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CERN 70-4Proton SynchrotronDepartment4 February 1970

ORGANISATION EUROPEENNE POUR LA RECHERCHE NUCLEAIRECERN EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

HIGH-FREQUENCY AND PULSE RESPONSE OF COAXIAL TRANSlviiSSION CABLESWI1H CONDUCTOR, DIELECTRIC AND SEMICONDUCTOR LOSSES

H. Riege

GENEVA1970

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Copyright CERN, Genwe, 1970Propriete litteraire et scientifique reseryee pourtous les pays du monde. Ce document ne peutetre reproduit ou traduit en tout ou en partiesans l'autorisation ecrite du Directeur generaldu CERN, titulaire du droit d'auteur. Dansles cas appropries, et s'il s'agit d'utiliser ledocument a des fins non commerciales, cetteautorisation sera volontiers accordee.Le CERN ne revendique pas Ia propriete desinventions brevetables et dessins ou modelessusceptibles de depot qui pourraient etre decritsdans le present document; ceux-ci peuvent etrelibrement utilises par les instituts de recherche,les industriels et autres interesses. Cependant,le CERN se reserve le droit de s'opposer atoute revendication qu'un usager pourraitfairede Ia propriete scientifique ou industrielle detoute invention et tout dessin ou modele decrits dans le present document.

Literary and scientific copyrights reserved inall countries of the world. This report, orany part of it, may not be reprinted or translated without written permission of thecopyright holder, the Director-General ofCERN. However, permission will be freelygranted for appropriate non-commercial use.I f any patentable invention or registrabledesign is described in the report, CERN makesno claim to property rights in it bu t offers itfor the free use of research institutions, manufacturers and others. CERN, however, mayoppose any attempt by a user to claim anyproprietary or patent rights in such inventionsor designs as may be described in the presentdocument.

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ABSTRACTThe distortion of arbitrary pulses is computed for coaxial transmission cables with

different kinds of losses. Starting from cables with normal skin effect losses, formulaefor attenuation and rise-time of rectangular and nearly rectangular pulses are developed.The theory of combined conductor and dielectric losses is reviewed. Furthermore, thecontribution of additional conducting or semiconducting layers to the losses in high-voltage pulse cables is investigated. By use of Maxwell's equations, the propagationconstants in th e frequency domain are calculated. For coaxial cables with semiconductinglayers also an approximate solution in the time domain presented, by which the distortion of arbitrary pulse shapes can be numerically computed.

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CONTENTS

1. INTRODUCTION2. TRANSMISSION LINE EQUATIONS3. HIGH-FREQUENCY APPROXH1ATION FOR COAXIAL CABLES WITH LOSSES

3.1 Coaxial cables with conductor losses only3.2 Coaxial cables with conductor and dielectric losses

4. LOSSES BY CONDUCTING AND SEMICONDUCTING LAYERSIN COAXIAL HIGH-VOLTAGE PULSE CABLES4.1 Skin effect in conducting layers4.2 Losses in semiconducting layers

APPENDIX I : FREQUENCY ANALYSIS OF SOME TYPICAL PULSE SHAPESAPPENDIX II SKIN EFFECT IN TWO CONDUCTING LAYERSWITH DIFFERENT CONDUCTIVITIESAPPENDIX Ill CALCULATION OF THE Cm1PLEX PROPAGATION CONSTANT YsIN THE CASE OF AN IDEAL TRANSMISSION LINE WITH ASEMICONDUCTING LAYERREFEREI1CES

Page

22

11

1314152327

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1. INTRODUCTIONIn the fast ejection systems of high-energy accelerators the transmission of pulses

through coaxial cables plays an important role. One problem is the power transmissionfrom the storage line pulse generators to the fast kicker magnets, which deflect the particle beam. Rise-time and attenuation of the transmitted pulses have to fulfil certainminimum conditions in order to guarantee a clean ejection. A second point of interest isthe distortion of pick-up signals (beam diagnostics, monitoring), which need to be knownif one wants to eject efficiently. I t is , therefore, very useful to have availabletheoretical methods, which allow the computation of the pulse response for differenttypes of transmission cables.

