SCIENCE

Coupling between jacketed braided coaxial cables

J.M. Tealby, BEngP.A. Cudd, BSc, PhD,Prof. F.A. Benson, DL, DEng, PhD, CEng, FIEE

Indexing terms: Coaxial cables, Interference, Modelling, Crosstalk

Abstract: A model that predicts coupling betweentwo or more jacketed braided coaxial cables innonflat pack configurations in free space is pre-sented. The model will also predict the couplingbetween jacketed cables over a ground plane. Themethod of moments technique is used to deter-mine the external line parameters between thecables, and an eigenvalue method is employed tofind the coupling between the cables. The theoreti-cal predictions are compared with experimentalresults in the frequency range 1 MHz to 1 GHz.

List of principal symbols

P(x) = voltage on cable i at a distance x from thesending end

P(x) = current on cable i at a distance x from thesending end

co = angular frequencyVs = vol tage at the sending end (x = 0)/s = current at the sending end (x = 0)[I/] = matrix whose kth column is the kth eigenvectorUl = f c t h r o w o f [ C / ] - 1

Uk = Euclidean normalised kth eigenvectorif/k = square root of the feth eigenvalueZt = surface transfer impedanceZO,ZC = characteristic impedancesp T = second row of [Zc] ~1

yk = propagation constant for the kth cables/ = cable run lengthrci = outer radius of the ith cable's braidrdi = radius of the ith cable's jacketa = surface chargeUltk = first entry of Uk

X = wavelengthfj. = permeabilitye = permittivity

1 Introduction

Many installations use radio-frequency coaxial cables totransmit from one device to another. These cables areusually manufactured with a braided outer shield. A

Paper 5635A (S8), first received 24th June 1986 and in revised form 23rdDecember 1986The authors are with the Department of Electronic & Electrical Engin-eering, University of Sheffield, Mappin Street, Sheffield SI 3JD, UnitedKingdom

braid will allow a certain amount of signal from theinside of the cable to leak through, and the inductiveleakage per unit length has been quantified in terms ofthe surface transfer impedance Zt of the cable. Althoughcapacitive leakage occurs, it has been found [1] to benegligible in comparison with the inductive leakage forthe kind of cable configuration to be considered here.The leaked signal will cause a current to flow on theoutside of the braid which will also induce currents on tothe outside of any other cables present. These in turn willcause signals to be introduced into the inner circuits ofthe other cables. It is assumed that the currents flowingon the outside of the braids are consistent with a quasi-TEM mode of propagation. The external circuits formedby the outside of the cable braids can be considered toform multiconductor transmission lines. Each braid, withthe chosen return path, forms a twin line with associatedline parameters of resistance R, conductance G, induc-tance L and capacitance C. Mutual terms between eachof the twin transmission lines give rise to the induced cur-rents present on the cable outer surfaces. The ratio of thecoupled signal received at one end of the victim cable tothe input signal of the excited cable, when expressed indecibels, is called the coupling factor. Depending onwhich end of the victim cable is analysed, the near-end(i.e. the termination closest to the input of the excitedcable) or far-end coupling is determined. Much of thework on coupling between braided coaxial cables hasshown that an assumption of weak coupling, betweencable inner circuits and the external twin line circuits ispermissible [1-5] .

Coupling between parallel braided coaxial cables hasbeen studied for two cables [2, 3] and more recently fortwo or more coaxial cables, in free space and over agroundplane [4, 5]. However, in all these investigations,the jackets of the cables were removed. The couplingwhich then occured could be modelled in a much morestraightforward manner. When the jackets are present thecapacitances between the cables are altered, while theinductance terms are unchanged. The effect of the jackethas been studied for solid-core transmission lines [6, 7]and a method of moments technique has been applied tofind the capacitances between the conductors [8]. Thistechnique was then extended to calculate the capac-itances between jacketed coaxial cables [9] in a flat-packconfiguration. This work indicated that no effect on thecoupling due to the jackets was observed up to a fre-quency which corresponds to the cable run length / beingequal to A/10. The alteration in the capacitances, due tothe jacket material, produces a change in the propagationcharacteristics of the quasi TEM wave. The latter changemanifests itself as a shift in coupling resonances to lower

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 10, DECEMBER 1987 793

frequencies than those observed for cables without theirjackets.

This paper describes the prediction and measurementof the far-end coupling factor for jacketed coaxial cables(JCC) in the cable configurations shown in Fig. 1. In thisstudy the losses of the jacket material are ignored overthe frequency range of interest, which is 1 MHz to1 GHz. Unless large values of tan S, e.g. greater than0.01, are encountered, the loss in the jacket materials islikely to affect the resonant coupling level only.

