SCIENCE

Field distortion parameters for anonaxisymmetric cylindrical

transmission lineC.B. Edwards, B.Sc, Ph.D., D.I.C., A.R.C.S.

Indexing terms: Measurement and measuring, Electromagnetics, Coaxial cables

Abstract: The electrostatics analysis package PE2D has been used to investigate the electric potential distribu-tion in cylindrical transmission lines under conditions of progressive displacement of the inner conductor fromthe 'coaxial' case. The implications for the high voltage insulation characteristics of such deformed lines isdiscussed.

1 Introduction

The computer code PE2D* has been used to study theeffect on the local electric field and potential distribution ina quasicoaxial transmission line, as the inner conductor isprogressively displaced from the axisymmetric position.This work was originally undertaken to investigate engi-neering tolerances for the construction of Sprite, the highpower rare-gas-halide laser now in operation at theRutherford Appleton Laboratory [1]. Sprite employscoaxial transmission lines, of both 10 Q and 5 Q character-istic impedance, with water dielectric in the high-speedhigh-voltage-pulse forming sections. Water is particularlyuseful as a dielectric material in short-pulse (< 250 ns)high-voltage generators as it exhibits a high relative per-mittivity (er ~ 80) at high frequencies, and deionisation anddemineralisation can give sufficiently high parallel resist-ance for use as an insulating medium for such pulses.However, the design of water-filled transmission lines iscomplicated by strong polarisation effects, and the break-down strength at the positive electrode is only half thatexhibited at the negative electrode under uniform fieldconditions. Consequently, an accurate, quantitative studywas required to optimise the performance of such com-ponents, and PE2D is ideal for this purpose.

The calculation of electric field in various geometries isfacilitated by the introduction of a dimensionless par-ameter /, known as the field enhancement factor [2],whereby the net electric field E on an electrode situated ina specific geometry G is given as follows:

= fa JM (1)

where fG is the field enhancement factor obtained from theliterature [2], and EM is the mean electric field evaluatedas the applied voltage divided by the physical separation ofthe electrodes. This paper essentially extends the availableenhancement factors by the introduction of a secondenhancement factor to treat the case of displaced cylin-drical lines.

* PE2D is available on the interactive computing facility at the Rutherford Apple-ton Laboratory. It is available to industry world wide through Compeda Ltd.

Paper 2628A, first received 3rd September 1982 and in revised form 11th March1983Dr. Edwards is with the Rutherford Appleton Laboratory, Chilton, Didcot, Oxon,England

2 Notation

The notation employed is illustrated in Fig. 1. R and r arethe major and minor radii of the transmission line, respec-tively, and x is the separation of the electrodes at the point

inner conductor

outer conductor

dielectric

Fig. 1 Definitions of symbols describing geometry of displaced coaxialtransmission line<5(.x) = x/(r - R)

of interest (x = R — r in the nondisplaced coaxial case).3(x) is a measure of the relative displacement from thecoaxial case, normalised for application to lines of differingoverall size, given by

S(x) =R-r

(2)

X{d) is the subsidiary enhancement factor, required in addi-tion to / above, to take account of distortion due to dis-placement S. Thus the net field on the required conductoris given by

— Jcoaxial X(6) (3)

where the mean field EM is given as the applied voltagedivided by the conductor separation for zero displacement,i.e.

£ « . =R-r

(4)

fcoaxial is available from Reference 2 for various values ofR/r, and xid) is obtained from data reported here.

322 IEE PROCEEDINGS, Vol. 130, Pt. A, No. 6, SEPTEMBER 1983

3 Results and discussions

Figs. 2 and 3 shows the values of x as a function of distor-tion 3 for the inner and outer conductors for R/r = 2 andR/r = 8, respectively, and other values of R/r may be

6 -

trated in Fig. 4, which is a cross-section through the poten-tial distribution of a grossly distorted line with R/r = 8.

0 0.2 1.6 1.8 2.0

Fig. 2

0.4 0.6 0.8 1.0 1.2 1.4normalised displacement 8(x)

Field enhancement x as fraction of displacement 5,r = R/2

obtained by extrapolation for reasonable values of 3. Itwill be noted that as 3 decreases, the resulting net field onthe outer conductor rises more rapidly than that on theinner {xouter > xinner for 3 < 1). For the case of R/r = 2, ofparticular relevance to the Sprite 5 Q lines, with a negativeinner conductor, the nondisplaced case (5 = 1) gives anoptimum electric field distribution for water dielectric as inthis configuration Einner = 2Eouter. Consequently any dis-placement will eventually cause positive electrode failure.

From Figs. 2 and 3 the mechanical concentricity toler-ance can be obtained for any required dielectric break-down strength in a particular system.

It should be appreciated that as 5 approaches zero, theenhancement factor x tends to infinity. This may seemincorrect because as the electrode separation decreases theelectric field must become less enhanced and eventuallyreach a plane parallel limit with an enhancement of unity.However, due to the definitions above (exprs. 3 and 4), themean field is constant for constant R and r, and so x alsotakes account of reduced electrode separation: it isobvious that x must approach infinity to compensate avanishing electrode separation!

It is of interest that as 3 approaches 2 (the maximumpossible displacement), the value of x includes an increas-ing component due to the focusing effect of the outer con-ductor. Thus, in the case of r <̂ R (Fig. 3), x actually riseswith 3 despite the increasing electrode separation whichtends to obscure the effect for r ~ R (Fig. 2). This is illus-

| 3

0 0.2 0.4 0.6 0.8 1.0 12 1.4 1.6 1.8 2.0normalised displacement 6(x)

F i g . 3 Field enhancement x as fraction of displacement S,r = R / 8

) . 0

0 .8

0.6

| 0.4

0.2

6=0.3, 6=1.7

0 R 2R

Fig. 4 Potential distribution through displaced line, r = R/8

The electric field is given by the gradient of the plot, andfield enhancement is the derivative of the field, i.e. thecurvature of the plot. To the right of the inner conductorwhere 3 = 1.7, the curvature, and hence enhancement, islarge enough to compensate for the increased separationassociated with the displacement giving a higher gradient,and hence field, at the inner than might be expected; to theleft, where 3 = 0.3, the enhancement is small, but thereduced separation accounts for the high value of x-

In conclusion, the distribution of electric field in non-concentric cylindrical transmission lines has been quanti-fied in terms of a supplementary field enhancement factor,X and the data presented enable net electrode field to becalculated over a wide range of geometrical parameters.

4 References

1 SHAW, M.J., O'NEILL, F., EDWARDS, C.B., NICHOLAS, D.J., andCRADDOCK, D.: 'Sprite, a 250 Joule KrF laser', Appl. Phys. B, 1982,28, p. 127

2 ALSTON, L.L. (Ed.): 'High voltage technology' (Oxford UniversityPress, 1968)

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