The exact calculation of the response of coaxial cables to an input signal with anarbitrary frequency spectrum is analytically a complex problem. Fortunately, however, thefrequency content of signals practically applied is usually concentrated within a limitedfrequency range (see Appendix I). Then i t is often possible to find rather simple solutions,which describe the cable behaviour with a good approximation.

2. TRANSMISSION LINE EQUATIONSThe purpose of tl1is Section is to review briefly the well-known general transmission

line theory [see, for example, Johnson1), Matick2), and Guillemin3)] and to give some definitions which are used in the following Sections. The time response of a transmissionline to an input signal can be calculated from the transmission line equations fo r voltageV and current I as functions of time t and space coordinate z:

(1)

Here R (resistance/length), L (inductance/length), G (conductance/length) and C (capacity/length) are constant parameters.

thatFor the spectral amplitudes of voltage and current, V , I , i t follows from Eq. (1)w w

dVw(z)~ =- (R + jwL) Iw (z) =- Z5 (w) Iw(z)diw(z)d- ; - = -(G + jwC)VJz) = -Yp(w) Vw(z)These equations are also valid if the series impedance/length, z , and the parallelsadmittance/length, Yp, are arbitrary functions of w. The general solutions of Eq. (2)are given by

Vw(z) = v; exp (-Yz) + v:, exp (+Yz)lw(z) = f [v: exp (-Yz)- v:, exp (+Yz)] ,

0

(2)

(3 )

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- 2 -

where V+, V- are the voltage amplitudes of the waves moving in the +z or -z direction,w wrespectively. The characteristic impedance Z0 is

and y represents the complex propagation constant, which is given by

If Z and Y are simple functions of frequency w, sometimes a solution can be found ins pthe time domain with the aid of Laplace or Fourier transformation.3. HIGH-FREQUENCY A P P R O X I ~ l A T I O N FOR COAXIAL CABLES WITH LOSSES

For coaxial cables with small losses, Z0 can nearly be regarded as a real constant and y is a rather simple f1mction of frequency in the frequency range betweenseveral hundreds of kHz and the cut-off frequency. From Eqs. (2) and (4) we see, i fR

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- 3 -

where Ri is the resistance of the conductors per length, and Li and La are the internaland external inductances per length.

OUTER CONDUCTOR

INNER CONDUCTOR

DIELECTRIC INSULATOR

Fig. 1 Configuration of a coaxial cable

The approximate expression fo r Zsi obtained with the ! ~ e l l equations is given by

with r . , r, a., a a n d ~ - , ~ being the radii, conductivities and permeabilities of the1 a 1 a 1 ainner and outer conductor, respectively (see Fig. 1). From Eqs. (4), (5), (10) and (11)and with the well-known formulas fo r La and C

c 2mo:

where E = dielectric constant of the insulator= permeability of the vacuum permeability of the dielectric

(11)

and furthermore with = ~ a ~ ai = aa = ac we get the following expressions fo r y, aand B

y = Aj; + Bjw (12)a = ~ A (13)B = Bw + 11. A , (14)

where A = l /E(_l_+_l_) 12 V ra r i ln(ra/rJ (15)B = ~ (16)

These formulae are valid as far as the skin depth 6 1 2 / ~ w a c is small with respectto the radial thickness of the inner and outer conductors.

