Magnetron Theory

Magnetron theorySpilios Riyopoulosa)Science Application International Corporation, McLean, Virginia 22102

~Received 28 August 1995; accepted 17 November 1995!

A guiding center fluid theory is applied to model steady-state, single mode, high-power magnetronoperation. A hub of uniform, prescribed density, feeds the current spokes. The spoke charge followsfrom the continuity equation and the incompressibility of the guiding center flow. Included are thespoke self-fields~DC and AC!, obtained by an expansion around the unperturbed~zero-spokecharge! flow in powers ofn/V1 , n, andV1 being the effective charge density and AC amplitude. Thespoke current is obtained as a nonlinear function of the detuning from the synchronous~Buneman–Hartree, BH! voltageVs ; the spoke charge is included in the self-consistent definition ofVs . It isshown that there is a DC voltage region of widthuV2Vsu;V1 , where the spoke width is constantand the spoke current is simply proportional to the AC voltage. The magnetron characteristic curvesare ‘‘flat’’ in that range, and are approximated by a linear expansion aroundVs . The derivedformulas differ from earlier results@J. F. Hull, inCross Field Microwave Devices, edited by E.Okress~Academic, New York, 1961!, pp. 496–527# in ~a! there is no current cutoff at synchronism;the tube operates well below as well above the BH voltage;~b! the characteristics are single valuedwithin the synchronous voltage range;~c! the hub top is not treated as virtual cathode; and~d! thehub density is not equal to the Brillouin density; comparisons with tube measurements show the bestagreement for hub density near half the Brillouin density. It is also shown that at low space chargeand low power the gain curve is symmetric relative to the voltage~frequency! detuning. Whilesymmetry is broken at high-power/high space charge magnetron operation, the BH voltage remainsbetween the current cutoff voltages. ©1996 American Institute of Physics.@S1070-664X~96!00303-4#

I. INTRODUCTION

Although magnetrons are the earliest developed sourcesof high-power coherent radiation, they have remained theleast well understood. The theoretical description is compli-cated by the various time scales involved in the wave–particle interaction, the high space charge, and the funda-mentally two-dimensional nature of the instability: thegrowth rate is tied to the field gradients~both DC and AC!transverse to the wave propagation direction. Among otherdifferences, magnetrons convert potential energy to radia-tion, opposed to the rest of the unbound electron devices~traveling wave tubes, cyclotron masers, free electron lasers!,which convert kinetic energy. Furthermore, it has been re-cently observed experimentally, and shown theoretically,that, in the low gain, low space charge regime, the gain curveis symmetricrelative to frequency detuning from synchro-nism; most other devices exhibitantisymmetricgain versusdetuning.

The first theoretical efforts in parametrizing high-powermagnetron operation resulted in the scaling laws,1 introducedby Slater~1943! and documented by Collins~1948!. Theycomprise a set of equations that scale the operation param-eters for a desired tube design based on the operation param-eters for an existing design. In effect, one deals with similar-ity transformations among members of the same tube family.One cannot predict the parameters for a novel design, if thelatter does not extrapolate from the existing tubes.

The next systematic effort on theoretical modeling was

undertaken by Hull.2 The magnetron characteristic curvesobtained in Refs. 2–3, relating the DC anode currentI andthe RF powerP to the applied DC voltageV at given RFfrequency, are of the form

I5AV~V2Vs!2, ~1!

P5BV~V2Vs!2@11CV1/2~V2Vs!#. ~2!

The currentI is positive for current reaching the anode. TheconstantsA, B, and C depend only on the anode circuitparameters, vane geometry, and RF frequency, whileVs isthe synchronous~Buneman–Hartree, BH! voltage, inducingE03B0 velocity at the top of the hub that matches the RFphase velocity. The least satisfying feature of Eqs.~1!–~2! isthat both anode current and RF power~thus RF gain! go tozero when the resonance conditionV5Vs is met, contrary toexperimental results and particle simulations of crossed fielddevices. In addition, the operation voltageV is a double-valued function of the currentI according to~1!. As depictedin Fig. 1 there exist two, nearly symmetric operation voltagesaroundVs at the same currentI . To exclude this possibility,one must either reject operation belowVs on theoreticalgrounds, or postulate that the second branchV,Vs is un-stable and thus not observed during actual operation. Yetmagnetrons and the related crossed-field amplifiers~CFAs!are known to exhibit stable operation with single valuedV2I characteristics well belowVs , which cannot be ac-counted for by~1!–~2!. Current cutoff is also observed atvoltages above the BH voltage; Eq.~1! provides no currentcutoff on the high-voltage side.a!Electronic mail. [email protected]

1137Phys. Plasmas 3 (3), March 1996 1070-664X/96/3(3)/1137/25/$10.00 © 1996 American Institute of Physics

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A guiding center fluid approach has recently been devel-oped and applied4–6 in the study of CFAs with distributedemission~thermionic and/or secondary! cathodes. Simulationresults,6 from the numerical implementation of the guidingcenter GC fluid model, have showed good agreement be-tween the GC model predictions and experimental character-istic curves for the SFD 262 tube. In the present work weapply the guiding center fluid approach to address the per-formance of the magnetron. The GC fluid equations can ac-tually be solved analytically for a magnetron at steady-stateoperation, taking advantage of the uniformity and periodicityaround the tube. The results lead to the following new char-acteristic equations to replace~1!–~2!:

I5V

Rs@11B~V2Vs!#, ~3!

