Current conduction in coaxial cable braidsat radio frequencies

A.H. Badr, B.Sc, M.Eng., Ph.D., Prof. F.A. Benson, D.Eng., Ph.D., C.Eng., F I.E.E., and M.M. Rahman,B.Sc, M.Eng., Ph.D.

Indexing terms: Coaxial cables, Radio-wave propagation, Measurement and measuring

Abstract: Results of braiding-factor measurements on two coaxial cables, one with a wire braid and theother with a tape braid, are reported here. The investigations, performed in an attempt to understand moreclearly the mechanism of current conduction in the braided outer conductors of cables at radio frequencies,establish the existence of two different levels for the braiding factor at the higher and lower radio frequencies,respectively. The determinations of quantitative values for these levels are extremely dependent on the toler-ances of braid dimensions. In the transition region between the two frequency limits braiding factors arefound to be anomalous.

1 Introduction

The mechanism of current conduction in solid-jacketedcoaxial cables is well understood, except at the higher micro-wave frequencies. The complex geometrical construction ofbraided cables, however, presents an inherent difficulty inany attempts at formulating theories of current conduction inthem. The mode of current conduction in the braid of a cablehas been thoroughly studied [1-3] in the region 1-40 GHz.Hopkinson [1] measured the attenuation of a variety of cableswith polythene dielectrics, both braided and solid outer-jacketed, in the frequency range 4—40 GHz. Bayrak [4]extended this work to include the study of some solid outer-jacketed cables containing a PTFE (polytetrafluorethylene)dielectric. Teperek [2,3] and Benson [3] examined speciallymanufactured cables to see the effects of varying parameterssuch as braid-wire materials, braid-wire diameter and lay length,including one cable with a tape braid, i.e. each spindle of typi-cally four or five wires running alongside each other, in thecase of wire-braided cables, is replaced by a single strip or tape.Goldberg and Slaughter [5] performed tests on cables withidentical cores but differing braids to try to obtain more infor-mation concerning braid conduction up to a frequency of10 GHz. Earlier work on braided cables up to 10 GHz has alsobeen reported [6].

A literature survey of the investigations on braided cablesbelow about 100 MHz did not disclose much published workat these frequencies. Blackband [7, 8] has made importantcontributions in this region which include the formulation oftwo theories for current conduction in braided cables at radiofrequencies. Teperek and Benson [9] have measured theattenuation/frequency characteristics for three wire-braidedcables and a tape-braided one in the frequency range 100 kHzto 100 MHz, and have presented the results as braiding-factor/frequency curves. Some geometrical characteristics of braidsare discussed in a paper by Degauque et al. [10], who suggestedthat to provide two-way mobile communication at 7 MHzalong mine tunnels a few kilometres long a leaky braidedcoaxial cable be used with an inductive coupling to the trans-mitter and the receiver.

The studies reported here have been carried out in anattempt to understand more clearly the mechanism of currentconduction in braided cables below frequencies of 100 MHzand, in particular, to see how well Blackband's predictions[7, 8] for wire and tape braids agree with braiding factors

Paper 1369A, first received 2nd December 1980 and in revised form18th March 1981Dr. Badr- and Prof. Benson are with the Department of Electronic &Electrical Engineering, University of Sheffield, Mappin Street, SheffieldSi 3JD, England, and Dr. Rahman is now with Bangladesh Universityof Engineering & Technology, Dacca, Bangladesh.

354 0143-102X1811050354 + 04 $01.50/0

determined from measurements. The investigations follow onfrom the Teperek and Benson work [9] but contain improve-ments in that:

(a) The frequency range is extended downwards.(b) The cable dimensions are measured to give more accu-

rate theoretical braiding factors.(c) Tolerance estimates for the theoretical braiding factors

are made.

2 Measuring techniques

The braiding factor, as defined by Blackband, is the ratio ofthe braid resistance to the resistance of a solid tube of thesame material as the braid, and of thickness twice the diameterof a braiding wire or the thickness of a braiding tape. At thelower radio frequencies (below about 100 kHz), currentdistribution can be considered to be uniform over the wholecross-section of a braiding wire/tape. But at the higher radiofrequencies (above about 10 MHz), the diameter of a braidingwire or the thickness of a braiding tape becomes large com-pared with the depth of current penetration due to skin effect,and the losses in the braid increase, for which Blackbandaccounted by considering the presence of circulating currentsin addition to the normal longitudinal current. Thus thebraiding factor exhibits two different levels at the higher andat the lower radio frequencies.