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- 4 -

For a matched or infinitely long coaxial cable i t is now possible to calculate thedistortion of arbitrary input signals V.(t), the frequency content of which does not extendlconsiderably beyond the limits of validity of Eqs. (13), (15) and (16). From Eqs. (3) and(12) one gets the following relation between the Fourier components Vw(i) and Vw(O) of theoutput voltage Vout(t,i) and the input voltage Vi(t)

(17)If V.(t) = 0 fo r t< O, we find the output voltage V t l t , i ) appearing after the length il ouof cable by use of the convolution theorem 5 ) as

t-BVout (t,i) = A ~ J Vdt-B- x) exp ( - (A)2) dx H(t-Bi) ,2v 1T 4x x 2

0

where H(t) is the Heaviside function. If one neglects the pure cable delay, Bl, andrestricts oneself to the distortion of the signal and if the "cable rise-time" To = (A) 2is introduced, one can write Eq. (8) withy= t/T 0 , z = x/To also as

yVout(Y) = ~ J i (y-z) exp (-lz)

0

(18)

(19)

The output voltage Vout' according to Eqs. (18) and (191 can be numerically evaluated fo rany input signal by a FORTRAN program on the computer. However, in several cases one caneasily find an exact solution.

A very simple response comes out for a delta pulse of weight 1, Vi(y) = o(y)/T 0 , inthe input of the cable. By definition of the delta function o(y) the delta response d (y)cof a cable with conductor losses is expressed as

(20)

The delta response dc(y) is a good approximation for rectangular pulses of weight 1,i f the pulse length is much smaller than T0 In Fig. 2, dc(y) is plotted fo r To = 1. Theresponse uc(y) to a unit step input pulse, Vi(y) = H(y), follows directly from Eq. (19) as

Vout(Y) = uc(Y) = erfc ~ ) H(y) = [ 1 - erf ( z ~ ) ] H(y) ,where

erf(x) 1 - erfc (x)

is the well-known error function.Figure 3 shows a plot of uc(y). The response sc(y) to a square pulse of length T

can be expressed with a = T/To as

Vout(Y) = sc(Y) = erfc (2}y) H(y)- erfc (u : - a ) H(y-a)

(21)

(22)

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- 5 -

de (y)1.0

0.9

0.8

0.7

0.6

0.5

01+

0.3

0.2

0.1

00 0.5 1.0 5.0 10.0 y

Fig. 2 Delta response of a cable with 2 conductor losses only. y = t/

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- 6 -

1,0 -------------------------------0.8

0.6

0.4

0.2

o ~ - - - - - - - - - - - - , - - - - - - - - - - - - - - r - - - - - - ~ - - - - - r , - ~ - - ~ - r - - r - - r ~ - - , - ~ - - ~ . -0 5 10 15 20 40 60 80 100

Fig. 3 Unit step response of a cable with conductor losses only.y = t/T 0 =normalized time; t time; To = ( A ~ ) ~ = c a b l e length.

Sc (t iT)

. o + - - - - - - - - - - - ~ T ~ / ~ T ~ = ~ o o ~ - - - - - - - - - - ~

0. 8

0.6

0.4

0.2

0.5 1.0 1.5 2.0 (tiT)Fig. 4 Distortion of a square pulse af ter different lengths of a cablewith conductor losses only. T =pulse length; t = time; To = ( A ~ )= cable length.

y

TI1e distortion of a trapezoidal pulse with a'/a = 0.2 can be seen in Fig. 5. Figure 6shows the response to a parabolic input pulse (see Appendix I), which is computed numerically from Eq. (19). For small rise and fall times, Eq. (24) is also a good approximationfor the parabolic pulse response. Finally we regard the distortion of a pulse withoutflat top, namely Vi(y) = sin 2 (rry/a), the response of which is presented in Fig. 7.

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- 7 -

For practical purposes i t is often not necessary to know the total shape of the distorted signal. Generally i t is sufficient to know the attenuation and the rise-time ofthe pulse. Since the pulse, after i t has passed through a certain length of cable, doesnot anymore present a flat top, the most convenient definition of attenuation of squarepulses with and without ini t ial rise-time is att = 1 - Vmax/V0 , where V0 is the flat topvoltage of the input pulse and Vmax the maximum voltage of the distorted pulse.