P5hs

Vs2

Rs@11A~ I2I s!#, ~4!

whereA, B, andh are constants,R has units of resistance,and (s) stands for values at synchronism. These equationshold within the voltage rangeuV2Vsu<V1kD/sinh(kD)around the synchronous voltage; operation in that range ischaracterized by the maximum achievable spoke current at agiven AC voltage. Outside that range the departure from syn-chronism limits the current that reaches the anode. The cur-rent cutoff voltagesVc , located well above and belowVs ,are also estimated using the GC theory.

The central result from the GC fluid model is that theoperation range for magnetrons~as well as CFAs! is centeredaround the~BH! voltage Vs ; finite AC power and anodecurrent are drawn at synchronismV5Vs . Although zero gainat synchronism applies to other microwave devices~travelingwave tubes, TWTs; free electron lasers, FELs!, recently it

has been shown experimentally7,8 and theoretically9,10 thatthe low space charge, small signal operation exhibits sym-metric gain versus voltage detuningV2Vs , with maximumgain at resonance. The symmetry is broken at high-power,high space charge operation, but the fact remains that mag-netrons operate well below, as well as above, the Hartreevoltage. The range of below-synchronism operation becomesmore extended when the spoke charge effects are included inthe definition of the Hartree voltage; that pushesVs to highervalues, moving synchronism farther up from the low-voltagecurrent cutoff point.

Rather than a point-by-point comparison between thepresent approach and Refs. 2–3, some of the main differ-ences that are responsible for the new results will be empha-sized. First and most obvious is the assumption in Refs. 2–3that the hub surface acts as a perfect conductor for the RF;the RF mode structure is then taken similar to the vacuumstructure, with the hub surface treated as a virtual cathode.To see why this is misleading, consider the linear equationfor the small amplitude RF mode structure in the presence ofa uniform hub,11–14

]

]x S 12vp2

@v2ku0~x!#22@V22vp2#

D ]V 1

]x

2k2S 12vp2

@v2ku0~x!#22@V22vp2#

D V 1

5kV 1

v2ku0~x!

]

]x S vp2V

@v2ku0~x!#22@V22vp2#

D , ~5!

where u0(x)5cE0(x)/B0 is the local drift velocity andvp254pe2n0(x)/me . For a uniform hub density truncated at

x5d, one hasdn0/dx5n0d(x2d) and the coefficient on theright-hand side~RHS! becomes singular, 1/[v2ku0(d)]→`, when the drift velocity at the hub top approaches syn-chronism. The singularity forces the local RF voltage to zero,V 1(d)50, in order to balance the left- and right-hand sideof ~5!. However, when the hub top is nonsynchronous,[v2ku0(d)]Þ0, the RF voltage profilesV 1(x) exhibit amaximum at the hub surface, as one should expect, since thatis where the surface charge perturbation peaks. It has beenrecently shown15 that the singularity at resonance is an arti-fact of the usual perturbative expansion, and is removed withthe proper mathematical treatment of the resonant fluid mo-tion. Both the corrected linear RF voltageV 1 and the paral-lel RF fieldE1y52]V 1/]y exhibit a local maximum at thehub top, regardless of synchronism. In the nonlinear regime,including the effects of fully developed spokes, the local ACmaximum is shifted near the spoke center; the AC voltage atthe hub top remains finite and, in fact, larger than the emptycavity value. The treatment of the hub surface as a virtualcathode, rooted in the singular behavior of~5!, is removedfrom the present work.

A second major difference from previous efforts is thetreatment of the spoke charge. At high-power operation, onemust address the effects from the spoke self-fields, both inthe DC potential and the AC mode structure. To that end,Ref. 2 replaced the spoke charge with a point chargeq at a

FIG. 1. The typical magnetronV2I curve according to earlier results, givenby Eq. ~1!, against experimental operation points~X’s!, taken from Ref. 2,Fig. 6.44~Litton 4J52 atQL5166!.

1138 Phys. Plasmas, Vol. 3, No. 3, March 1996 Spilios Riyopoulos


specified location within the A–K space; a sequence of im-age charges enforce the conducting wall boundary conditionsat the anode and cathode. To determine the equivalent spokechargeq, the following constraint was introduced: the sumof the vacuum fields and the space charge corrections mustequal the synchronous field at a fixed point termed ‘‘synchro-nous base point.’’ One may raise the question of why thesynchronism condition should always be observed at thesamelocation inside the hub, regardless of the external DCvoltage and the amount of space chargeq. This assumptionis not necessary in the present work, where the spoke chargedensity is determined naturally from the hub density and thecontinuity equation. The AC potential from the spoke chargeis obtained in closed form using the Green’s function ap-proach. The formal solution of the loaded cavity potential isthen approximated by an effective potential, which preservesthe geometric similarity with the vacuum AC solutions.

The present approach uncovers the relation betweenvoltage detuning and maximum spoke current. A resonantregion of finite width exists around the synchronous voltagewhere the current channel to the anode~spoke! maintains aconstant width, equal to the maximum possible~saturated!width. TheV2I curve turns out almost linear in that area.Far from synchronism, the anode current is limited quicklyby a narrowing of the spoke channel, as an ever increasingfraction of the orbits starting at the top of the cathode layer~hub! cannot reach the anode. This is marked by pronouncedbreaks on both sides of theV2I curve, until current cutoffpoints are reached at large detuning. The potential due to thespoke charge is included in the precise definition of the syn-chronous~BH! voltage, marking another difference from theearlier treatments.

