22 IEEE TRAbSACTIONS ON ELECTRON DEVICES, VOL. ED-24, NO. 1, JANUARY 1977

Starting in the Cold Cathode Distributed Emission Crossed Field Amplifier

ELDEN K. SE[/iW, MEMBER, IEEE

Abstract-The role of back bombarding electrons in the “starting” process of distributed emission crossed-field amplifiers is examined in this paper. Simple closed-form expressions are dl?- rived which show how the various parameters of the crossed-fie Id amplifier affect the back bombardment potential and phase shift of the electrons in the initial (space-charge-free) current build-1.p region. A comparison of the results of this paper with a more a(:- curate computer analysis is made for a specific case. The agreement between the two methods is shown to be excellent. Several curves are plotted which make it possible to easily find the back bonl- bardment potentials and phase.shifts for a wide range of tube pn- rameters.

I I. INTRODUCTION

NTEREST in crossed field amplifiers of the distributed emission or emitting sole type has contirl-

ued over a number of years due to the fact that they haTae very high power capability, while at the same time, they offer both moderate gain and bandwidth.

Another very attractive feature of this type of amplifier is the so called self-triggering feature. During the time that the RF input power is off, the beam current is zero. HOW- ever, when RF power is applied at the input, the beam current builds up very rapidly. Thus the amplifier is only “on” when there is an RF input power.

There are at present two different configurations of this amplifier which are undergoing development: the reen- trant type with circular format and the nonreentrant type with a linear format. In the reentrant amplifier the beam travels a more or less circular path until it interacts suffi- ciently to be collected on the slow-wave circuit. In tke nonreentrant amplifier the beam travels longitudinally down the tube where a portion of the beam is collected an a collector, rather than reentering the- interaction re- gion.

In order for these distributed emission CFA’s to “start ” the RF input power must be large enough so that an elec- tron which is emitted from the cathode by field emission will return to the cathode and cause a buildup of current by the process of secondary emission.

This means that in traversing one cycloidal path an electron must gain enough energy from the RF field to strike the cathode with a velocity for which the secondary emission ratio is greater than unity. In order for the current to continue to build up to the full operating value it is also necessary that the back bombardment process be cumu-

The author was with Varian Eastern Tube Division, Union, NJ. He s Manuscript received June 23, 1975; revised July 9,1976.

now with the Department of Electrical Engineering, San Jose Stale University, San Jose, CA 95192.

lative. That is, an electron must undergo “many” impacts with cathode before being moved into a position relative to the RF wave where the back bombardment potential is too small to give a secondary emission ratio greater than unity.

In this paper, the back bombardment potential is cal- culated in closed form using approximations which are shown to give accurate results when compared to a more exact computer analysis. The phase shift of back bom- barding electrons is then calculated using appropriate approximations which also lead to closed form solutions. Again, a comparison with more exact computer results gives excellent agreement. Several figures are included which make it possible to easily calculate back bombard- ment potentials and phase shifts. These figures also make it possible to see how the various CFA parameters affect the potential and phase shift of back bombarding elec- trons.

In the final section of this paper some general conclu- sions and recommendations are made concerning the de- sign of this type of amplifier to improve the self-triggering feature.

Vaughan [l] has also considered the problem of beam buildup in distributed emission CFA’s. However, his analysis was made using a computer program which makes it necessary to calculate separately each specific case. As a consequence, the effect of the various CFA parameters on the starting mechanism is somewhat obscured.

11. BACK BOMBARDMENT CALCULATIONS

In the following section back bombardment calculations are made taking into account both the longitudinal and transverse components of RF electric field. Space-charge forces between the‘electrons are neglected since in this initial stage of current build-up the current density is very small. An approximate formula for back bombardment voltage is derived which applies to the cases of high mag- netic field and low RF to dc electric field values and which includes the effect of the initial phase of the electron rel- ative to the RF field.

The configuration which we use for this analysis is shown in Fig. 1. The electric field is shown a t one instant of time. An electron is assumed to start from rest a t y = yo, x = 0, and t = 0 and to travel in a cycloidal path as shown by the dotted line.

The fundamental components of the RF electric fields in the interaction region are given by [2]

Ey = -El sinh Px cos (ut - Py) (1)

SHAW: CROSSED FIELD AMPLIFIER 23

Fig. 1. Sketch of CFA geometry used in the analysis of back bombard- ment potential and phase shift.

E, = +El cosh Ox sin ( w t - py) (2)

where /3 is the propagation constant of the fundamental component of the fields of the slow-wave structure.

An electron starting from rest will travel a cycloidal path and depending on the value of yo will either be accelerated or decelerated by the electric field. When we make the assumption that the RF electric field is much smaller than the dc electric field so that during the first cycloid the shape of the electron trajectory is not much affected by the RF fields, the back bombardment potential becomes a function only of the RF electric fields and can be written as

where T, is the cyclotron period. Neglecting the effect of the RF field on the shape of the

electron trajectories, the instantaneous velocities and positions of an electron are [3]

EO U,O = - sin w,t

B EO B

uyo = - (1 - cos act) (5)

Y = yo + - t - - sin w,t Eo Eo B w,B

and

Eo 1 B wc

where Eo is the dc electric field, B is the z-directed mag- netic field, and w, is the cyclotron frequency.