2 Theoretical analysis

The investigation into the coupling between two braidedcoaxial cables has shown that, for cables which are elec-trically short (1 < A/4), a lumped circuit can be used tomodel the propagation characteristics of the quasi TEMwave which exists on the outside of the cables [10].However, if the cables are electrically long, a distributed-line model must be used [10]. For the external twintransmission circuits formed by (n + 1) parallel conduc-tors, the distributed line equations are given by

dV(x)dx

dl(x)dx

(1)

(2)

•d"-

o oTgroundplane

Fig. 1 Cable configurations examinedE = excited cable V = victim cabled, = 4.5 mm d = I cm d' = 4 cm h = 3 cm

The transmission-line parameters of the external circuitsfor (n + 1) cables are given by the entries to the n x nmatrices, resistance [_K], inductance [L], conductance[G] and capacitance [C], which are indicated in Fig. 2.

T

Fig. 2 The entries to the transmission line parameter n x n matrices[K], [L], [G] and [C]Cable O is the reference conductorRo — Rtj for all i andj

When eqns. 1 and 2 are differentiated, and after somesubstitution, eqns. 3 and 4 result:

dx2

d2l{x)dx2

x)

Let

= (M

(3)

(4)

(5)If only two coaxial cables are present then \_A~] is givenby y2, for which y is uniquely defined. Any n x n matrix[A] has an infinite set of matrices whose square is equalto [A], therefore it is impossible to find a uniquelydefined propagation matrix [y]. Consequently an eigen-value approach is invoked to solve eqns. 3 and 4. Oncethe boundary conditions have been satisfied the solutionsare found to be

V(x) = £ U&Vs cosh \pkx - [ZJIs sinh il/kx)Uk (6)*=i

^[zj-1!

x cosh if/kx — Vs sinh \j/kx)Uk (7)

Following a similar procedure in which the couplingexpression is found for two braided cables, expressions 6and 7 are used to determine the far-end coupling factor Ffor n cables and one return path. However, consider thecoupling between two cables within a system of (n + 1)cables. If the first cable (k = 1) is taken as the excitedcable with an input of Vin applied, and the second cable(k = 2) is taken as the victim cable, with an output signalof Vfar received, taking the reference cable as the (n + l)thcable, the coupling factor relating these cables is given byexpr. 8 for identical excited and victim cables:

Yis.7' t^1

2Z~o fc=iL . k (

2y2 cosh \jtkl — cosh y/

\ — y2)2 sinh ^/klUk (8)

794 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 10, DECEMBER 1987

If the excited and victim cables are not identical, thenexpr. 8 should be modified [11]. The eigenvalues relate tothe physical propagation modes that are present on thecables. This method can now be used to determine thecoupling if the line parameters [/?], [L], [G] and [C] areknown.

To determine the entries in the [C] and [L] matrices,the method of moments is used. This technique has beenstudied by Paul [8] for solid conductors with andwithout their jackets, the same method has been used bySali [9] to produce the line parameters for the externalcircuits of coaxial cables in a flat-pack arrangement.

Method of moment techniques are used in manyapplications concerned with electromagnetic field prob-lems [12]. They allow for a conducting surface to begiven a 'weighting' function to describe the surface chargedistribution (<r) which can then be applied into the wellknown equations relating potential and charge:

V2</>= -a{r)/e (9)

It has been shown [13] that the 'weighting' functionwhich is best suited to model a cylindrical surface is theFourier series.

The per-unit length charge distributions around theconductor jacket surface are modelled as the followingFourier series:

= ai0 + cos + sin (10)

and for the jacket/free-space boundary:m m

Udd = aio+ £ aik cos (kOd + I £» sin (fc0,) (11)

The total charge per unit length #,- on the ith conductorsurface is taken as the free charge qfi minus the boundcharge q{, where

q{ = qn — ,̂- (1/j

so

q.= 1^(6^36+ lJo Jo

Once the charge distribution on the conductor surface isknown, the potential </> due to this distribution can alsobe determined from

r0) =- 1

4ne0

on{ln)\n\r-r0\2dln (14)

The moment method will determine (/> at specific pointson the conductor's surfaces due to all the charge distribu-tions in the system. It has been found [8] that thenumber of points on the conductor surface is to be takenas the same number of expansion terms in the Fourierseries given in eqns. 10 and 11. From this process thegeneral capacitance matrix [Cff] is found from therelationship

qf = (15)