Finally, the Brillouin equilibrium picture for the hub isabandoned. In the nonlinear operation regime the spoke cur-rent cannot be considered as a perturbation of a pre-existingBrillouin flow; rather, it is the RF action that determines, to alarge extent, the hub density. While the approximation of auniform density cathode layer is still applicable, the steady-state hub density should be determined by the balance be-tween the current emission from the cathode~secondary orthermionic! and the spoke current drawn at the hub top. Forsimplicity, the hub is presently treated as a ‘‘black box’’ ofprescribed density. The hub height is then found from thehub density and the applied DC potential in the A–K gap.That treatment marks a break from the tradition of consider-ing the hub equilibrium as a well-known entity and treatingthe spoke flow rather heuristically. Here, instead, the spokedynamics is rigorously developed from the assumed hubdensity via the GC flow equations, the continuity, and thePoissons’ equations. That approach is validated by compari-son with experiments.

II. GUIDING CENTER FLUID MODEL

We adopt the picture of a uniform density cathode layer~hub! coupled to current filaments~spokes! to the anode, asshown in Fig. 2. This is not to say that there exists a sharphub boundary; rather, there is a drop in the density of thespace charge cloud marking the base of the spokes. Further-more, the character of the electron motion is different above

and below the hub top. TheE3B drift velocity above thehub top remains nearly synchronous with the AC phase ve-locity, v2ku0(x).0. Thus, it can be shown that the GCdrift is the dominant component of the synchronous electronmotion in the spokes. Two consequences follow:~a! the GCflow in the spokes is incompressible; and~b! the currentassociated with the cyclotron rotation has a null contribution6

to the energy exchange with the RF.Inside the subsynchronous hub, going quickly away

from the drift resonancevÞku0(x) due to the high sheardu0/dx, the motion turns out to be more complicated. TheRF-induced changes in the Larmor radius, associated withthe polarization drift, cannot be neglected near the drift-cyclotron resonance,v2ku0(x).V. Strictly speaking, theperturbed hub flow is compressible and the cyclotron contri-bution to the energy exchange is not exactly zero. The pic-ture is further complicated by the electron collisions with thecathode acting both as an electron source and sink.

It can be argued that the energy exchange inside the hub,localized at the drift-cyclotron layer, is much smaller thanthe energy exchange due to the synchronous spoke motion,because of the small width of the cyclotron layer comparedto the spoke size and because the RF field strength decreasesexponentially toward the cathode. The main approximationsunderlying the GC fluid model therefore are the following:

~i! A uniform density hub feeds the spoke density butdoes not contribute directly to the energy exchangewith the RF.

~ii ! The RF gain results from the spoke current associatedwith the GC motion.

Again, ~ii ! does not imply cold fluid with zero Larmorradius. It is shown in Appendix A that the contribution of thecyclotron motion to the energy exchange with the wavej c–E1averages to zero for any Larmor radius and AC frequency.From ~i! it follows that the spoke is the only source of ACcharge generating the AC self-potential. Remarkably, the for-mation of the fully developed spokes~a nonlinear effect!

FIG. 2. Schematic illustration of the magnetron interaction space. HereXandY are the GC coordinates in the frame moving with the wave phasevelocity. The spoke profile shown corresponds to perfect synchronism andneglecting spoke charge fields.

1139Phys. Plasmas, Vol. 3, No. 3, March 1996 Spilios Riyopoulos


rather simplifies magnetron theory, compared to the linearstability theory for the hub. The full electron dynamics~GCplus cyclotron! must be retained in the latter, where the en-ergy exchange from small ripples in the hub surface~thespoke precursors! is comparable to the energy exchanged inthe drift-cyclotron layer inside the hub.

The hub height and hub density must be determined. Upto the present day, the question of a self-consistent equilib-rium profile for the hub is far from resolved, let alone thehub profile in the presence of RF interaction. A Brillouinflow equilibrium16,17 involving pure-drift orbits at the localE03B0 velocity yields a constant hub density, but seemshard to achieve via thermionic or secondary emission anddisagrees with the finite Larmor radius observed in simula-tions. Double-stream-type equilibria,18,19 involving large cy-cloidal motion around a drifting guiding center, yield adouble-peaked density at the bottom and the top of the hub,that is not supported by numerical observations. It is argu-able that none of the above models succeeds in the presenceof strong RF interaction and/or secondary emission.

For simplicity, we assume a prescribed hub density. Inaddition, the GC density inside the hub is taken equal to theactual hub density. The spoke GC density then follows fromthe GC motion and the continuity equation. The electric fieldat the cathode is set to zero,

Ec[E0~x50!50, ~6!

assuming conditions near space-charge-limited flow. For anygiven DC voltageV0 the hub heightd is expressed in termsof the hub densitynH via Poisson’s equation. General ex-pressions for the magnetronV2I curve, as well as the othercharacteristic curves, are derived in terms of the hub densityas a free parameter. The theory is validated when a givenchoice of hub density yields good agreement with the experi-mentally measured characteristic curves for a given magne-tron. We compare the measurements for three tubes of dif-ferent designs, and find that all three are well matched by thetheory predictions using a hub density nearly 0.5 times theBrillouin density.

The self-consistent inclusion of the spoke self-fields is achallenge in analyzing magnetron operation. The slow-wavestructure in a magnetron cavityvp/c!1 is given by the elec-trostatic approximation, E152“V 1, B1.0, where¹2V 1524pn1 , andn1 is the spoke charge density. The self-consistent~‘‘loaded cavity’’! AC mode profiles cannot berepresented by a superposition of vacuum solutions, becausethe solutions of Poisson’s equation in vacuum¹2V 150 donot form a complete set for the space-charge solutions20 un-der given boundary conditions. Instead, a boundary valueproblem is solved in the anode cathode space using theGreen’s function approach. By prescribing a monochromatictraveling wave potential2V1 sin(ky2vt) as a boundarycondition at the anode, the corresponding vacuum solution~‘‘cold cavity’’ mode! is

V 1~x,y,t !52V1

sinh~kx!

sinh~kD!sin@ky2vt#. ~7!