Using(6) and (7) in (1) and (2) with the approximation w l w , smiU to obtain the instantaneous fields seen by the electron, and then using (4) and (5) together with these instantaneous RF fields in (3), we obtain

x = -- (1 - cos w,t) (7)

where K is the interaction impedance and P is the input power. Several interesting results are immediately ap- parent from (8):

1) The maximum value of the back bombardment po- tential is obtained when an electron leaves the emitting

Io00 800 600

400

Vb 2oo (VOLTS)

100 80 60

40

20

0 IO4 2 4 6 8 IO5 2 4 6 8 IO6

PK (WATT-OHMS)

Fig. 2. Maximum back bombardment potential as a function of CFA parameters.

surface a t yo = 0. That is, when the IRF field provides maximum acceleration in the y or longitudinal direc- tion.

2 ) To the degree that the approximations apply, the maximum back bombardment is only a function of four parameters:

d w , , pa, P, and K .

It should be noted that approximate synchronism between the RF wave and the average electron velocity has been assumed for this calculation.

In order to facilitate calculation of back bombardment potentials the curves of Fig. 2 have been prepared. Only the maximum values of back bombardment potential are plotted. To obtain the back bombardment potential for electrons starting a t other than maximum accelerating field or other values of sinh pa, the results of Fig. 2 need only be multiplied by cos pyolsinh pa.

In order to check the accuracy of the approximate equations, a more exact computer analysis was used [4]. The results of using the computer program with 400 iter- ations per cyclotron period are compared with the ap- proximate formula (8) in Fig. 3. The results show excellent agreemeah even for this case of a relatively large w/wc = 0.52.

111. PqASE SHIFT OF BACK BOMBARDING ELECTRONS

In order for the interaction to build up to full operating current and power, the back bombardment process must be cumulative. That is, an electron must undergo enough impacts with the cathode to build up the current to the full operating value. The relative phase of an electron which impacts the cathode is being continually changed by the action of the RF fields and any asynchronism which may exist between the RF wave and the beam. This means that after a certain number of impacts, an electron may move into a “favorable” RF phase and no longer strike the cathode. It is thus important to be able to calculate the phase shift of back bombarding electrons to see if they

24 IEEE TRANSACTIONS ON ELECTRON DEVICES, JANUARY 1977

600 I I I I I I

> v

t

- "EXACT"CURVE, FROM COMPUTER CALCULATION

0 I I I I I -90 -60 -30 0 t30 f 6 0 +90

STARTING PHASE, B y , (DEGREES)

Fig. 3. Comparison of computer calculated and approximate back bombardment potentials.

undergo sufficient impacts to make the amplifier "start."

Below, the phase shift of back bombarding electrons is calculated by relatively simple approximate expressions which are shown to agree well with a more exact computc?r calculation. A. Analytical Calculation

The y distance an electron moves in one cyclotron period is given by

where T, - At is the time it takes an electron to complete one bombarding cycloid and At is the amount this time differs from one cyclotron period T,. In the calculation which follows we make the assumption that At << T,.

We now let u y = uyo + U y l (10)

where uY0 is the undisturbed cycloidal velocity given by (5) and uyl is the change in the y velocity due to the RF fields. Using (10) in (9) and performing the integration, we ob- tain

In this same time, the RF wave will have traveled the di:+ tance given by

(12)

The difference in phase between the back bombardkg electron and the RF wave is thus given by

The first term in (13) is due to the asynchronism be- tween the wave and the beam. If we define the asynchro- nism parameter b as

b = ( E o B - w / P ) / ( w / P )

then the asynchronism phase shift term may be written as

W Ab', = 360b -. (14)

The second term in (13) is due to the fact that the cy- cloidal shape of the trajectory is altered by the RF fields such that the electron strikes the cathode before the cy- cloid is complete. In order to find the phase shift due to this term, we must find the magnitude of At. In order to cal- culate At, we again use the assumption that the cycloidal trajectories of back bombarding electrons are only slightly affected by the RF fields. The velocities of impact are then related to the back bombardment potential by

we

For t = T,, the velocities are given approximately by

and

Using (16) and (17) in (X), solving for At, and then cal- culating the corresponding phase shift from (131, we ob- tain

where El is the peak amplitude of the RF field and Eo is the dc electric field in the interaction region.