The open- and short-circuit input impedances of each cablesample were measured at different frequencies using bridgesand the results used for the calculation of attenuation andcharacteristic impedance from which braiding factors weredetermined (see Appendix 7.1) DC values of braid resistanceswere measured for comparison with the braiding factor at thelower radio frequencies. An analysis of possible errors in thebraiding factor calculated from experimental results wasperformed on the basis of inaccuracies of the bridges and thevariations in the cable dimensions.

Four different Uniradio cables [UR43, URM43, URM57(all wire-braided) and a tape-braided version of UR43] andtwo TV distribution cables (TR104/022 wire-braided andTR112/022 a tape-braided version of TR104/022) weremeasured.* A number of impedance and admittance bridgesranging from 1 kHz to 100 MHz, in conjunction with a gangedsource and detector, were used for the input impedancemeasurements.

3 Results and discussion

Of the results obtained for the different cable samples, thosefor Uniradio 43 wire- and tape-braided cable are presented in

•Made and supplied by BICC Ltd.

IEEPROC, Vol. 128, Pt. A, No. 5, JULY 1981

Figs. 1—4 as typical. Fig. 1, shows values of characteristicimpedance of Uniradio 43 wire-braided cable (UR43W)obtained from measurements against frequency. It also shows,for the sake of comparison, the nominal value and calculatedvalues including internal inductance effects. Fig. 2 shows theattenuation/frequency characteristic for UR43W. Figs. 3 and 4

a 65

0.01frequency , MHz

Fig. 1 Characteristic impedance against frequency for cable UR43(wire-braided)

• measured valueo calculated values

nominal values

show values of braiding factors obtained from measurementdata against frequency for UR43W and UR43T (the tape-braided version of UR43W), respectively. The latter two graphsalso show theoretically predicted values of braiding factorsusing Blackband's formulas. The DC value of braiding factor isalso shown. The dotted lines illustrate the limits of error in thetheoretical calculations of the braiding factor due to the vari-ation of the braid dimensions (see Appendix 7.2). The error inthe braiding factor calculated from experimental results isestimated to be within ± 15%.

It is seen from Fig. 1 that all the experimental values ofcharacteristic impedance above about 500 kHz are scatteredaround the nominal value and lie within the nominal tolerancelimit of ± 3 £2. Higher values below this frequency can beattributed to the effect of internal inductance due to the finitedepth of current penetration at these frequencies. This is con-firmed by comparison of the measured values with the calcu-lated values (including internal inductance effects) shown bythe solid curve. A point-by-point comparison, however, is notjustified since the calculated values were determined fromnominal cable dimensions.

The attenuation/frequency characteristic of Fig. 2 can beroughly divided into three separate regions. Up to about200 kHz, where the depth of current penetration due to skineffect is about the same as the diameter of a braiding wire,the slope of the curve is least. Beyond about 20 to 40 MHz,

ior

oo

•o

'0.1

0.010.001 0.01 0.1 " 1 10 100

frequency , MHz

Fig. 2 Attenuation against frequency for cable UR43 (wire-braided)

IEE PROC, Vol. 128, Pt. A, No. 5, JUL Y1981

where the depth of current penetration is of the order of atenth of the braid-wire diameter, the gradient is a maximum.Between these lower and higher frequency limits, which can beconsidered as the transition region, the slope is within the twolimiting values.

