the primary cablepararpeters R, L, C and G. These quantities charac-terize the cable for an infinitely small element oflength (i.e. at a cross-section); because of the twistingof the wires, they will not be truly constant over thelength of the cable but will depend on the location ofthe cross-section. Owing to the complicated geometrythis first step in the calculations, in which the effects inTable I enter into the picture, is especially difficult tocarry out; it is the subject of a separate area ofresearch in which a great deal of theoretical work isunder way, as mentioned earlier.At the second level the model is what is called a 2N-

port, a network circuit built up from the quantities R,L, C and G calculated - at many cross-sections -from the first level, and referred to at this level as'distributed network elements'. A network of this typecan be used to form N independent circuits. A circuitconsists of two lines, with a 'port' at the input end andat the output end of the network. In each circuit thereare two potential drops: one across the input terminalsand one across the output terminals, so that there arealso two current levels. The network establishes alinear relation between all the currents and voltages atthe input and output. This relation is mostly clearlydescribed by a 'section matrix', a matrix of 2Nx 2Nelements (jJmn, the secondary cable parameters.

In the simple case of two conductors without a .shield the section matrix consists of four elements. Ifthe conductors are parallel and the primary param-eters thus independent of the length coordinate, thelinear relation between the current (Is) and the poten-tial difference (Vs) at the input and the current (IR)and the potential difference (VR) at the output of acable section of length ~z is not difficult to determine,and is:

.[ :OSh(Y~Z) Ze Sinh(Y~Z)] [VR]. (1)

- sinh(y~z) cosh(yó.z) IRZe

[6) V. Belevitch, Theory of the proximity effect in multiwirecables, Philips Res. Repts 32, 16-43 and 96-117, 1977.See also G. C. Groenendaal, R. R. Wilson and V. Belevitch,Calculation of the proximity effect in a screened pair and quad,Philips Res. Repts 32, 412-428, 1977.

[7) V. Belevitch, R. R. Wilson and G. C. Groenendaal, Thecapacitance of circuits in a cable with twisted quads, PhilipsRes. Repts 32, 297-321, 1977.

[8) V. Belevitch and R. R. Wilson, Cross-talk in twisted multiwirecables, Philips J. Res. 35, 14-58, 1980.

It will be evident that the pair (VR, IR) can be used inturn as the input signal to the next cable section. Thecoefficient Ze is the characteristic impedance, andy (=a + jP) is the complex propagation constant.The real part a is known as the attenuation constantand the imaginary part P as the phase constant.In this case Ze and y are given by

Ze = (R + jwL)/(G + jwc)]l (2)

and

y = a + jp = ((R + jwL) (G + jwc)Jl. (3)

The quantities Ze, a, P are known from conventionalcable measurements, and are also frequently referredto as secondary cable parameters (the name reservedin this article for the elements of the section matrix).

The objective at the second level is therefore to cal-culate the elements (jJmil. These can be derived fromthe primary cable parameters by means of ordinarynetwork theory. The 2Nx 2N elements characterize ashort length of cable (a 'cable section'). In the generalcase the derivation is much more complicated than inthe example given above, in which there is only onecurrent and one potential difference at the input andoutput.At the third level the transfer matrix of a complete

cable is calculated by treating the cable as a cascadedcircuit of the separate cable sections and then multi- .plying the appropriate section matrices in the propersequence. This results in the 2Nx 2N matrix elementsifJmn, the 'system parameters' of the complete cable.

Calculation of transmission and crosstalk

The primary cable parameters (R, L, C and G infig. 3) are quantities that only really become electric-ally significant when we consider a length of cable andnot a cable cross-section, and treat that piece of cableas an electrical network. Fig. 4 shows a network thatcan serve as equivalent circuit for a piece of cable oflength ó.z. Longer pieces are equivalent to cascadedarrangements of such equivalent circuits.A telephone cable, especially a modern high-quality

cable, generally provides such good transmissionchannels that the intricate problem of analysing thetotal network, via the determination of both transmis-sion behaviour and crosstalk, can with advantage besplit into two separate problems. The transmissionbehaviour and the crosstalk are then determined sep-.arately. Splitting the total problem in this way makesthe calculation much simpler and more efficient. It ispossible to make such a split because the energy lostduring transmission as crosstalk is at least an order ofmagnitude smaller than the energy used for the trans-mission.