If jackets are present then the boundary conditions at thejacket/free-space periphery have to be included whendetermining [C9]. Once the general capacitance matrix isfound the transmission-line capacitance matrix [C] canbe produced by making one of the cables the referenceconductor and referring all the potentials on the otherconductor surfaces to this [14]. The inductance matrix[L] is obtained from the capacitance matrix [Co],

(determined for the same cable configuration but with thejackets removed), then, using the Maxwellian equation,

so that

(17)

Now consider a cable configuration where a groundplaneis present. The image cables produced [3] will havecharge distributions that will be equal in magnitude butopposite in sign to the corresponding wire above thegroundplane.

When [C9] is found, no further transformations arenecessary as the charge distributions on the image con-ductors have taken into account the potentials producedon the real conductors with reference to the groundplane.If a groundplane is present it is taken as the return pathfor any currents produced by the leakage. The latterstatements are consistent with the theoretical analysisoutlined in eqns. 1-8 and their underlying principles, pro-vided that no significant propagation of non-TEM modesoccurs on the multiconductor transmission line.

3 Experimentation

The experiments on multiple JCC in free space werecarried out on apparatus arranged as in Fig. 3. A free-

brass bulkheads

Fig. 3 Experimental apparatus used for JCC in free space

space environment was attempted by placing the twobulkheads on wooden benches. Good electrical contactwas ensured between the cable braids and bulkheads. Thecentre-to-centre separation, when their jackets were incontact, was kept constant by looming the cablestogether with lacing cord which is often used in practicalsituations. Experiments incorporating a groundplanewere then carried out for two JCCs using an apparatussimilar to that shown in Fig. 3 but with the bulkheadsplaced on a copper sheet. The cables were kept adjacentto the groundplane along the length of the cable run bymeans of thin strips of Sellotape. Care was taken to makesure that good electrical contact existed between thebulkheads and the ground plane.

The measured coupling levels were determined byinjecting a signal Vin at a specific frequency (in the range1 MHz to 1 GHz) into one end of the excited cable. Thecoupled signal Vfar received at the other end of the victimcable was fed into a spectrum analyser and noted. Thecoaxial cables were then disconnected from the arrange-ment and the signal generator was connected to the spec-trum analyser, via two-step attenuators which arecalibrated to an accuracy of 0.5 dB. The attenuators werethen adjusted until the same level of signal Vfar wasreceived at the spectrum analyser. The level of coupling,for that particular frequency, could then be read offdirectly from the attenuators. The latter process wasrepeated for the next frequency and so on. The Zf ofURM108 cable used in these experiments was measuredprior to the coupling experiment over a frequency range

1EE PROCEEDINGS, Vol. 134, Pt. A, No. 10, DECEMBER 1987 795

of 1 MHz to 50 MHz. At 50 MHz the measured value ofZf for URM108 was 267 mQ/m. The Z, values at fre-quencies higher than 50 MHz were determined from theassumption that a linear relationship between Zt and fre-quency for URM108 existed. Investigations by Sali [15]have shown that, for a braided coaxial cable that exhibitslinear variation of Zf with frequency up to 100 MHz, thisrelationship between Zt and frequency continues to becorrect up to 7 GHz.

4 Results and discussion

Fig. 4 displays the measured and predicted couplings forfour JCCs in free space in the configuration shown in Fig.la. The major effects due to presence of the jacket areclearly seen, such as shifted resonances and the formationof more resonances. The accuracy of the prediction ofcoupling levels is within 10 dB of the measured valuesand errors in the frequency of the resonances are less

-20r

than 1%. If the presence of the jacket is ignored, then theerror in predicting the coupling level could be as high as60 dB, and the resonant frequencies may be in error byas much as 15%. If the cables are given a 1 cm centre-to-centre separation, as shown by Fig. \b, the shift in theresonant frequencies is reduced to 5%.

Two cables were placed on the groundplane with acentre-to-centre separation of 4 cm as in the configu-ration shown in Fig. Id. The off-resonance coupling canbe seen to be low (less than 110 dB) yet the level of theresonant frequency coupling remains high at about60 dB, as given in Fig. 5. The effect of the jackets can beobserved. If the jackets were neglected for this case thenthere could be a possible error in the prediction of theresonant frequencies again of up to 15% and in the coup-ling level prediction of up to 80 dB. When the cables areelevated to a height of 3 cm, with the same centre-to-centre separation, as shown by Fig. le, the effect of thejackets is minimal but still present.