The loaded cavity solutionV 1 can be cast in the form

V 1~x,y,t !52V1L~x!sinh~kx!

sinh~kD!sin@ky2vt1C~x!#,

~8!

where the smooth functionsL(x) and C(x) are formallyexpressed in terms of the Green’s function convolution withthe space-charge density.

Even the formal solution~8! is, in general, impossible tomanipulate analytically. It is more tractable and physicallyinsightful to introduce theeffective fieldapproximations,characterized by constant effectiveL and C. In this ap-proach, the spoke self-field is parametrized by a meanchange in the RF voltage amplitude and a mean shift in theRF phase, relative to the empty cavity values for the sameboundary conditions. While retaining the essential space-charge effects, the profiles of these approximate solutionspreserve the geometrical similarity with the vacuum cavitymodes, and are easier to handle.

The complex growth rate is obtained from the time-averagedJ–E1. The AC field is the sum of the vacuum solu-tion plus the spoke self-field,E15E11dE1. In most micro-wave devices, it is usually true that when the growth rate issmallG/v;e!1, the space charge fields are also small com-pared to the vacuum solutions,dE1;eE1; hence, since theleading contribution to the growth rateG;J–E1/W is of ordere, the termJ–dE1/W is of ordere2 and can be dropped. Inmagnetrons and CFAs the AC space-charge field can be com-parable to the empty cavity fielddE1;E1, even when thegrowth rate is small. The self-field must be retained insideE1for the growth rate computation, asdE1 significantly altersthe value~but not the order of magnitude! of G. A largespace-charge potential does not necessarily lead to a largegrowth rate, because most of the wave energy is stored in thespace above the vane tips~‘‘slow wave circuit’’ or ‘‘delayline’’ !; a large field modification in the A–K space makes asmall overall difference in the total energy stored in the cav-ity, for given AC amplitudeV1 at the anode.

The remainder of this paper is organized in the followingmanner. A general closed set of GC fluid equations forsingle-mode, steady-state magnetron operation is derived inSec. III. In Sec. IV, the basic magnetron equations are solvedand the spoke topology is obtained in relation to the voltagedetuning from synchronism, neglecting altogether spokecharge effects. The spoke self-field effects are introduced inSec. V, via an expansion of the effective space-charge fieldsaround the vacuum mode profiles. At high AC power oneexpands in powers of the inverse RF amplituden/V1 , wheren is the ratio of the effective spoke density to the hub density.Despite the changes in the field-line topology induced by thespace charge, the spoke current changes little compared tothe absence of space-charge fields, as long as one operatesnear the BH resonance. The set of characteristic equationsfor the magnetron is derived and discussed in Sec. VI. Thelinearized version of the characteristic equations for nearlysynchronous operation is also derived in that section. Thetheoretical predictions are compared with experimental re-sults in Sec. VII. A final summary and discussion of the GCfluid model is given in Sec. VIII.



III. SINGLE-MODE, STEADY-STATE MAGNETRONEQUATIONS

A closed, nonlinear set of equations is introduced basedon the GC fluid model to describe single-mode, steady-statemagnetron operation. Formal self-consistent solutions for thespace-charge fields are first given in closed form; these ex-pressions are then reduced to analytically tractable effective-field approximations. The computation of the GC motion andthe space-charge-induced fields is performed in a referenceframe moving with the AC phase velocity. Finally, the com-plex growth rate for the AC wave in the Lab frame is ex-pressed in terms of the fields and currents in the synchronousframe.

From now on we introduce dimensionless variables, bynormalizing length to the AC wave numberk21, time to theAC frequencyv21, mass tome , voltage to (ueu/mevp

2), fieldsto (kueu/mevp

2), and current tomvp3/ueu; the normalized

plasma frequency is given byvp25(4pe2n0/me)/v

2, thenormalized cyclotron frequencyV5(ueuB0/mc)/v, and thenormalized charge density isn52vp

2/4p. To avoid prolif-eration of symbols, all quantities encountered in the texthereforth assume their normalized value, unless otherwisenoted.

A. Guiding center orbits

We consider particle motion under a DC potentialV 0(x)and an AC potential of the general form,

V 1~x,y,t !52LV1

sinh x

sinhDsin@y1C2t#. ~9!

The expressions forL andC will be specified later with theintroduction of the effective fields. By going to the synchro-nous frame, moving at the AC phase velocityvp5v/k51,and by averaging over the gyroangle, the fast time scale iseliminated from the equations of motion. The remainingslow guiding center motionu5(dX/dt,dY/dt) is then givenby

dX

dt5

1

V

]H

]Y,

dY

dt52

1

V

]H

]X. ~10!

The trajectories from Eqs.~10! are given byH(X,Y)5const,where the HamiltonianH is defined by

H~X,Y!5LV1

sinhX

sinhDsin~Y1C!2@ V 0~X!1EsX#

1O S V12

V2 ,k2r2D . ~11!