92 J. VELDHUIS

The transmission problem of a cable with N con-ductors inside a conducting shield (sh) thus reduces tothe simple transmission problem - although it has tobe repeated N times - of a circuit with only two wiresin free space (shown in red in fig. 4). The solution to

Philips tech. Rev. 40, No. 4

agation of voltage and current waves in the entire net-work. To find a general solution to these equations- thus giving all the secondary cable parameters <pmll(fig. 3) - is so massive a task that numerical methodshave to be used.

Fig. 4. Simplified equivalent circuit of a short piece of telephone cable (length Az) in which N con-nection circuits are formed from N numbered wires inside a shield (sh), which at the same timerepresents the zero level for all the electric potentials V. The circuit (p,q) consisting of the wires pand q is shown in full. Only the network elements printed in red need be known to calculate thesignal transmission in the circuit. The dashed network elements form the 'environment' ofthe cir-cuit; they are necessary for calculating the crosstalk in (p,q). R, L, C, G are primary parameters,whose significanee is defined in fig. 3. LM mutual inductance per metre. All network elements referto separate conductors.

this problem is provided by the equations (1), (2) and(3) discussed earlier. In reality, of course, the wiresare not in free space, and if this simple calculation ofthe transmission behaviour using equations (1), (2)and (3) is to be sufficiently accurate, then the primaryparameters of this single circuit (the red network ele-ments in fig. 4) must be calculated with due allowancefor all the interactive effects in Table I, as they occurin the complete cable. As mentioned earlier, the twist-ing of the wires makes the primary cable parametersfunctions of z, the coordinate of length along thecable. The values of the primary constants used inequations (1), (2) and (3) are averages over the lengthof the cable.

The separate crosstalk problem is a much more dif-ficult problem in network theory, since the 'environ-ment' does have to be taken into account by includingall the network elements shown dashed in fig. 4. Thiswould in fact become the problem of solving thegeneralized telegraphist's equations [9], a large arrayof coupled differential equations describing the prop-

The quality of modern telephone cables is so goodthat the coupling between the different circuits is suffi-ciently small to allow a considerable simplification ofthe network structure of the 'environment' in fig. 4.In our approach to the calculations this meant that weneglected any feedback effects from a circuit receivinginterference to the circuit producing the interference,as effects of higher order.

'Indirect crosstalk', due to the induction of signals in a thirdcircuit as an interfering intermediate stage, and often discussedpreviously [2], has also been omitted from our network calcula-tions. This is because its contribution to the total crosstalk in high-quality telephone cables is negligible. The effect of the 'other' cir-cuits is included in the calculation of the primary parametersthrough the cage effect. The contribution from a cage, which causesa marked increase of the capacitances between the circuits enclosedby the cage, is much more important than this crosstalk via thirdcircuits. It would be more accurate, of course, to take both formsof crosstalk into account in the calculations; however, it is sufficientto include the cage effect alone if a first-order approach to crosstalkis considered acceptable.

Philips tech. Rev. 40, No. 4 TELEPHONE CABLES 93

Á signal-carrying circuit therefore only experiencessignificant interference as a result of direct crosstalkfrom the other main signals on the other circuits in thecable.

We have calculated both the Near-End CROSSTalk(NEXT) and the Far-End CROSSTalk (FEXT), forthe near and far ends of the circuit receiving inter-ference, by inserting the primary cable parametersin N. A. Strakhov's equations [101. The couplingparameters of importance in crosstalk (Cmn and Lmn)depend on the length coordinate (z), This dependencewas taken into account by substituting our calculatedcoupling parameters in the appropriate equations.Since many conversations are transmitted simultane-ously in a telephone cable, the crosstalk is often thesum of many contributions (with complicated statis-tical aspects). The annoyance caused by crosstalk in atelephone circuit depends of course on its magnituderelative to the desired speech signal. The annoyancedue tothe crosstalk increases with the recognition ofthis unwanted signal as speech.