200 400 600frequency, MHz

800

Fig. 4 Four URM108 cables in free spacepredicted coupling with jacketpredicted coupling without jacket

* * * measured values

1000

-20r

-60CD

2 -80o

:=-100Q.

-120

-140

-160.0 200 400 600

frequency, MHz

Fig. 5 Two URM108 cables over a groundplanepredicted coupling with jacketpredicted coupling without jacket

• * * measured values

800 1000

796 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 10, DECEMBER 1987

Fig. 6 displays the measured and predicted couplingsfor three JCCs in free space for the configuration of Fig.lc. The solid line on the graph shows the coupling pre-diction with a value of 2.1 for the relative permittivity erof the jacket material. As can be seen this gives poor

agreement with the measured values of coupling. The seffvalue of 2.1 is the sr of pure FEP over the frequencyrange 10 kHz to 1 GHz [16]. To achieve better couplingpredictions an eeff value of 1.85 has been used, and isshown on Fig. 6 as a dotted line. A possible explanation

-20r

-40

-60

2 -80

= -100

-120

-K0

-160200 400 600

frequency, MHz800 1000

Fig. 6 Three URM108 cables in free spaceCable length / = 1.001 m

predicted (er 1.85)coupling (er 2.1)

* * * measured values

Table 1: Effects of jacket material on coupling between braided coaxial cables in the high fre-quency region (from a frequency which corresponds to a cable run length of X/10 to 1 GHz)

1 Two cables in free spaceMajor effectFor cables with their jackets touching thecoupling resonances are shifted to a lowerfrequency as shown below:

Comments and other effectsAs the separation of the cables is increasedthe shift in the resonances is reduced. Fortwo URM108 cables:

Cable jacket shift in resonances.* Separation (cm) shift in resonance.*

URM108URM43

FEPPVC

15%25%

In both cases the maximum couplingobserved was -54 dB

5.23.93.02.01.8

2 Multiple cables in free spaceMultiple peaks at each resonant positioncould occur. The position and number ofpeaks are dependent on the cableconfiguration.

For cables in nonflat pack arrangements, in closeproximity, the prediction technique will require a largememory allocation. For 20 cables the size of the memorymust be able to cope with an array of size 360 by 360.

3 Cables over a ground plane.Shifts in resonances still occur, owing tothe presence of the jacket even forcentre-to-centre separations greaterthan 3 cm.

Given below are some experimental resultsfor two URM108 cables over a groundplanein different cable configurations.

Separation

cm2rd2.04.02rd2.04.0

height

cmrdrdrd2.02.02.0

shift in*resonance

%18+15 +1516 +3.5+2.3+

max. couplingobserved

-dB45.547.560.044.045.049.0

+ double resonances occur, and the peaks with thegreatest shift are noted* percentage of the appropriate resonant frequencyin free spacerd radii of the jacketed cable (for URM108, rd is 2.2 mm)

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 10, DECEMBER 1987 797

for the eeff of the jacket not equalling er of the constitu-ent material may be that braided nature of the outer con-ductor alters the effective permittivity seff of the cableinsulator from that of the pure material [17]. As anexample consider URM108. A value of 1.85 for the effec-tive permittivity of the jacket material and a tan d valueof zero were used in the predictions shown in Figs. 4 and5. In contradiction of these results for a speciallydesigned URM43 size cable which had a pure polythenejacket it was found that eeff = er within measurementaccuracy. For the latter case the jacket was thicker thanthe FEP one on the URM108 cable. So any effects ofairgaps would be smaller.

The additives used in the manufacture of cable jackets,such as plasticisers, stabilisers, fillers and antioxidants,have been kept to a minimum in the FEP jacket ofURM108, and so the jacket material can be taken as99% pure. The loss tangent of FEP over this frequencyrange is 0.0002, consequently the losses in the jacketmaterial were neglected. This may not be possible forcables which are coated with PVC, which is only about60% pure and has loss tangents of 0.1 and below [18].

Although the method of moments technique is used todetermine the line parameters and has been shown toproduce accurate predictions, there is a great cost incomputer memory. If a large number of cables arebunched together in a nonsymmetric geometry (especiallynon-flat-pack arrangements) then the computer memoryrequired may be too large for many computer systems.The case given in Fig. 6 required an array of size 72 by72. This means that if the number of cables was 20 thesize of the array would be 360 by 360. If the loss of thejacket material is also to be included, the size of the arraydoubles owing to the complexity of each entry.