1. 0

0.8

0.6

0.4

0.2

1. 0

o.8

0.6

0.4

0. 2

trc (t iT)

TRAPEZOIDAL INPUT PULSE

0.2 0.4 0.6 0.8 1.0 1.2 1.4

a= T/t0 = 35a'= T' /T = 0. 2

1.6 1.8 2.0

Fig. 5 Response of a cable with conductor losses only to a trapezoidalinput pulse and a square pulse of f la t to p length T. T' = r ise- t ime;t = time; To = ( A ~ ) = cable length.

VoutP A R A S O L ~ INPUT PULSE

DISTORTED PARABOLIC PULSE

SQUARE PULSE AFTER MELENGTH OF CABLE

t IT= 1

a:T/t0 :3 5a'= T' IT = 0.2

ti T

0 4 L - - - - + - - - ~ ~ - - ~ - - - - ~ - - - - - r - - - - ~ - - - - ~ - - - - ~ - - - - r - - - ~ - - - - - - - . .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Fig. 6 Response of a cable with conductor losses only to a parabolicinput pulse and a square pulse of f lat to p length T. T' = r ise- t ime;t = time; To = ( A ~ ) = cable length.

ti T

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- 8 -

Yout

0 0.5 1.0 1.5 ti TFig. 7 Distortion of a sin 2 (nt/T) input pulse after different lengths ofa cable with conductor losses only. T = pulse length; t = time;To = (A) 2 ; t = cable length.

A rough idea of the rise-t:iJne of a distorted pulse is given by the "cable rise-t:iJne"To = (M) 2 For a unit step To indicates the t:iJne after which the output signal is reach-ing 48% of the input voltage V0 With this value, however, one does not take into accountthe strong bending of the response function between V0/2 and V Therefore i t seems moremaxconvenient to define the rise-t:iJne t as the t:iJne in which the output signal is increasingrxfrom 0 to a certain ratio x of the max:iJnum voltage Vmax. For a square pulse of length Tthe maximum nearly coincides with the starting point of the second term of sc(y) [seeEq. (22) and Fig. 4] provided T > T0 Therefore i t holds approx:iJnately with q = T/T 0that

q > 1

and a tt erf ~ ) , q > 1 We can also calculate the relative rise-t:iJne fo r a rectangular pulse YrxEqs. (22) and (25).

t /T , fromrx

(25)

(26)

Yrx ~ [(1 - x) .fTici + xT 2 , (27)valid only for q > 1, qyrx > 1, x > 0.5.

In Fig. 8 the attenuation of a square pulse is plotted as a function of q. Figure 9shows a graph of normalized rise-t:iJne Yrx = trx/T versus q fo r different ratiosx = V(y )/V When using these graphs practically one only has to determine therx max

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0.1

0.01

ATTENUATIONot t-

I It '

i II

: i: I1 I I I i' ' I

! L ' r::: ( )3/2 s D I s D r ;;2vTI 0 t - X yXwhere I 1 is the modified Bessel function of order 1. TI1en the response to an arbitraryinput pulse V.(t) can be computed from the convolution1

(48)

(49)

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ATTENUATIONo! ' Hl::u .II l , . r ,,.., II i( j BILE:: T t ' r t ' , , ~ ~ ~ LHfi ~ '"l'n.l':j r8.m1 ! !!H li"' lljjil j!: ljiJit .: ' !1' lii''''"' 'ji.'i!f"''lj f 1-+ JPI!W t ~ r - ! t H ti! 11 1ril11 1:;1t ,lfii:ti 1 1 1 1;: 1 ;it ! : l ; ! ~ n ~ ; t ! t + f 1 ~ i 11 Ll'ill!i I ' ~ H ' ! :i :;l; i i i ' I 'I u J1 '" I I !i ,,, l'lil!l 11