Thèoretical background

The theory used here for calculating the primaryand then the secondary cable parameters is J. R. Car-son's well-known quasi-stationary theory [111, whichhas also been widely used elsewhere. The twisting ofthe conductors introduces fundamental difficultieswiththis theory, but we were able to find a way round thedifficulties.

Carson's theory shows that for perfectly parallelconductors signal transmission is possible in the formof waves of the quasi-TEM type, virtually transverseelectromagnetic waves. A subsidiary condition is thatthe signal wavelength should be large compared withthe distance between the conductors in a cross-section.Table 11 lists a number of field-strengths and currentdensities, etc., for cables carrying waves of the quasi-TEM type. These waves closely resemble pure TEMwaves, which can only be excited if the materialbetween the conductors is completely homogeneousand the wires are perfect conductors.Transmission with quasi-TEM type waves is effi-

cient. The mean lateral radiation of energy is virtuallynegligible and the attenuation in the direction of prop-agation is low for wires that are good conductors.

In applying the quasi-stationary theory there are two separatesteps in the calculation of both the transmission and the crosstalkbehaviour. The first step gives the primary cable parameters (thefirst level in fig. 3), by calculating them from the transverse com-ponents of the field vectors E and H. To do this, Maxwell's equa-tions of the electromagnetic field, with their boundary conditions,are solved for a cross-section of the cable. The second step gives thesecondary cable parameters (the second level in fig. 3), by deter-

mining the wave propagation along the cable. As mentioned earlier,the determination takes the form of an analysis of voltages andcurrents in the network model, the complete diagram in fig. 4.In actual telephone cables there is the complication.> already

discussed, that the conductors are not parallel, which means thatthe quasi-stationary theory does not apply. The distances betweenthe twisted wires vary periodically with the coordinate (z) alongthe length of the cable.To find the primary cable parameters at a cross-section we have

kept to the local values of the distances between the wires - definedbetween their centres. In doing so we have assumed that the wiresare locally parallel to the axis of the cable and that the cross-sectionmay be treated as an infinitely short length of an infinitely longcable with parallel wires (this is our stratagem). The quasi-station-ary theory is then applicable and it can be used to solve the fieldequations with their boundary conditions for the cross-section. Thetwisting of the wires can then be taken into account by slightlyaltering the position of th~ cross-section along the cable and solvingthe field equations with their boundary conditions again.

We have used this method of approximation to cal-culate the primary cable parameters; they turn out tobe periodic functions of the coordinate of length. Thisis in fact due to the periodic variation of the trans-verse distances between the wires.In the calculations of the crosstalk problem we

must take accurate account of the periodic variationof the primary cable parameters. For calculating the

Table 11. Field-strengths and current densities for electromagneticwaves of the quasi-TEM type (virtually transverse), in a cable with

. parallel high-conductivity wires. The longitudinal direction of thecable coincides with the direction of the z-axis. Inside and outsidethe conductors the z-dependence of these quantities is given byexp( - yz), where I' is the propagation coefficient.

~y

e

Inside conductors Outside conductors

EElectricfield-strength s, [a] ~ Ex, s, s,«Ex, s,

HMagneticfield-strength Hz «Hx,Hy Hz «Hx,Hy

JCurrent density J~O ""0

e,»Displacementcurrent density latDI «J =I' 0 [b]

[a] Ez inside the conductors is orders of magnitude smaller than Exand Ey outside the conductors.

[b] The value depends on the frequency and remains small up to themicrowave range.

[9] S.~. Schelkunoff, Conversion of Maxwell's equations intogeneralized telegraphist's equations, Bell Syst. tech. J. 34,995-1043, 1955.

