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Electron-beam distortionin theregionof aklystrongapA. H. Falkner, M.A.,Ph .D. C.Eng., M.I.E.E.

Indexing terms: Electron beams, Electric distortion, Klystrons

ABSTRACTSome klystrons incorporate a very dense magnetically focused electron beam. Such a beam is distorted in theregion of each tunnel gap by the local changes in the elec tric-f ield patte rn. The effect has been evaluated in away allowing its magnitude to be found in any par ticular case by reference to a set of graphs. The solution in-cludes the effect in the tunnel near the gap and considers both radial motion and axial velocity reduction. Theres ult s can be applied to annular and solid beams. The distortion is shown to be small in practical cases, but itdoes not follow that the overall effects are small.

LIST OF SYMBOLSe0 = dielectr ic constant of free space = charge/mass ratio of electron6 =angle in cylindrical polar co-ordinates

x{r,z) in the potential in the tunnel at the gaphas been found by a relaxation method assuming an artificialresonator shape, confined between discs at z = 1and extend-ing radially to 4a. The effect i s assumed to be confined to adistance 4 into the tunnel. These are considered reasonableapproximations. 01(r, z) is found to be of the form(Fa) P(r/a,z/l, L), where L = /a. The term (Fa) containsthe dependence on the parameters of the beam and thus thesolution depends on finding P as a function of position and theone parameter L. Fig.2shows the solution obtained forL = 0'5. The corresponding electric-field pattern is super-imposed showing the direction of the forces on electrons inthe beam. This shows that the force is outwards close to butoutside the gap and inwards approximately in the gap region.A similar result has been found for other L values in therange 0-05-1-0.

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The radial motion of an electron in this distorted electricfield and the focusing magnetic field B z is given by2

3r

002 004 006 008 0 10Fig.2Potential and field patterns for L =0-5

where $ 0 is the potential in the undistorted beam. The radialmotion causes w to change, but the type of magneticallyfocused beam considered has the property that a>dependson radial position and not on the path to that posit ion. Thisapplies during formation of the beam from a cylindricalcathode 11 '12 and is still true if the beam is subsequentlydistorted. Thus the value of u> during the distorted motion atthe gap can be evaluated from the equation for u> in the steadybeam at the same radius. Therefore3

an d r = 77- drDefine Q as the normalised fieldthen f = 7]FQ

4

5Eqn. 5 is solved to find the radial velocity r and radial dis-placement Ar in the normalised forms

uo

and W =* Ar

6

7where u0 is the undisturbed uniform axial velocity and thechange in this is not taken into account so that

t=oAr is derived similarly.

2.2 Validity of approximationsThe validity of the approximation of taking no further stagesof the iteration depends on:(a) the rad ial excursion being small compared to the tunnelradius a(b) the proportional change in density being smal l.These conditions are satisfied if

WL2(- M (1-51 104p) < 1\vc/ V\ 3/2and P ITT - (151 104 p) 1

PR0C.IEE, Vol.120, No. 10, OCTOBER 1973

(9)

(10)

The approximations in eqns. 6 and 7 also require the conditionof eqn. 10.The interpretation of the results is simplified when the poten-tial depression (px appears mainly as a reduction in the axialvelocity. The condition for this is

f2\ / y \ 3 / 2 (0-757 104p) IDSuggested l imi ts on the expressions of eqns . 9, 10 and 11 are0-l as an average, or 0-2 taking the peak values during motionthrough the gap. This should give a solution with er ro rs ofthe order of 10%. As an example consider the case L =0-7 5,^- = 3-0, p = 10 5 A/V 3 / 2 , - = 0-9. Using the solutionsvc aobtained this leads to values of 0-075, 0-22 and 0*025, respec-tively, for the maxima of eqns. 9,10 and 11. This is apossible but probably extreme case, suggesting that theapproximations are normally acceptable.2.3 Symmetry of solutionIt is interesting to note that the radial velocity at z = 0 isalways nearly zero within the approximations used. Thisfollows from the symmetrical nature of the field patternQ about z = 0 and consideration of Gauss's theorem forany cylinder contained in the tunnel and terminated byz = oo,z0 = 0.

Then / Qdz = 0, implying T(0) = 0. This result has been ooconfirmed in the calculations. Fur the r it follows that thedistortion is symmetrical about z = 0, and in particula r thatthe beam retu rns to it s undistorted shape after leaving thegap region. The approximations used will result in a residualripple that is small compared to the maximum excursion.

3.1RESULTS AND DISCUSSIONPresentationof results

The results are presented assuming the symmetry ofSection 2. 3 and are there fore shown up to the gap centre only.For pract ical applications, the functions P giving the potentialdepression and W giving the radial excursion a re requi red.Figs. 3 and 4 show these functions for L = 0-25. The peakvalue of P for R = 1-0 is approximately proportional to Lover the range L = 0-1-1-0. As L increases in this range theradial variation of P decreases, and when L = 1-0, P variesfrom 0-3 on the ax is to 0-4 at R = 1-0. The value of W atR = 0-9 is almost independent of L in the range L = 0-1-1-0,but the radial variation decreases with increasing L. Know-ledge of the peak value of T is also required and this issufficiently well represented by the formula:T= 0 026 12)