} : I W ; ~ 1 : 1 : i r ! i . , ~ ~ ; ; : : : : , i :'1'1:,'1:: ~ : ~ : , 1 : : i''F : ~ ~ ! ' : : ~ ,, 'iil;i.;:;: I [i[F' wl!Jli:i:,w;l::::jr'j'[[)1i1!!1 j 1 ~ ! - ; - t t ~ ~ ~ I i;: 11 tl t, I ;:i; ' ' " ,,1f+H , 1.. . . , ; ; r . ~ ~ . , , . ~ ~ ~ ~. . . . . i ., ..., . , i!: . . . . .. .. ,:::u . I . :1 11 q . ,,lt . . .. I 11 .. , 11 . I 1, ' 'I ' lt , l i ' 1 I 'I ,j,, , ' 1 1 :i; ttd'i I'0 1 I I : ; ; . ~ """'I,, ! ' :! .,, 1,,, 1. I ' ! ,, l:!ill ,r,,,, I! . ' ::i , . ' ' ' : .. ljf,, I '; 1 ' 1 I i ..0.1 10 100 1000 FREQUENCY ( MHz)

Fig. 16 Real and theoret ical ly calculated attenuation constant a as function of frequency fo r BICC 40P3/20 cable with a semiconducting layer.

f- '00

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- 19 -As an example, Eqs. (45) and (49) are applied to a BICC 40P3/20 high-voltage pulse

cable, which has a semiconducting layer of d = 0.7 mm nominal thickness between the in sulator and the inner conductor. The characteristic parameters of this cable are givenby r. = 6.3 mm, r = 10.6 mm, 0 2 x ( 1 / ~ ) , s/s 0 = 2.3. Since the conductivity 01 a c sof the semiconducting layer was not exactly known, the frequency response (attenuation)of the cable was measured (see Fig. 16). A theoretical curve according to Eq. (45) wasfitted to the experimental one by choosing the parameters w = 5 x 10 9 (1/sec) ors 10s = 10- 1 ( 1 / ~ ) , r e s p e c t i v e l y , ~ = 0.23 and A= 2.70 x 10- 7 (sec 2 /m). The effectivevalues of and A are higher than those calculated from the nominal values of the characteristic cable parameters. This may be mainly due to the braided conductors.

With the parameter values given above the unit step responset

V ut (t,) = J r(x,) dx (SO)was computed fo r a cable length of 80 m (To = 0.46 nsec, TD = 46 nsec). Figure 17 showsa comparison between the computed response and the response experimentally measured afterthe same length of cable.

Figure 18 shows a few distorted pulses computed with different linear input risetimes. As fo r cables with conductor losses only [Eq. (29)], i t holds approximately thatthe rise-time of the output pulse (e.g., from 10 to 90%) is the sum of the cable rise-timefor a unit step pulse plus half of the initial r i s ~ - t ~ e of the input pulse.

From Figs. 17 and 18 i t appears that the pulse rise-time is much more deterioratedby a cable with a semiconducting layer than by a cable with conductor losses only. Besidesthe normal delay Tn' a considerable additional delay is produced, which can be quantitatively

UNIT STEP RESPONSECALCULATED DISTORTION BY CONDUCTOR LOSSES ONLY

1.0 -----------0.8

0.6

0. 4

0.2

EXPERIMENTALLY MEASURED PULSE jIIl

------CALCULATED DISTORTION BY CONDUCTORAND SEMICONDUCTOR LOSSES

0 ~ - - - - ~ - - - - ~ - - ~ - - - - ~ - - - - - r - - - - - r - - - - . - - - - - . - - - - - ~ - - - - - - - - ~ - - - - - r - - ~ ~0 10 20 30 40 50 60 70 so 90 100 110 120TIME ( nsec)

Fig. 17 Real and theoret ical ly calculated unit step response of an 80 mlong BICC 40P3/20 ~ c a b l e . Characteris t ic parameters according to Eq. (48):To = 0.46 nsec, TD = 46 nsec, ws = 5 x 109 1/sec.