[10] N. A. Strakhov, Crosstalk on multipair cable - theoreticalaspects, in: NTC 73, Conf. Rec. Nat. Telecomm. Conf.,Atlanta 1973, Vol. I, pp. 8B/1-7.

[11] J. R. Carson, The rigorous and approximate theories of elec-trical transmission along wires, Bell Syst, tech. J. 7, 11-25,1928.

ElIBLIOTHEë:K NAT. LABIN.V P ïLlPS'

GLOEIU. "FABRIEKENPOSl ....;_S 80.000

5600 JA EINDHOVE

94 J. VELDHUIS Philips tech. Rev. 40, No. 4

transmission behaviour, on the other hand, this varia-tion is not so important. Consequently in the trans-mission calculations the cable parameters can first beaveraged over z, so that we can after all substitutevalues for the primary cable parameters that areindependent of z in the network equations (1, 2, 3).

A diagram showing the method of solving the field equations,with their boundary conditions, for a cable cross-section is givenin fig. 5. It can be seen that the calculation of the primary cableparameters Rand L is kept separate from the calculation of Cand G.

The first calculation aims primarily at determining the N mag-netic vector potential fields ([Adm, m = 1,2, ... ,N in fig. Sa),

I parallel outside SOLVE LAPLACE"s EQUATION(J finite cond~ magne t ic vectoriJt 0=0 -

! potentials {Az}mm=1,2, .. ,N

given I t transformation~ . Ltransformation w- eire y;;=Rmn+Jw mn

N conductors I-- bOUndary; I -- {Az}m ~ Om f---- N circuits f---currents I; conditions

Um- Umresistance /m

n =1, 2, . ,N 1 vottoçelm i; _.,.In inductance /m

SOLVE HELMHOLTZ's EQUATION

~ magnetic vector r---inside potentials {Az}m

conductors P,S m=1,2. ,N

Q

given

N conductorscharges än (per metre)

n= I, 2,. ,N

transformationW_Cfrc

potentials Vm "---___,~I N circuits

A 12 N Vm- Vmq m=". Ön-Qn

SOLVE LAPLACE's EQUATION

Fig,S, Block diagram for calculating the primary cable parameters R mn, Lmn, Ci-» and Gmn for atelephone cable with quad geometry (fig. I). J. R. Carson's quasi-stationary theory is used [IlJ

so that the calculation can be divided into two stages. In both stages the most important part ofthe calculation consists in solving the equations of the electromagnetic field with their boundaryconditions, for a cross-section of the cable. a) Calculation of the resistances per metre (R",n) andof the self-inductances (Lmm) or mutual inductances (L",n), per metre. The matrix consisting ofthe elements Rmn + jwLmn is the impedance matrix (fig. 60). The essential feature of the calcula-tion is the solution of Laplace's equation outside the conductors and Helmholtz's equation insidethe conductors. The solution to Helmholtz's equation also provides as a result the proximity effectP and the skin effect S (Table I). Cf conductivity. a,D displacement current density. The symbol -indicates that the quantity has been defined for a conductor (without the symbol it has beendefined for a circuit). w -+ circ transformation of quantities for a conductor to correspondingquantities for a circuit. b) Calculation of the capacitances per metre (Cmn) and the conductancesper metre (Gmn). The essential feature of the calculation is the solution of Laplace's equationoutside the conductors. One of the results is the electrostatic proximity effect Pq. corr correctionfor the cage effect (K in Table I). ,) loss angle of the material outside the conductors. The cablesheath acts as a shield at zero potential. (Note: Rmm, Lmm, Cmm, Gmm are called R,«, L,«, C""Gm in Table I.)

The resultant loss of accuracy in the transmission cal-culations is insignificant.