The graph of braiding factor against frequency (Fig. 3)would be best considered in terms of the different frequencybands identified by Blackband [7]. At the higher frequencies(above about 10 MHz), where skin-depth is small compared tobraid-wire diameter, the experimental values of braiding factorare scattered around the theoretical value and lie well withinthe tolerance limits shown by AKBWH. At the lower fre-quencies up to about 50 kHz, the experimental values of

3.0-AK BWH

en AKBWL

1.5-

IDC

AKBWH

1.0-

0.001 0.01 0.1 1frequency , MHz

10 100

Fig. 3 Braiding factor against frequency for cable UR43 (wire-braided)

braiding factor clearly show a level distinctly below the high-frequency level. The discrepancy between these values and thepredicted level can possibly be put down to tolerances in braiddimensions, since effects of these tolerances are found to belarge as seen from the graph. The braiding factor calculatedfrom DC measurements agrees very closely with the experi-mental value at 1 kHz. In the transition region between thetwo frequency limits the results are anomalous. Anomaliesnear resonant frequencies are to be expected, since smallvariations in the measured open- and short-circuit impedancesat these frequencies would give rise to large errors in theresults. The dip in the curve of Fig. 3, however, extends overat least one decade in the frequency range (unlike a resonanteffect) and does not exhibit the random scatter one wouldanticipate from abnormally high measuring errors. The sameanomalous dips appear in two of the Figures in the Teperekand Benson paper [9], but no specific mention of the effectwas made there because the dips appeared close to the lowestfrequency employed, and so were less noticeable than in thepresent case. It is also interesting to note that at the frequencieswhere the measured braiding factors are decreasing one mighthave expected the measured attenuation characteristic ofFig. 2 to show a somewhat steeper gradient. All of this strongly

AKBTH'

,2.5

§2.0- AKBTL :DC

f15i"° 1.01-

• • AK BTH

AK BTL

0.001 0.01 10 1000.1 1frequency , MHz

Fig. 4 Braiding factor against frequency for cable UR43 (tape-braided)

355

suggests that the anomalous behaviour in the transition rangeis a valid phenomenon.

A possible qualitative explanation for the anomalouseffect may be that at the transition frequencies there is lessrelative braiding loss for the following reasons:

(a) The average nonuniformity of current distributionacross a wire radius could be less pronounced than that acrossthe wall thickness of the corresponding solid reference sheath,especially as the latter thickness is four times the wire radius.

(b) The effective surface area of braid wire towards whichthe current flow is beginning to be forced at these frequenciescould well be rather greater than the initial surface area of thesolid reference sheath.The overall mechanism may thus be somewhat akin to theaction of litz wire in high-Q, RF inductors. It must be pointedout, however, that some measurements carried out duringthe present studies on cable samples of different lengths haveestablished a relationship between the anomalous behaviourand sample length rather than just to braid-wire diameter.

In Fig. 4 the experimental values of braiding factor forthe tape-braided cable at low frequencies are within the theor-etical tolerances of KB. In the high-frequency range the valueof the braiding factor has increased, but is not actually withinthe lower tolerance limit until the frequency gets to about25 MHz. Once again the anomalous effect is evident in thetransition region between the two frequency limits.

Comparing the experimental results of the braiding factorin the high-frequency range in Figs. 3 and 4 with the theor-etical predictions confirms that Blackband's predictions forthe wire-braided cables are near enough to the experimentalresults to lend support to his physical representation of acirculating current in braid conduction. However, the sameapproach does not give good agreement for the tape-braidedcable tests which suggests that the circulating current [11]does not occur in this cable. Since the thickness of the tape isapproximately half the braid-wire diameter of cable UR43W(the equivalent wire-braided cable), and since one tape replacesa spindle of four or five wires lying side by side, a closer-knitbraid can result, and thus current may pass from one tape toanother. The extra loss here of the tape braid over that of asolid-outer-jacketed cable may, therefore, be due to a slightresistance associated with crossing from one tape to another.In wire braids, the resistance between adjacent wires is large,owing to the smaller area of contact of circular surface, soBlackband's circulating current conduction mode occurs.

Comparison of the measured results with those obtained byTeperek and Benson [9] between 100 kHz and 100 MHz showsfairly good agreement at the high-frequency end for bothUR43 wire- and tape-braided cables. The results of Teperekand Benson [9] at the low-frequency end are widely scatteredso no justifiable comparison is possible.