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- 20 -described by TD = 6 B ~ / 2 [for definition see Figs. 19 and 20; compare also with theapproximation of Eq. (47) fo r small p]. The attenuation of a long pulse is mainly determined by the conductor losses (Fig. 20). The semiconductivity as influences only the risetime of the pulse (Fig. 21). Figures 19, 20 and 21 present the unit-step response fo r constant cable length, but different sets of the parameters TD' T0 and ws.

Vi /V o1.0 - - -T - - - . - - - - - - ,. - - - - - - - - - - - - - -- - - - - - - - - - - - -1 I /0.8

0.6

0.4

0.2

(J (J/ "-'/:'tl PI ~ "'' ,, /"/ ''/ ~ . ; , v

~ - - , //I I /I I /

I / / /I I /I I /// I /II /

0 ~ - - - - T - - - - - r - - - ~ ~ - - ~ - - ~ - r - - - - - r - - - - , - - - - - ~ - - - - r - - - ~ ~ - - ~ - - - - - r - - ~ ~0

1.0

0.8

0. 6

0.4

0.2

0

10 20 30 40 50 60 70 80 90

Fig. 18 Distortion of pulses with l inear r ise- t ime inwith conductor and semiconductor losses. Parameters:TD = 46 nsec; ws = 5 x 10 9 1/sec, representing a BICC

UNIT STEP RESPONSE

..l..o = 42.5 nsec. lo =4 7.5 nsc

_1_ 0 = 52.5 nsec

'to

10 20 30 40 50 60 70 80 90

100 110 120TIME (nsec)

a coaxial cableTo = 0.46 nsec,40P3/20 S"l cable.

100 110 120TIME (nsec)

Fig. 19 Unit step response of coaxial cables with semiconducting layersof different thickness. (Constant parameters: To = 0.23 nsec,Ws = 2.8 x 10 9 1/sec.) The parameter TD = 6 ( d ) B ~ / 2 quanti tat ively describes an additional delay, which, for small T0 , is counted from 0 to 50%of the f la t top voltage.

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- 21 -

Increasing 6, which means the thickness of the semiconducting layer, results in an increasing additional delay only. Higher skin effect attenuation (A) leads to more attenuationand higher rise-time, whereas decreasing the semiconductivity a increases the rise-timesonly.

UNIT STEP RESPONSE

1.0

0.8

0.6

0.4

0.2

to

0 : 0 . 2 3 n ~ c0 :0.31nsec0 :0.52nsec= .44nsec

0 + - - - ~ - - - - ~ - - ~ _ . ~ r - - - ~ - - - , - - - - ~ - - ~ - - - - ~ - - ~ - - - - ~ - - ~ - - - - ~0

1.0

0.8

0.6

0.4

0.2

0

10 20 30 40 50 60 70 80 90 100 110

Fig. 20 Unit step response of cables with a semiconducting layer anddifferent amounts of skin effec t losses represented by the skin effect"cable rise-time" co . Constant parameters: 'n = 47.5 nsec;w = 2.8 x 10 9 1/sec.s

UNIT STEP RESPONSE

(IJ =2.8x10 9 (1/sec)

10 20 30 40 50 60 70 80 90 100 110

120TIME (n&ec)

120TIME (nsec)

Fig. 21 Unit step response of cables with a semiconducting layer withdifferent conductivit ies (w =a /E), Constant parameters: co = 0.23 nsec,s s'n = 47.5 nsec.

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- 22 -Acknowledgements

I would like to thank !llr. B. Kuiper fo r encouraging this work and fo r his helpfulcriticism.

I t is also a pleasure to make acknowledgement to Mr. A. Messina for several stimulat-ing discussions and his help with experimental work. I am grateful to !ltr. U. Berger fo rthe preparation of the figures.