A limiting condition in the calculations is that allthe pitch lengths in the cable have to be small withrespect to the signal wavelength, So far this conditionhas always been satisfied, because in practice attenua-tion and crosstalk increase so rapidly as wavelengthdecreases that a cable becomes useless at frequencieswell below those at which our approximate quasi-stationary theory loses its validity.

capacitance/m

conductance;'

where N quasi-direct currents in the wire cores (in) form the inputdata. The voltage drop per metre (Um) along the conductors can becalculated from the magnetic fields. Dividing by the current givesthe required complex impedance, which can then be used to find theresistance per metre, the mutual inductance per metre and the self-inductance per metre.

The main purpose of the second calculation is to determine Nelectrostatic potential fields (V"" m = 1,2, ... ,N in fig. Sb), withthe charge per metre on each wire (Qn) as the input data. The cageeffect, which only occurs in wires that are twisted, is included in thecalculation as a special boundary condition, which states that cir-cuits forming part of a cage must everywhere have the constant

Philips tech. Rev. 40, No. 4 TELEPHONE CABLES 95

potentialof the cable shield (zero). As soon as the complete poten-tial distribution is known, all the capacitances (per metre) can nowreadily be calculated.The separation of the calculations can be seen as a further sim-

plification of the quasi-stationary theory. In this approach thetransverse field associated with the TEM waves is approximated bytaking the individual contributions from a static electric field and aquasi-static magnetic field. The two fields are thus effectively 'de-coupled' by omitting from Maxwell's equations contributions fromthe magnetic field and from the electric field that are very small.The first field, the electric field, is therefore calculated for thesimpler case of perfect conductors (0- = 00 ); this is a good approx-imation, since the z-component of the electric field inside the con-ductors is extremely small (see Table II). The second field, themagnetic field outside the conductors, is calculated with the dis-placement currents set equal to zero (a,D = 0, in fig. 5a); the field isthen due entirely to the currents in the conductors. This condition iscorrect for all signal frequencies below those in the microwaveband. It should be noted in passing that the current-density distri-butions of the quasi-direct currents in the calculations in fig. 5a arenot uniform over the cross-sections of the wires: the current con-centration resulting from the proximity effect and the skin effect areboth found when solving the equations for the magnetic vector

. potential (Helmholtz's equation) inside the conductors.

The program

The strategy

In designing our cable-simulation program we fol-lowed the distribution of the calculations over threelevels as illustrated in fig. 3. The programming methodadopted is known as 'stepwise refinement' [12], whichmeans that a further redistribution of the separateparts of the calculation is introduced at each level bysplitting it up into increasingly refined modules. Oneof the main objects of such a strategy is to make theprogram as 'user-friendly' as possible; the extent towhich this has been achieved was checked against fivequality criteria for readily communicable and user-friendly software (Table 111). Since, as we have seen,there are two distinct objectives in research on tele-phone cables, it was a challenge for us to ensure thatthe program would be user-friendly in regard to eitherobjective. The manufacturer can incorporate anylikely modifications to a cable design into the programwithout too much difficulty, while the engineer whohas to consider the possibilities for data signals caneasily calculate the transmission behaviour of existingcables.

The stepwise-refinement procedure offers some ap-preciable advantages. One is that a module can bemodified, considerably if necessary, without intro-ducing the need for significant changes in the rest ofthe program. We found many advantages in usingmodules, especially while the program was being writ-ten; the scheme adopted made the program easy to

[12] N. Wirth, Program development by stepwise refinement,Comm. ACM 14,221-227, 1971.

grasp and therefore easy to communicate. Anotherimportant aspect of the modular structure is that it iseasy to check that the program is correct. The checkhas two aspects: making sure that the relations be-tween modules are as intended, and ensuring that theindividual modules are operating to specification. Wefound that such checks took little time or trouble.To meet the criteria for 'efficiency' and 'corn-

prehensibility' (Table Ill) we obviously had to givecareful consideration to the choice of programminglanguage. The entire progam was written in PASCAL,except for a few modules in FORTRAN (see next sec-tion). The program was run on a VAXll/780 mini-computer, which operates with words of 32 bits. Theversion of PASCAL used had already been providedby the computer manufacturer with facilities for com-piling each of the modules separately into machineinstructions. This saved a great deal of time when wecame to write our program (and it still saves timewhenever we have to make modifications) .