Table 1 outlines the major effects observed and pre-dicted during the course of this investigation. Theseeffects are tabulated for the high-frequency region only,since the jacket has no discernable effects in the low-frequency region.

5 Conclusions

A coupling prediction model now exists for multiple JCCin free space and two JCC over a groundplane which willcope with cable configurations that are closer to practicalarrangements than those previously studied. When JCCare in close proximity to each other or a groundplanethen the effects of the jacket must be included. If the res-onances produced in the coupling are to be accuratelypredicted, a knowledge of the eeff of the jacket materialmay be required and not just a value of er. If the methodof moments technique is to be used to determine the lineparameters for a large number of cables in nonflat packarrangements, then some means of making it more effi-

cient will be required or the use of computational tech-niques whereby the storage of large arrays is not needed.

6 Acknowledgments

The work described has been carried out with thesupport of the Procurement Executive of the UK Minis-try of Defence. The authors would like also to thank N.Wyatt and colleagues at BICC (Helsby) for supplying thecable samples tested.

7 References

1 CUDD, P.A., SALI, S., and BENSON, F.A.: 'Calculation of coup-ling between coaxial cables'. Proc. Int. Conf. on EMC, IERE (56),pp. 141-149, Sept. 1982

2 SCHELKUNOFF, S.A., and ODARENKO, T.M.: 'Crosstalkbetween coaxial transmission lines', Bell Syst. Tech. J., 1937, 26,pp. 144-164

3 BADR, A.H., BENSON, F.A., and SITCH, J.E.: 'Coupling betweencoaxial cables over a ground plane at low frequencies', IEE Proc.A, 1980,127, (8), pp. 549-552

4 SALI, S., BENSON, F.A., and SITCH, J.E.: 'Coupling betweenmulti-coaxial cable systems. Pts. 1 and 2', ibid., 1982, 129, (3), pp.162-171

5 BADR, A.H., BENSON, F.A., CUDD, P.A., and SITCH, J.E.:'Coupling between a pair of coaxial cables in a multicable system',ibid., 1981,128, (8), pp. 547-551

6 CRAGGS, J.W., and TRANTER, C.J.: 'The capacity of two-dimensional systems of conductors and dielectrics with circularboundaries', Quart. J. Maths., 1946,17, pp. 138-144

7 DECRETON, M.C., GARDIOL, F.E., and MAILLEFER, G :'Numerical analysis of the line capacitance and crosstalk factor forinsulated wire pairs', Arch. Elektron. UEB, 1974, 28, pp. 415-420

8 PAUL, C.R., and FEATHER, A.E.: 'Application of momentmethods to the characterization of ribbon cables', Comput. & Elec-tron. Eng., 1977, 4, pp. 173-184

9 SALI, S., BENSON, F.A., and SITCH, J.E.: 'Coupling in adielectric-coated multicoaxial cable system; an analysis based on aquasi-TEM model', IEE Proc. A, 1984,131, (1), pp. 67-73

10 BADR, A.H., BENSON, F.A., and SITCH, J.E.: 'Coupling betweencoaxial cables at radio and microwave frequencies', Radio Electron.Eng., 1982, 52, pp. 177-180

11 CUDD, P.A.: 'Coupling between braided cables in a multiconductorenvironment up to microwave frequencies'. Ph.D. thesis, Universityof Sheffield, UK, 1982

12 HARRINGTON, R.F.: 'Field computation by moment methods'(Macmillan, NY, 1968)

13 CLEMENTS, J.C., PAUL, C.R, and ADAMS, AT.: 'Computationof the capacitance matrix for systems of dielectric-coated cylindricalconductors', IEEE Trans., 1975, EMC-17, pp. 238-248

14 PAUL, C.R., and FEATHER, A.E.: 'Computation of the transmis-sion line inductance and capacitance matrices from the generalizedcapacitance matrix', IEEE Trans., 1976, EMC-18, pp. 175-183

15 SALI, S.: 'Transfer impedance of braided coaxial cables at radio andmicrowave frequencies', M.Eng. thesis, University of Sheffield, UK,1979

16 FINK, D.G.: 'Standard handbook for electrical engineers'(McGraw-Hill, 1979), Table 4-67, p. 148

17 DUMMER, G.W.A., and BLACKBAND, W.T.: 'Wires and r.f.cables' (Pitman, London, 1961), p. 171

18 VON HIPPEL, A.R.: 'Dielectric materials and applications' (Wiley,NY, 1954)

798 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 10, DECEMBER 1987