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- 23 -APPENDIX I

FREQUENCY ANALYSIS OF S D r ~ E TYPICAL PULSE SHAPESThe harmonic content of the transmitted pulse gives an indication for the applica

bility of a certain transmission line theory. Therefore, the frequency spectra A(w) ofsome typical pulse shapes V(t) computed by

are presented.a) Rectangular pulse

-12tt/T -8tt/T

+ooA(w) = ln JV(t) exp (-jwt) dt

- T/2

-4tt/T

ll' vVo

0

+T/2

4tt/T

(AI.l)

a)

Stt/T 12tt/T W

Fig. A.l The rectangular pulse in (a ) th e time and (b) th e frequency domain.

An ideal rectangular pulse of length T and amplitude V0 has the spectrum

A(w) = V sin w :!:2 TTW (AI. 2)

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- 24 -The main frequencies are concentrated in the f irs t few peaks of the frequency distri

bution (Fig. A.l). The frequency spectrum of a delta pulse, which is a special case of asquare pulse with V0 + oo, T + 0, extends from 0 to w + oo with the same amplitude,A(w) = V0T/2n = const.b) Trapezoidal pulse

vVo

T a)

-(T/2+T") + (T/2+T') t

Vo(T+T') /2 n

b)

-12n/T -8n/T 2n/T' -4n/T 0 4n/T 2n/T' 8n/T 12n/T W

Fig. A.2 Trapezoidal pulse in (a ) th e time and (b) the frequency domain.

The frequency spectrum of a trapezoidal pulse with a linear rise and fall of length T'can be expressed as

A(w) = --0- sin w - - - sin w - V (T + T') T'TIW2T' 2 2 (AI.3)Compared with the ideal square pulse the finite rise-time T' leads to a concentration of

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- 25 -the harmonic content in the first few peaks of the frequency distribution (Fig. A.Zb).This is even more true. for a parabolic pulse.c) Parabolic pulse

A parabolic pulse (Fig. A.3) is an even better, and rather simple, mathematicalrepresentation of a real "rectangular" pulse.

T

-(T/2+T')

- 1 2 ~ t 1 T -81t/T 2 ~ t ! T ' -4Tt/T 0

v

lA (w)l

Vg(T +T' )2Tt

a)

+

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- 26 -

Then the frequency spectrum can be written

A(w) = sin2 (wT'/4) sin [w(T + T')/2] TIW T (AI .5)

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- 27 -

APPENDIX IISKIN EFFECT IN TWO CONDUCTING LAYERS WITH DIFFERENT CONDUCTIVITIES

The series impedance Zsi of the arrangement shown in Fig. 12 can be calculated withMaxwell's equations

_,. a _,.rot E = - ll at H (AILl)_,. _,. a _,.rot H = a E + s at E (AIL 2)

_,.Elimination of the magnetic field vector H delivers_,. a _,. a2 _,.liE = ]l

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- 28 -

where 11' = 82 - 81, l :' = 82 + 81, v = d/8 1 and DN is the denominator of A1.current J flowing in the two conductors (1) and (2) is calculated by

The total

(Ail. 7)

The total series impedance Zsi = E0/(J!1x) of a strip of width !1x per unit length can beexpressed as

(1 + j)[z' exp (v) exp (jv) +11' exp (-v) exp (-jv)J(AILS)

where Rs(w) is the real part of Zsi and Li the internal series inductance pe r length.

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- 29 -

CALCULATION OF THE CDr1PLEX PROPAGATION CONSTANT y5 IN THE CASE OF ANIDEAL TRANSMISSION LINE WITH A SEMICONDUCTING LAYER

APPENDIX III

We apply the Maxwell equations to the coaxial structure shown in Fig. 15. Conductorsand insulator are assumed to be ideal. Only the longitudinal electrical field componentE , the radial component E and the azimuthal magnetic field component are differentz rfrom zero. All derivatives a/a in the Maxwell equations (AII.l) and (AII.2) are equal tozero.