Choice of language

When we were choosing the programming languagethere were two main conditions. The first was to havea language that could easily handle data types, so thatwe could specify the geometrical structure of a tele-phone cable as freely as possible. The second condi-tion was that we should be able to solve a wide varietyof numerical problems simply and rapidly, since thecalculations for determining the primary cable param-eters are extensive, There were many languages suit-able for 'structured' programming, and because ofthe conditions just described we finally decided to usePASCAL. One important consideration was thatPASCAL has much to offer for type classification,thus helping to ensure the comprehensibility of the

Table Ill. Quality criteria used in designing the cable-simulationprogram, mainly to ensure smooth communication and 'userfriendliness' . .

Criterion

Each module should meet its specificationexactly.

If properly used the program should run cor-rectly; each error should be clearly signalled.

Small program modifications should be pos-sible 'on the spot', and should not thereforerequire extensive rewriting of other parts of theprogram.

The program should be readable and under-standable.

For all task implementations fast standardalgorithms are preferable to 'clever' (but un-clear) ad hoc solutions; for interactive opera-tions the time needed per task unit should be ofthe order of seconds.

Description

Correctness

Completeness

Flexibility

Clarity

Efficiency

96 J. VELDHUIS Philips tech. Rev. 40, No. 4

program. PASCAL does have some disadvantages,such as the absence of variables with the type of acomplex number, so that calculations with complexnumbers required extra programming. Another dis-advantage was that the limits of declared arrays can-not be dynamically changed during execution, andthis was also compensated for by additional program-ming work.

At the places in the program where extensive cal-culations are required, e.g. to solve a large set oflinear equations, a separate processor, the 'array pro-cessor', was used for greater efficiency. This proces-sar, which is exceptionally fast, only accepts moduleswritten in FORTRAN. At these places the programcalls up routines in FORTRAN, even though it is writ-ten in PASCAL itself. This is possible largely becauseof the modular structure of the program.

Example of a module

Fig. 6 shows a module, called ZMA TRIX, whosecomplete program text is printed in fig. 6a. The mainoperation carried out in this module is the determina-tion of the impedance matrix at a cross-section ofa cable. In other words, this module establishes, fora particular cross-section of a telephone cable, theequations - in the form of a matrix - relating thevoltages on the individual wires to the currents inthese wires (fig. 5a). The calculations are complicatedand include a number of steps, each occupying anindependent submodule. The flow chart in fig. 6bshows the sequence of these sub-operations. We shallnow see how a number of the quality requirementslisted in Table III reveal themselves in this typicalprogram module.Correctness. The program module (fig. 6a) starts atthe top with the specifications of the module. In thisparticular case merely specifying the main operation,as is done here, is sufficient. The specification shouldalways be kept as simple as possible, because the cor-

Fig. 6. Example of a module in the program for simulating tele-phone cable (fig. 3). This module can be used for determining theimpedance matrix (fig. Sa) for a cross-section of the cable. a) Theprogram of the module. The program starts with a list of the speci-fications that the module has to meet; then the procedures andvariables are declared; the third segment, between BEGIN andEND, contains the calculations, for which (b) represents the flowchart. In addition to the specifications the program later gives ex-planations in naturallanguage as well, i.e. alilines between (. and ..).The program itself is written in the high-level language PASCAL.The instruction explained by ( • solve Ax = b in Array Processor s )

brings in a second processor. This is faster than the ordinary type,which improves the efficiency. It requires FORTRAN software,however. The subroutine used solves the set of linear equationsAx = b with the aid of a standard algorithm in FORTRAN; the textis not given here. b) Flow chart for the calculation in (a). The dif-ferent operations are executed sequentially where possible, for sim-plicity and clarity. The only branching in this example is includedto provide a check (red blocks) on fundamental difficulties orcomputing errors in solving the equations Ax = b.