R=1 O

008

006

004

OO2

000

Fig. 3Normalised potential depression for L =0 25

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The function P is interpreted as a proportional reduction inV\3

the axial velocity of 1-51 x 104 x p x f^ -j p andW as a radial excursion Ar

= 1-51 x 104 x p x L2U^H Wa\Vc/

0 2 rR=09

0 7

-4

Fig. 4Normalised

-3

radial

-2z/

motion for

L

-1

= 0-25

0-605

0

3.2 Discussion of resultsAs expected, the effect is concentrated near the wall when Lis small and permeates towards the axis as L increases.In the region where | z| t.In contrast, the radial excursion always increases withinitial radial position at every point.The example given previously implies that the approximationsare normally satisfactory. The conditions for this alsoimply that the beam distortion is small . It is difficult tocalculate the effect on the overall device performance.Rook's6 method of considering an equivalent gridded gapdoes not allow for al l the mechanisms involved. The followingmay all be relevant:(i) the change in resonator-bea m coupling(ii) the change in beam loading on the re sonator(iii) modified space-charge waves due to the differentialaxial velocity across the beam at the gap(iv) changes in the modulation pattern as the beam is distort-ed leaving the first resonator and entering the second

resonator.The single-mode theory for annular-beam klystrons givenby Ber s1 3 shows that small changes in the beam configura-tion would have little effect on the overall gain if thesechanges continued uniformly into the tunnel. Evaluation ofhis expressions for high-density beams using typical beamgeometries and gap lengths show that the gain reductioncaused in this way is usually less than 2 dB. The multimodetheory of Beck and Deer ing9leads to similar conclusions, im-plying that effects (i) and (ii) are unimportant. The differen-tial axial velocity is estimated from the radial variation ofP and is very small, particularly for thin beams. The theoryof Siegman14 then suggests the overall effect is also small.However, it is not certain that a change in beam geometrythat occurs in a length much less than the space-chargewavelength will allow a smooth transition of the wave patternwithout signal los s. Using the example given ear lier , theradial distortion at the maximum, while being small com-pared to the radius a, would be 50% of the beam thickness

for a beam of thickness 0-15a. The sudden change to theoriginal dimensions might seriously affect the performanceand this problem requi res further study.The effect is probably distinct from those due to imp rec isefocusing and beam scalloping because these do not causesuch sudden changes.

4 CONCLUSIONThe magnitude of the effect has been evaluated for the gener-al case, and the approximations used are valid in practicalexamples . Presen t theory does not allow evaluation of thesubsequent effects, but it appears that these will probablydepend mainly on the sudden nature of the distortion of thebeam as it passes the gap. In any particular design the mag-nitude of the distortion should be considered, and it may benecessary to reduce the effect by reducing the gap lengthor the beam wall thickness.

5 ACKNOWLEDGMENTSThe author is grateful to Professor A.H.W.Beck for intro-duction to the importance of the problem discussed in thispaper, and to the Lanchester Polytechnic for making avail-able the necessary computing facilities.

REFERENCES1 ANDERSON, L. B., and BERS, A.: 'A broadband megawatthollow beam multicavity klystron'. Proceedings of the4th International congress on microwave tubes,Scheveningen, 1962, pp. 60-622 MANDICS, P. , and BERS, A.: 'High perveance hollowelectron beam study'. Massachusetts Institute of Tech-nology Quarterly Progr ess Report. (Research Lab ora-tory of E lectronics) Report 70,1963, pp. 43-573 MANDICS, P., and BERS, A.: 'High perveance hollowelectron beam study'. Massachusetts Institute ofTechnology Quarterly Progress Report (ResearchLaboratory of E lectronics ) Report 71,1963, pp. 45-504 POELTINGER, A.: 'Experimental study of a hig h-perve-ance, hollow electron beam' . M.Sc. disserta tion ,Massachusetts Institute of Technology, Sept. 19625 MIHRAN, T. G., BRANCH, G. M., and GRIFFIN, G. J.:'Design and demonstration of a klystron with 62%efficiency', IEEE Trans., 1971 EDr l8. , pp. 124-336 ROOK, C.W.: 'Approximate techniques for evaluationof electronic loading in a klystron gap in the presence ofa potential depression'. Massachusetts Institute ofTechnology Quarterly Progress Report (ResearchLaboratory of Electronics) Report 65,1962,pp.45-497 PEASE, R. L.: 'Calculation of the potential depress ionin klystron gap s'. Lincoln Laboratory Technical Report316, 19638 FALKNER, A.H.: 'Radio frequency wave phenomena onannular electron beams'. Ph.D. disserta tion, CambridgeUniversity 19669 BECK, A. H. W., and DEERING, P. : 'Space charge waveson annular beams in drift tubes', Pro c. IEE. 1958,105Bpp.635-64110 BERS, A., and PAWULA, R.: 'Dense space charge theoryof gap inter actio n'. Massachusetts Institute of Tech-nology Quarterly Progress Report. (Research Labora-tory of Electronics) Report 62,1961,pp. 117-12011 SAMUEL, A. L.: 'On the theory of axially symmetricelectron beams in an axial magnetic field', Proc. Inst.Radio Eng., 1949,37,pp.1252-125812 NATRASS, H. L.: 'An experimental study of beams form-ed by magnetron guns '. Ph.D. dissertat ion, CambridgeUniversity 1962, pp. 154-16013 BERS, A.: 'Kinematic theory of gap inte ract ion forrelativistic electron beams'. Massachusetts Instituteof Technology Quarter ly Pr ogress Report (ResearchLaboratory of E lectronics) Report 66,1962, pp. 29-32

14 SIEGMAN, A. E.: 'Analysis of multivelocity electronst reams by the density-function method. J . Appl.Phys., 1957,28, pp. 1132-1138

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