We can write the field components:Ez = E2 ( r) exp (- y 5 Z) exp (jwt)Er = Er(r) exp (-y5 z) exp (jwt)

Hq, = Hq,(r) exp (-y 5 z) exp (jwt) ,

(AIII .1 )(AIII.2)(AIII.3)

where Ez, Er and H are the amplitudes of the waves spreading in z direction with the propagation constant-ys. For the component Ez i t follows from Eqs. (AII.3) and (AIII.l) that

(AIII.4)

where (AIII.S)(AIII.6)

The solutions E (l), E (2) in the two media 1 (semiconductor) and 2 (insulator) can bez zwritten with the Bessel function J 0 and the Neumann function N0 as) ( r ) = A1 J 0(jA. 1r) + B1 N0(jA. 1r)

~ ( z ) ( r ) = A2 J 0 (jA. 2r) + B2 N0(jA. 2r ) ,where A. 1 and A. 2 are valid for medium 1 and medium 2 respectively.

Using the boundary conditions(z)= Ez (raJ = 0

the integration constants B1 , A2 and B2 can be expressed by A1 as

(AI II. 7)(AIII.S)

(AIII.9)

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- 30 -(AIII.lO)

(AIII .11)The boundary condition fo r the magnetic field component H at the interface of insulatorand semiconductor, -

(l) (2)( )H ( i + d) = H r i + d , (AIII.l2)gives the equation for the determination of Ys With the aid of the Maxwell equations(AII.l) and (AII.2), one can verify

(AIII.l3)

and together with Eqs. (AIII .7) to (AIII.l2) the following equation for ys is obtained:

No(jA.lri) Jo[jA.l(ri +d)] -Jo(jA.lri ) No[jA.l(ri +d)]x No(jA.2ra)Jo[jA.2(ri+d)]-Jo(jA.2ra)No[jA.2(ri+d)] (AIII.l4)

J 1 is the Bessel function and N1 the Neumann function of order 1. Generally i t is dif-ficuLt to solve this complex transcendental equation, which contains also the higher modesof wave propagation. For our case we try to find an approximate solution. For an idealdielectric insulator (2) i t holds that

and for the semiconducting layer (1)(AIII.l6)

Both media are assumed to have the same E and We now consider only the principal propagation mode at frequencies not higher than a few hundred ~ l l i z . For semiconductivities osof less than 10- 1 (1/Qm) and usual cable dimensions, we can use the following approximations of the Bessel functions for small arguments Jzl

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- 31 -w i t h ~ = ln (1 + d/ri)/ln (ra/ri). Solving fo r ys and observing Eqs. (AIII. lS) and(AIII.l6) results finally in

(AIII .18)or fo r 6. 1

(AIII.l9)

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- 33 -REFERENCES

1) W.C. Johnson, Transmission lines and networks 04cGraw Hill, New York, 1950).2) R.E. Matick, Transmission lines for digital and communicative networks U4cGraw Hill,New York, 1969).3) E.A. Guillemin, Communication networks, Vol. 2 (John Wiley and Sons, New York, 1953).4) G. Fidecaro, The high frequency properties of a coaxial cable and the distortion offast pulses, Nuovo Cimento Suppl. ser. X, 15, 254 (1960).5) G. Doetsch, Anleitung zum praktischen Gebrauch der Laplace-transformation (Oldenburg,Mlinchen, 1961).6) G. Brianti, Distortion of fast pulses in coaxial cables, CERN 65-10 (CERN, Geneva, 1965).7) M.J. Lorrin, Transmission d'une impulsion tres breve par cable coaxial, Bulletin de

la Societe F r a n ~ a i s e des Electriciens, 7e serie, tome 8, No. 94, 1958.8) Jahnke-Emde, Tafeln hoherer Funktionen (Springer, Berlin, 1962).