(*------------------------------------------------------------------*)(* In a cable cross section the impedance matrix ZMAT is calculated *)(* ZMAT r e l a t e s the (per unit Length) Longitudinal voLtage drop and *)

(* the current in the .....-i r e s , *)(*------------------------------------------------------------------*)

ï.HlCLUDE 'gLobaL.dat'

PROCEDURE f i Lla (VAR mat :typ_A;VAR w . t yp Warr;VAR l :typ-Larr) ; EXTERN;

PROCEDURE fi llb (VAR b :typ-rhs;VAR w :typ-Warr) ; EXTERN;

PROCEDURE lamn (VAR l :typ= La r r ) ; EXTERN;

PROCEDURE screen (VAR w r t yp war r); EXTERN;PROCEDURE fi t lzmat (VAR zmat :typ-z;

VAR w :typ-warr;

VAR rhs : typ -rhs) ; EXTERN;

PROCEDURE APso Lv (VAR bar :typ- A;VAR rhs :typ -rhs;

n,m,st :;nteger) ; FORTRAN;

PROCEDURE zrna t r i x CvàR ZMAT: typ_z);

CUNST UK = 1;

VAR A typ A;b typ-rhs;Lns typ-Larr;Wp typ-Warr;st,N,M: integer;

BEGINN:=di m;M: =5 tmax;

(* gLobaL constants dim and stmax *)

lamn(Lns) ;

screen(Wp) ;

(. fill the lambda-mat r i x Lns .)

(. fill screen factor array Wp .)

(. fill mat r- i x A (NxN) us i ng Wp, Lns .)

(. fill mat ri x b (NxM) u s i ng wp .)

(. sol ve Ax=b in Array Processor .)

(. x is returned in b , status in st .)

fi lLa(A,Wp,Lns);f i l Lb(b,Wp);

APso Lv (A ,b, N,t·l,s t i ,

IF st=OK THENf i llzmat (H1AT ,Wp,b);

ELSE8ELJIN WRITELN('ERROR in APsoLv');

WRITELN('Status ',st:2);

HALTEND;

END; a

b

Philips tech. Rev. 40, No. 4 TELEPHONE CABLES 97

rectness of the module can then be dernonstrated bymeans of simple tests. Any limiting conditions ap-propriate to. the use of a module, which thereforeform part of the specification, are included at the top.The flow chart in fig. 6b illustrates another objective,which is to compose the modules from a very limitednumber of sub-operations, whose interrelationshipscan be represented by a diagram with as few branchesas possible,

the program is of course improved by the choice of alanguage such as PASCAL; with such a high-levellanguage it is easier to relate the formulation of theproblems to naturallanguage than it would be with acomputer-oriented language (e.g. an assembly lan-guage). This again increases the comprehensibility.Efficiency. An example of the introduetion of a fast,standardized algorithm to improve the efficiencyof theprogram is the use of the array processor mentioned

Table IV. Comparison of calculated and measured values of attenuation and phase shift in a tele-phone cable with strong proximity effect [131. .

Attenuation(dB/km)

Measured Calculated(Lenahan (131) (Lenahan)

2.553.307.4011.2327.9641.95111.73166.25

2.563.287.4611.2227.5341.83112.72168.02

5

1050

100500

1000

5000

10000

Completeness. The blocks 'status OK' and 'error sig-nal' in fig. 6b are necessary to ensure that the text ofthe program is complete in this respect. In solving theset of linear equations Ax = b the first step is to deter-mine whether all the basic conditions (e.g. detA =1= 0)have been satisfied and then to check whether the cal-culations are proceeding correctly. Any errors are sig-nalled and this stops the program. If there are noerrors, the matrix ZMAT can then be set up or 'filled',completing the operation of the module.Flexibility, The modular structure is the best guaranteeof flexibility. In the case of fig. 6 the set Ax = b issolved by the numerical routine APsolv, a separatesub-module. If another method of solution seemedbetter; and assuming that it could also be embodied ina sub-module, called SOLV here; then we need onlyreplace sub-module APsolv by its counterpart SOLV,and compile the text of the module ZMATRIX again.(At the level of the ZMATRIX module the availablePASCAL version offers separate compilation.)Comprehensibility, In addition to the specification theprogram text in fig. 6a also indicates the sub-opera-tions that have to be carried out in succession. This isin fact a kind of description of fig: 6b in words, whichimproves the comprehensibility. In addition 'to thesimplicity of the flow charts, the comprehensibility of