The third term in (13) is due to the phase focusing effect of the RF fields. In order to calculate this phase shift term we must first calculate the perturbed velocity uYl. The equation of motion for the perturbed y velocity is

where E, and E, are the RF electric fields. Solving this equation for uyl and then using this value of u y l , in (13), we obtain the phase shift due to RF phase focusing to be

The total relative phase shift of an electron in one back bombarding cycle is the sum of (141, (18), and (201, giv- ing

180wAt 360 Tc- At +- + x u y l d t . ( 1 : l )

T + 360 --sin @yo. (21) w El

wc Eo

SHAW: CROSSED FIELD AMPLIFIER

20 I I I I I

25

w / w , = .52 - b = .007

o o o o o CURVE CALCULATED - USING EQUATION NO. 20 -

-‘EXACT” CURVE, FROM COMPUTER CALCULATION -

w / w , = .52 -I b = .007

o o o o o CURVE CALCULATED - USING EQUATION NO. 20 -

-‘EXACT” CURVE, FROM COMPUTER CALCULATION -

0 I I I I I -90 -60 -30 0 f30 +60 +90

STARTING PHASE, pro (DEGREES)

Fig. 4. Comparison of computer calculated and approximate values of phase shift.

u)

y t40 a

n

- b = O

0 W

v

+EO - b- LL

I I -90 -60 -30 0 +30 +60 +90

STARTING PHASE, Py, (DEGREES)

Fig. 5 . Phase shift of back bombarding electrons as a function of initial phase position.

Notice that the phase shift of a back bombarding elec- tron only depends on the amount of asynchronism b , the relative magnitudes of the RF and dc electric fields EIIEo, the relative magnitudes of the RF and cyclotron frequen- cies w/wc , and the initial phase of the electron relative to the RF wave pya.

Although (21) is an approximate expression, the agree- ment with the more exact computer calculation is excel- lent. Fig. 4 shows a comparison of the phase shift calculated using (21) and the phase shift calculated using the exact computer program.

In Figs. 5 and 6 the phase shift of back bombarding electrons is plotted as a function of the starting phase for various values of the parameters w l w , and EIIEo. To correct these curves for asynchronism, they need only be moved up or down by the asynchronism phase shift given by (14).

111. DISCUSSION OF RESULTS

A. The Formation of Bunches in the Out-of-Phase Region

Vaughan [l] has pointed out that if the beam and the RF wave are not in synchronism, there is a phase position

STARTING PHASE, @yo (DEGREES)

Fig, 6. Phase shift of back bombarding electrons as a function of initial phase positions.

where the electrons can accumulate (a stable phase posi- tion). Such a position occurs at Point “A” in Fig. 7. Point “B” in Fig. 7 also is a point of zero phase shift but is not a stable position. In order to enhance the buildup of beam current it might be desirable to allow a bunch to form a t Point “A” and then to reduce the asynchronism so that the bunch would move quickly into the favorable phase. The amount of asynchronism necessary to give a stable phase position can be found by solving (21) for the asynchronism parameter b with the phase shift set equal. to zero. For zero phase shift

From (22) we see that the asynchronism parameter must be negative. This means that wl/3 must be greater than EoIB. In order to reduce EoIB, either Eo could be made smaller or B larger. Since an increased magnetic field would reduce the back bombardment potential, it appears that reducing the dc electric field near the input region of the CFA is the best way to form a bunch of electrons in the unfavorable phase. A relatively easy way to decrease the

26 IEEE TRANSACTIONS ON ELECTRON DEVICES, JANUARY 1977

-90 -60 -30 0 +30 +60 +90

STARTING PHASE, By, (DEGREES)

Fig. 7. Phase shift of back bombarding electrons with negative asin- chronism parameter.

dc electric field would be to increase the cathode anode spacing at the input of the tube.

B. Minimum CFA Starting Conditions

1 ) Back Bombardment Potential (BBP) and the Sw- ondary Emission Ratio: The secondary emission ratio of cathode material is equal to unity for two different valc.es of the BBP. Between these two values of BBP the secon- dary emission ratio is greater than unity. Obviously, the maximum BBP must be larger than the lower unity cross-over point. It is not quite so obvious but also true that the BBP must be greater than the unity cross-over value for electron initial starting phases of other than zero de- grees.

From (8) we see that in order to obtain electron multi- plication by secondary emission for Pyo = f60°, the maximum BBP must be twice the unity cross-over value. It would thus appear that in a conservative design the BBP must be at least twice the unity cross-over value.

2) Phase Shift of Back Bombarding Electrons: In order that the back bombarding electrons make many collisions with the cathode before moving into the favorable phase, it would be desirable to keep the relative phase shift given by (21) small. One way to accomplish this by deliberate asynchronism has already been discussed above. From (21) we see that the relative phase shift could also be reduced by lowering El/Eo or w/w,. Thus, in order to enhance starting, El/Eo and w / w , should be made as small as pos- sible consistent with the required BBP.

ACKNOWLEDGMENT

The author wishes to thank H. L. McDowell of the Varian/Eastern Tube Division for his many helpful sug- gestions and discussions concerning this subject.

REFERENCE^ J. Rodney M. Vaughan, “Beam buildup in a DEMATRON amplifier,” IEEE Trans. Electron Deuices, vol. Ed-18, pp. 365-373, June 1971. D. A. Watkins, Topics in Electromagnetic Theory. New York: Wiley, 1958, pp. 26-27. M. Chodorow. and C. Susskind. Fundamentals of Microwave Elec- tronics. New York: McGraw Hill, 1964, pp. 232-234. S. P. Yu, G, P. Kooyer, and 0. Buneman, “Time dependent computer analysis of electron-wave interaction in crossed-fields,” J . Appt . Phys., vol. 36, p. 2550, 1965.