Any leakage of energy through the cable braids has beenneglected in the Blackband theories [7, 8] . It is, of course,well known that the leakage of energy into and out of coaxialcables is an important source of interference and radiationlosses from braided cables increase with frequency. Rahman,Sitch and Benson [12] have measured the leakage propertiesof a large number of coaxial cables over a wide frequencyrange, up to 3.5 GHz in some cases, and including samplesfrom the two UR43 cables used in the present studies. Thesurface transfer impedance per unit length ZT, and the equiv-alent surface transfer impedance due to mutual capacitanceZF> were evaluated and compared with values computed usingformulas found in the literature. It might have been expectedin the present investigations that any errors due to leakagewould show themselves in discrepancies in either the ZT or KB

curves. The ZT measurements did not, however, show anyevidence to indicate that neglecting leakage introduces any

appreciable errors in the determination of braiding factorsfrom experimental data at the frequencies of interest. Anyunusual behaviour could, of course, have been masked by thelarge tolerances in braiding factors. The premise that radiationlosses are not significant in the present studies reported hereare supported implicitly by the fact that the measured high-frequency braiding factors do not exceed those predicted byBlackband.

4 Conclusions

The mechanisms of current conduction in wire- and tape-braided cables, at radio frequencies have been studied in detail,and results have been compared with Blackband's theoreticalpredictions. Qualitatively, the existence of two different levelsfor the braiding factors for wire-braided cables at the higherand at the lower radio frequencies have been established.However, the whole question of quantitative analysis needs tobe considered very carefully, since both the theoretical andexperimental values of braiding factor are extremely dependanton the variations in cable dimensions etc. Thus, although apoint-by-point correlation between theory and experiment hasnot been established, it is safe to conclude that current conduc-tion takes place longitudinally along the braid wires at thelower frequencies, but the presence of additional circulatingcurrents is necessary in explaining the conduction mechanismat the higher radio frequencies. For the tape-braided cable thebraiding factors obtained at the low-frequency end of therange employed agree with Blackband's predictions. Theresults at high frequencies, however, suggest circulating currentsdo not occur in this cable.

It has been demonstrated that in the transition regionbetween the two frequency limits the braiding factor resultsare anomalous and a possible qualitative explanation for theanomalous effect has been given.

5 Acknowledgments

The authors are grateful to the UK Ministry of Defence forsupporting the work and to BICC Ltd., Helsby for providingthe cable samples and extending some laboratory facilities.Thanks are also due to W.T. Blackband, formerly of the RoyalAircraft Establishment, Farnborough and to J.L. Goldbergof BICC Ltd. for useful discussions, and to the referees fortheir help in revising the manuscript. M.M. Rahman is gratefulto the UK Ministry of Defence for financial support.

6 References

1 HOPKINSON, P.R.: 'Attenuation characteristics of coaxial cables atmicrowave frequencies'. Ph.D. thesis, University of Sheffield, 1969

2 TEPEREK, R.J.: 'Microwave loss mechanism in braided coaxialcables'. Ph.D. thesis, University of Sheffield, 1972

3 TEPEREK, R.J., and BENSON, F.A. 'Attenuation/frequencycharacteristics of braided coaxial cables at microwave frequencies',Proc. IEE, 1973, 120, (10), pp. 1219-1225

4 BAYRAK, M.: 'Losses in coaxial cables and connectors at micro-wave frequencies'. M.Eng. thesis, University of Sheffield, 1971

5 GOLDBERG, J.L., and SLAUGHTER, R.J.: 'Braid construction andattenuation of coaxial cables at microwave frequencies' Proc IEE1966, 113,(6), pp. 957-962

6 WILLIAMS, B.C.: 'Some characteristics of coaxial cables at micro-wave frequencies'. Ph.D. thesis, University of Sheffield, 1959

7 BLACKBAND, W.T.: 'A theory of electrical conduction in wirebraids'. Technical Report 72153, RAE Farnborough Oct. 1972

8 BLACKBAND, W.T.: Losses in tape braid conductors'. TechnicalReport 72030, RAE Farnborough, March 1972

9 TEPEREK, R.J., and BENSON, F.A.: 'Losses in braided coaxialcables in the frequency range 100kHz-100MHz', Proc. IEE, 1973120, (12), pp. 1465-1468