2.563.287.4611.2127.5241.80112.87167.93

0.420.702.354.6819.4836.94168.60328.61

Phase shift(rad/km)

Calculated Measured Calculated Calculated(ours) (Lenahan (131) (Lenahan) (ours)

0.420.692.364.7119.4536.85168.70328.85

0.420.702.354.6919.5136.98168.74328.83

above for solving the large set of linear equationsAx = b in fig. 6b. At places such as this the programcalls up routines in FORTRAN, since the array pro-cessor can only work in FORTRAN. The incorpora-tion of program text in another language is facilitatedby the modular structure. In this way some parts ofthe cable-simulation program with a great deal of cal-culation can be handled rapidly and by standardizedmethods, which greatly improves the efficiency andalso improves the comprehensibility.

Results and their verification

We used the quasi-stationary theory described ear-lier 'and the simulation program to calculate theprimary and secondary cable parameters of a numberof experimental cables made by NKF. We verified theresults by comparing them with measurements on thecables. The calculations and the measurements were .found to agree well, both for cables with parallel wiresand for cables with twisted wires. (The differencesamounted to no more than 50/0 of the calculatedvalue.) Some of the differences are attributable tosmall variations in the manufacturing .processes,which are difficult to avoid. The conclusion is that thecable-simulation program gives results with an errorof no more than a few per cent.

98 TELEPHONE CABLES Philips tech. Rev. 40, No. 4

As a further check on our program we made a com-parative calculation of the attenuation and phase shiftin a cable made by Bell Laboratories on which T. A.Lenahan had carried out calculations and measure-ments [13]. Some of the results are listed in Table IV.In this cable, which contains only two wires inside aclosely fitting shield, the proximity effect is stronglypronounced. Here again the agreement was entirelysatisfactory. In the meantime a number of calcula-tions have been made on cables with much more com-plicated wire geometries, and these have also givensatisfactory results. The simulation of a 'real' tele-phone network using computer software in the man-

[13] T. A. Lenahan, The theory of uniform cables - Part I: Cal-culation of propagation parameters, Part II: Calculation ofcharge components, and also: Experimental test of propaga-tion-parameter calculations for shielded balanced pair cables, .Bell Syst. tech. J. 56, 597-610, 611-625 and 627-636, 1977.

ner illustrated in fig. 3 is therefore something thatstands a good chance of becoming reality in the nottoo distant future.

Summary. A study of multiwire telephone cables with quad geom-etry, with the twin objectives of improving the quality of the con-ventional copper cable with twisted wires, and of introducing digitalcommunications via the national telephone system, is most easilycarried out by computer simulation. The modular software forcable simulation, with 'stepwise refinement' and with emphasis ontransmission behaviour and crosstalk (which are particularly im-portant in local networks) is written in PASCAL for a VAXll/780minicomputer. Primary (R, L, C, G) and secondary cable param-eters and signal-transfer matrices of complete cables can be cal-culated with this software for given dimensions and materials. Twoproximity effects (due to currents and charges), the skin effect andMartin's cage effect in the generalized form given by Belevitch areincluded in the calculations. A program module described as anexample demonstrates the 'user-friendly' nature ofthe software, aswell as other qualities such as efficiency and error-signalling facili-ties. The method of calculation is based on Carson's quasi-station-ary theory of electrical transmission along wires, adapted to twistedconductors. Results of cable calculations are given along with theirverification by measurements.