The termV 0(X)1EsX is the Lorenz transformation~to firstorder in vp/c! of the electrostatic potentialV 0(X) in themoving frame;EsX is the inducedvp3B voltage. The syn-chronous fieldEs52V corresponds toE3B velocity equalto vp51. The GC velocities~10! can also be written as

u5“H3z

V⇒“–u50, ~12!

thus the flow is incompressible and follows the equipotentialsurfaces of the combined transformed potential

H52V 1(X,Y)2DV 0(X); in particular, when the DC fieldequals the synchronous field, thenDV 050 andu is nothingbut theE13B0 drift along the ‘‘frozen’’ AC equipotentials inthe synchronous frame. The velocity~12! can be split into

u5~u02vpy!1u1 , ~13!

where u05cE03B/B25yE0(X)/V is the DC drift, andu15cE13B/B2 is the RF drift. The Lab frame velocityU5vpy1u then becomes the sum of DC and AC drifts,

U5u01u152“V 03z

V2“V 13z

V. ~14!

In generalvpÞu0 anduÞu1, because the DC field is neitheruniform, nor equal toEs . The validity conditions for the GCequations, namely the separation in time scales,kX/V;kY/V;kV1/Er!1, and the absence of cyclotronresonancesku06v6nVÞ0 for nÞ0, break down insidethe hub. The detailed motion in the hub, however, is notrelevant to our model, as explained in Sec. II.

B. Space charge potentials

The Poisson’s equation for the total potentialV in thesynchronous frame is

¹2V 524pn, ~15!

wheren includes both the hub and the spoke charge density.Taking the Fourier transform in theY direction and lettingV (X,Y)5(V m(X)sin(mY)1V 2m(X)cos(mY), one has

d2

dX2V 0~X!524pn0~X!, ~16!

d2

dX2V 6m~X!2m2V 6m~X!524pn6m~X!. ~17!

Henceforth the subscript~1! signifies the sine~in phase! andthe ~2! the cosine~out of phase! Fourier components.

Because of the incompressibility of the GC flow, Eq.~12!, the density is invariant along the spokedn/dt50 andequal to the hub densitynH . The DC component of the spacecharge is equal to the hub densitynH for X,d, and to theaverage spoke density ford,X,D,

n0~X!5nH , X,d,~18!

n0~X!51

2p EY1~X!

Y2~X!

dY nH

51

2pnH@Y2~X!2Y1~X!#, X.d,

where the functionsY1(X) andY2(X) are the spoke bound-aries determined from Eq.~11!. Substituting in~16!, subjectto zero cathode fieldEc[E0(0)50 one obtains the DC field,

2V 0~X!5EcX12pnHX2, X,d,

~19!2V 0~X!5EcX14pnHdSX2

d

2D2dV 0~X!, X.d,

where the DC spoke potential is



2dV 0~X!54pEd

X

dX8Ed

X8dX9 n0~X9!. ~20!

The hub height d is determined from the conditionV 0(D)5VDC as a function of the hub density.

The AC potential can be written as a combination of thehomogeneous and nonhomogeneous solutions of Eq.~17!.Neglecting higher harmonicsumu.1,

V AC5@V 11dV 1#sin Y1dV 21 cosY. ~21!

The homogeneous solutionV 1 subject to the anode bound-ary condition yields the vacuum profile,

V 1~X!52V1

sinhX

sinhD~22!

The spoke self-field that contains both in- and out-of-phaseterms relative to the vacuum solution is given by

dV 61~X!54pE0

D

dX8 n61~X8!G1~X,X8!, ~23!

where the Fourier transformed charge densities inside~23!are

n61~X!50, X,d,~24!F n1~X!

n21~X!G5 1

p EY1~X!

Y2~X!

dY nHFcosYsin Y G5nHp

F2cosY2~X!1cosY1~X!

sin Y2~X!2sin Y1~X! G , X.d.

The nonlinear dependence ofdV 61 on the amplitudeV1 ishidden in the spoke boundariesY1,2(X). The Green’s func-tion,

G1~X,X8!5sinh~X,!sinh~D2X.!

sinhD, ~25!

whereX,[min(X,X8), X.[max(X,X8), satisfiesG1(0,X8)5G1(D,X8)50, yielding dV 1~0!5dV 1(D)50, and pre-serving the boundary conditions at the anode and the cath-ode. The complete AC solution~21! is then written as

V AC~X,Y!5V 1 sin@Y1C~X!#,~26!

V 1~X!5A~V 11dV 1!21dV 21

2 ,

or

V AC~X,Y!52V1L~X!sinhX

sinhDsin@Y1C~X!#, ~27a!

L~X!5AS 12dV 1 sinhD

V1 sinhXD 21S dV 21 sinhD

V1 sinhXD 2,

tanC~X!5~2dV 21 sinhD !/V1 sinhX

12dV 1 sinhD/V1 sinhX.

~27b!

The in-phase self-fielddV 1 can be either negative~shieldingof the AC potential!, or positive~reinforcing!, depending onwhether the spoke is centered aroundY5p/2 or 2p/2, re-spectively;dV 1 is minimum for a spoke centered aroundY50 ~i.e., near synchronism!. The opposite is true for the

out-of-phase self-fielddV 21. The self-potentialsdV 61, thecombined AC profile Eq.~26!, and the factorsL(X) andC(X) are plotted in Fig. 3 for the centered spoke configura-tion shown in Fig. 2. The structure of the ‘‘loaded cavity’’modes is discussed in more detail in Appendix A.

C. Effective-field approximation

The exact solutions for the AC fields from the spokecharge are used to obtain ‘‘effective-field’’ solutions for thepurposes explained earlier in the outline section. The soughtafter approximationV AC for the total RF fieldV AC , Eq.~27!, contains constantL andC,

V AC~X,Y!52LV1

sinhX

sinhDsin~Y1C!. ~28!