10 DEGAUQUE, P., DEMOULIN, B., FONTAINE, J., and GABILLARD,R.: Theory and experiment of a mobile radio communication intunnels by means of a leaky braided coaxial cable', Radio Set,1976, 11, pp. 305-314

356 IEE PROC, Vol. 128, Pt. A, No. 5, JULY 1981

11 DUMMER. G.W.A,, and BLACKBAND, W.T.: 'Wires and RF cables'(Pitman, 1961)

12 RAHMAN, M.M., SITCH, J.E., and BENSON, F.A.: 'Leakage fromcoaxial cables', IEEProc. A, 1980, 127, (2), pp. 74-80

13 SCHELKUNOFF, S.A.: "The electromagnetic theory of coaxialtransmission lines and cylindrical shields', Bell Syst. Tech. J., 1934,13, pp. 532-579

14 DEOBELIN, E.O.: 'Measurement systems application and design'(McGraw Hill, 1966), pp. 59-60

7 Appendixes

7. / Calculation of braiding factor from measured dataFor an open-circuited line, input impedance is given by

Zo c = Zo coth?/

For a short-circuited line

zsc = zo tanh yl

So

Zo = VZr^Z^ (1)

and

tanh 7/ = y/ZjZrZ (2)

Attenuation at high frequencies (i.e. R < to L and G < co C) is

given by

Neglecting the second term on the right-hand side of eqn. 3which would not [11] introduce any significant error belowabout 100 MHz

a = R/2ZC (4)

so that R could be calculated from a knowledge of aandZ o ,also, the inner conductor resistance Rt can be calculated [13],thus the braid resistance Rb—R—Rh and hence braidingfactor, could be found.

But for low frequencies, where R may be comparable toCJL, use of eqn. 4 could introduce serious errors and hence adifferent expression, given below, had to be used at these fre-quencies:

R = Re(7Zo)

7.2 Tolerance calculations for braiding factorThe braiding factors for wire-braided cable are given by [7]:

(i) at low frequencies (where skin effect is large comparedwith braid-wire diameter)

where

4sec20

= mnd/2 Isind

(5)

(ii) at high frequencies (where skin effect is small comparedwith braid-wire diameter)

sec2 6^BWH — 2Kfw

{tanh"1 (cos20)-cos20) (6)

The braiding factors for tape-braided cable are given by [8]:(i) at low frequencies (where skin depth is large compared

with tape thickness)

sec2fl^BTL — (7)

where

Kft = mw/2lsind

(ii) at high frequencies (where skin depth is small comparedwith tape thickness)

sec20^BTH —

sec20- 1

(8)

The analysis of possible errors in the calculated values of KB

was performed on the basis of the variations in the cabledimensions. By partial differentiation [14], the tolerance forKB is calculated as the following expressions:

(i) For wire-braided cables:

AKBW -dK,BW

dl^BW ^BW

dd,Ad,

(9)

where A/, Ado and Adw are the measured tolerances in /, dc

and dw, respectively.(ii) For tape-braided cables:

dKBT AldKBT

dKBT

dW

Adf

AW

dKBT

dtAt

(10)

where Al, Ado, At and AW are the measured tolerances in /,do, t and W, respectively.

A number of samples from each cable were measured forvariation in lay length /, dielectric diameter do, diameter ofbraiding wires dw thickness of tape braid t and width of thetape braid W. The limits of the table variation of each dimen-sion are shown in Table 1.

Table 1: Limits of variations in dimensions of cables

Item

Inner conductor diameter </,• mmDielectric diameter do mmBraid-wire diameter dw mmLay length /mmTape width (VmmTape thickness t mmBraid angle 6 degreesNumber of spindles mNumber of ends per spindle n

UR43 wire-braided

specifiedvalue

0.812.950.15

23.5——

23.7166

measured

min.

0.8133.0170.149

23.241————

-

max.

0.8253.0350.154

23.818————

—

UR43 tape-braided

specifiedvalue

0.812.95_

220.890.08

24.416

1

measured

min.

0.8122.97—

19.10.8370.083——

—

max.

0.8273.01—

19.50.9670.101——

—

IEE PROC, Vol. 128, Pt. A, No. 5, JULY 1981 357