Expression~27! preserves the geometric similarity with thevacuum RF profiles and makes the equations analyticallytractable. The goal is to selectL and C in order to yield

FIG. 3. Mode profiles~arbitrary units! across the A–K space for the cen-tered spoke in~a! vacuum AC potential~dashed!, and in-phase~solid! andout-of-phase~broken! spoke self-potentials.~b! Total ~‘‘loaded cavity’’! am-plitude profileV (X). ~c! Amplitude boost factorL and phase shiftC rela-tive to the anode phase.



amplitude growth rates and phase slippages that closely ap-proximate those obtained from the exact profiles. The com-plex growth rate depends on the fast-time, short-scale aver-agedJ–E1, where E1 is the AC field andJ is the spokecurrent density. SinceJ is determined from the parapotentialGC flow, J}2z3“V 1/V, we seekL and C producingE13B flow as close as possible to the exact solution abovethe hub. ThusL andC are obtained by the minimization of

D5Ed

D

dXE0

2p

dY$z3@“V AC~X,Y!

2“V AC~X,Y;L,C!#%2. ~29!

The variation of~29! in L andC, performed in Appendix B,yields

C5tan21S 2dV 21~d!

AV12dV 1~d! D , ~30!

L5F S 12dV 1~d!

AV1D 21S dV 21~d!

AV1D 2G1/2, ~31!

where A[@sinh(2D)2sinh(2d)#/2 sinhD coshd. Interest-ingly, L and C turn out to depend only on the value of thespace charge potential at the base of the spokedV 61(d),though no such provision was made initially.

D. Spoke current and AC power balance

The magnetron gain is computed in the lab frame. Asteady state is reached at saturation when the rate of ACpower production is balanced by the cavity losses. Thepower balance equation is given by

1

Q

V12

2Z5MbE

0

DdxE

2p

p

dy J–E1 . ~32!

The left-hand side is the loss rate for the energy stored in thecavity, Q being the loaded cavityQ, and the total cavityimpedance isZ 5 ALc /Cc/2N, whereALc /Cc is the imped-ance for each of theN anode segments. The power transferon the right-hand side is the integral over the interactionspace, whereM is the number of spokes,J is the spokecurrent density, 2p is the normalized wavelength, andb isthe axial height of the cavity. The nonlinear frequency shiftdv at saturation is given by

2 dvV12

2Z5MbE

0

DdxE

2p

p

dy J–E1 , ~33!

where E1(Y)[E1(Y1p/2) is the electric field 90° out ofphase relative to the AC. In the complex notationV 15(1/2i )V1 sinh(kx)/sinh(kD)e

i (ky2vt)1c.c. the left-hand sides of~32! and ~33! are, respectively, the real andimaginary parts ofJ–E1* . Accordingly, the complex growthrate at steady stateG[1/2Q1 i dv is related to the ‘‘in-phase’’ and ‘‘out-of-phase’’ components of the spoke currentI .

The spoke current density in the lab frameJ(x,y,t) isexpressed in terms of the synchronous frame coordinatesX5x, Y5y2vpt, vp51 by

J~x,y,t !5J0~X,Y!1J1~X,Y!, ~34!

whereJ0 andJ1 are, respectively, the contributions from theDC and AC drifts,

J0~X,Y!5nHV

z3E0 , E0~X,Y!52“V 0~X,Y!, ~35a!

J1~X,Y!5nHV

z3E1 , E1~X,Y!52“V 1~X,Y!. ~35b!

From the incompressibility of the GC flow the spoke densityequals the hub densitynH . The continuity equation]n/]t1“–J50 combined with the steady-state condition]n/]t50imposes current conservation“–J50. The currentI s(X)across any spoke cross section is then equal to the currentI 0at the base of the spoke,

I x~X!5bLEY1

Y2dY nHV1

V

sinhX

sinhDcos Y

5bLnHV1

V

sinhd

sinhD EY10

Y20dY0 cos Y0 , ~36!

whereY5Y1C and Y1(X), Y2(X) are the two streamlinesdefining the boundaries of the spoke, touching the hubX5datY10 andY20, respectively. Substituting expressions~35! forthe current inside~32!–~33!, and using Eq.~28! for the ACfield E152“V 1, the gain and phase-shift equations are re-cast as

1

Q

V12

2Zi5I 0E

d

DdX E0~X!, ~37!

2 dvV12

2Zi5E

d

DdX E0~X! I s~X!, ~38!

where, because of the spoke current conservation,I x(X) hasbeen factored outside the integral in the RHS of~37!, and

I 05MIx ~39!

is the total current forM number of spokes. Thus, the gainresults from the in-phase~‘‘resistive’’ ! current I x and turnsout proportional to the actual DC anode current, as if onewere dealing with a DC circuit. The frequency shift~39! isrelated to the out-of-phase~‘‘reactive’’ ! current I s , obtainedfrom the in-phase current, Eq.~36!, by the substitutioncosY→cos(Y1p/2)52sin Y; current conservation doesnot apply toI x , since it does not represent actual flow lines.In perfect synchronism we have mainly resistive current withsmall reactive contribution,I x50 anddv50. Far from syn-chronism there may be a pure reactive currentI x when theanode current is cut off,I x50. The nonlinearities in Eqs.~36! and ~37!–~38! are hidden in the spoke boundariesY1,2(X), which are nonlinear functions ofV1 according tothe GC orbits~9!.

E. Efficiency

The efficiency is defined ash5~power into AC!/~supplied DC power!. UsingP05V0I 0 , whereI 0 is the totalanode current and Eq.~37! for the AC power conversion, itfollows that



h5*dDdX E0~X!

V0.12

r1d

D. ~40!

Expression~40! shows that the energy transferred from eachelectron to the AC field equals the change in the GC potentialenergy between the top of the hubX5d and the GC locationX5D[D2r when the electron strikes the anode. The farright side of~40! is valid in the low space charge limit witha nearly constant DC field and has a simple interpretation. Ifthe electrons are emitted from the cathode with practicallyzero kinetic energy, the per-particle energy transfer to AC is

DW5V02K2E0r. ~41!

In ~41! K is the kinetic energy~drift plus cyclotron rotation!attained at the top of the hubX5d; the kinetic energy re-mains practically constant above the hub. The potential en-ergy converted to AC is further reduced byE0r, since theGC does not actually reach the anode. The average^K& isestimated from energy conservation in the absence of AC,^K&5(V2r21u0

2)/252E0d, assuming a uniform gyroangledistribution in the hub. SubstitutingK& into ~41! and divid-ing by V0I 0 recasts the far-right-hand side of Eq.~40! as

h512~V2r21u0

2!

2V02E0r

V0. ~42!

Finite Larmor radius effects reduce the electronic efficiencyrelative to the zero Larmor radius, Brillouin flow limith0512(u0

2/2V0), even though cyclotron motion does notenter theJ–E1 interaction directly!

Equations~36! for the spoke current,~37!–~38! for thegain and frequency shift,~40! for the efficiency, and~28! forthe effective AC potential, supplemented by the relations~30!–~31! for L and C and Hamiltonian~11! for the spokeboundaries, form a closed self-consistent set of nonlinearequations for steady-state magnetron operation.

IV. LOW SPACE CHARGE MAGNETRON OPERATION

The solutions of the steady-state equations developed inthe previous section are obtained here under the most sim-plistic assumption, namely, ignoring the effects from bothDC and AC space charge on the electron motion. Consider asituation with space charge density much below the Brillouindensityvp

2!V2 and hub height equal to the nominal Bril-louin heightd5dB , where

dB[D~12A122V0 /V2D2!.

V0

V2D. ~43!

The electric field in the A–K space is

E0~X!5Ec2vp2X, X<d,

~44!E0~X!5Ec2vp

2d2E0~X2d!

2n

2vp2~X2d!2S 12

w

2 D , X.d.

At low space chargevp2!V0/D

2, the cathode is not shieldedandEcÞ0 without loss of generality. The above scenario is

relevant to low-current, low gain operation; in particular, itapplies to low-power crossed-field amplifiers with thermi-onic emitting sole and low filament power.

The GC orbits in the vacuum AC potentials are given, inthe synchronous frame of reference, by

dX

dt5V1

V

sinhX

sinhDcosY, ~45!

dY

dt52

V1

V

coshX

sinhDsin Y2

E0~X!2Es

V, ~46!

whereEs52V is the synchronous field corresponding toE03B velocity equal tovp51. Sincevp

2D!V0 , one mayignore the variation in the DC electric field, takingE0(X).E0(d)5const. The orbits, plotted in Figs. 4~a!–4~d!for various values of the external potentialv0 , are then foundby H(X,Y)5const, where the HamiltonianH is

H~X,Y!5V1

sinhX

sinhDsin Y1~E02Es!X. ~47!

For exact synchronismE05Es , electrons within2p/2,Y,p/2 are drawn toward the anode forming a characteristic‘‘spoke’’ pattern, while the rest are driven back to the cath-ode. The spoke boundaries are formed by the streamlinestouching the hubX5d at Y56p/2. The boundary curvesfrom the streamline, Eq.~47!, are

Y1,2~X!57sin21S sinhdsinhXD . ~48!

A departure from synchronismDE0(X)[E02EsÞ0, eitherby detuning the external voltage or by the buildup of spacecharge, can cause drastic changes in the streamline topology,as demonstrated in Figs. 4~a!–4~d!. A boundary curve~sepa-ratrix! appears in space, separating trajectories that can reachthe anode from those that cannot. The separatrix is thestreamline passing through a fixed point,dX/dt5dY/dt50,of Eqs. ~45!–~46!. Given thatdX/dt50 at Y56p/2 fromEq. ~45!, a separatrix emerges when Eq.~46!, written as

6coshX2j50, ~49!

has solutions forX. The detuning parameterj is the ratio ofthe detuning above the hubE02Es to the AC amplitudeV1/sinhD,

j52E02Es

V1 /sinhD5v02vpuRF

, ~50!

where uRF[V1/V sinhD. Graphic solutions of Eq.~50!,showing the intersections of the curves6coshX with j, aredepicted in Fig. 5. As long asuju,1, the AC amplitude isstrong enough to overcome the mismatchuu02vpu. All par-ticles originating between2p/2,Y,p/2 reach the anodeand the flow is similar to the synchronous casej50 ~unre-stricted motion to the anode!. When uju exceeds 1, a separa-trix curve emerges passing through the unstable fixed pointQ1 at ~Y52p/2, X5S1! or ~Y5p/2, X5S1!, respectively.

The fraction of the GC trajectories that can reach theanode depends on the existence and the location of the sepa-ratrix curve relative to the hub top. Finite Larmor radii be-come important at this point; an electron will impinge on the



anode if theX position of the GC gets aboveD2r. Forpractical purposes one may considerD5D2r as a virtualanode. The following cases arise, starting from synchronismuju50 and moving toward increasing detuning:

~i! No separatrix curve, 0<uju,1 ~as in Fig. 2!, or sepa-ratrix curve with fixed pointS below the hub surface 1<uju,coshd @Fig. 4~a!#. All trajectories originating atX5d be-

tween2p/2,Y,p/2 can reach the anode. The width of thecurrent spoke channel is maximum and equal to half a wave-length Ds5p. The spoke boundaries are given by thestreamlines touching the hub atY56p/2,

sin Y1,2~X!57sinhd

sinhX1j

X2d

sinhX. ~51!

~ii ! The separatrix fixed pointS shifts above the hubuju.coshd @Fig. 4~b!#. The spoke width is reduced toDs5Y21p/2 whereY2(d) is the intersection of the separa-trix curve withX5d,

Y2~d!5sin21S sinhXs2uju~Xs2d!

sinhd D , Xs5cosh21uju.

~52!

~iii ! The entire separatrix curve shifts above the hub,Fig. 4~c!. The current goes to zero when there is no intersec-tion pointY2 of the separatrix curve with the hub. That oc-curs foruju.ujcu, where the cutoff detuningjc makes the sinargument in~52! equal to21,

ujcu~cosh21ujcu2d!2Ajc2215sinhd. ~53!

As shown in Fig. 4~c!, cutoff occurs when the minimumdistanceSm of the separatrix curve from the cathode exceedsd.

Cases~i!–~iii ! exhaust all possible spoke topologies inthe low space charge limit. A remarkable property is a sym-

FIG. 4. Guiding center streamlines in the A–K space, in the low space charge limit, for various values of the detuningj. ~a! j521.25; ~b! j522.50; ~c!j523.75; and~d! j52.50.

FIG. 5. Graphic solutions of Eq.~52!. The curves marked~a!–~d! corre-spond to the GC flows in Figs. 4~a!–4~d!, respectively.



metry of the GC phase space topology relative to the detun-ing j: opposite values6uju generate flow topologies that aremirror images relative to the midplaneY50, as demon-strated by a comparison of Figs. 4~b! and 4~d!. The symme-try can also be deduced from the~anti!symmetry property ofthe Hamiltonian ~47! H→2H under the transformationY→2Y and (E02Es)→2(E02Es).

The spoke current can now be determined from the gen-eral equation~36! with no space charge effectsL51 andC50,

I x5bnHV1

V

sinhd

sinhD EY10

Y20dY0 cosY05I sJ. ~54!

Above I s is the saturation current at maximum spoke widthY1052p/2, Y205p/2,

I s52bnHV1

V

sinhd

sinhD. ~55!

The ratio of the spoke current to saturation currentJ[ I x /I s 5 ( 12)*Y10

Y20 dY0 sinY0 is symmetric in detuningj and

given by

J~j!51, uju,coshd,

J~j!5S 12uju~cosh21uju2d!2Aj221

sinhd D ,coshd,uju,ujcu ~56!

J~j!50, uju.ujcu.

Figure 6 plots the ratioI x/I s for three different AC ampli-tudesV1 . For given AC voltage there is a flat-top region ofconstant current, symmetric around synchronismj50, i.e.E05Es . The saturated current is directly proportional to theAC amplitude,

I s5V1

L0,

1

L052b

nHV

sinhd

sinhD, ~57!

whereL0 plays the role of hybrid impedance, connectingDC current to AC amplitude. The width of the saturation

range increases with increasing AC amplitude, as the toler-ance to detuning increases with increasing RF bunching ac-tion.

In conclusion, the spoke current dependence on the ap-plied DC voltage is obtained analytically as a function of thedetuningj, Eq. ~48!; the latter is recast in terms of the dif-ferenceV02Vs between the applied voltage and the synchro-nous~BH! voltage as

j5V02Vs

V1

sinhD

D. ~58!

The resonance width is determined by the AC powerV1

5 A2ZP. As long as the detuninguVs2V0u is roughlysmaller than the AC amplitudeV1 , so thatj,coshd, thespoke current attains a nearly fixed maximum value for givenAC voltage; a weak dependence onV0 enters only via thedependence of the hub heightd on V0 , Eq. ~43!. Once ujuexceeds coshd there is a rather fast decrease in the spokecurrent with detuning. The break pointsuju5coshX in theI x(V0) curve, as well as the current cutoff valuesVc given byjc, Eq. ~53!, are nearly symmetrically located aroundVs .

The symmetry of the GC flow is reflected in the gainversus detuning frequency at low gain situations. The ACamplitude is related to the current via the power balanceequation

DPAC[DS V12

2ZD 5hV0MIx , ~59!

M being the number of spokes. In the low space charge limit,h given in Eq.~40! is practically independent ofV0 . In thelow gain caseG[DPAC/PAC!1, the AC amplitude may betaken constant and equal to the input amplitude, independentof the applied DC voltage. Combination of~59! withI x5JI s from ~54! shows that the gain under constant inputpowerPAC is symmetric relative to the detuning,

G5GsJ, Gs5hsVsMI sPAC

. ~60!

The dependence onV02Vs enters only throughJ(V02Vs)defined by~56! and ~58!. Symmetric gain versus detuninghas been confirmed experimentally7,8 by varying either thefrequency for given voltage or the voltage for a given fre-quency in low gain CFAs.

The gain symmetry is broken when space charge effectsand/or high-power operation are considered. First, the spacecharge fields modify the GC flow; there is still a currentsaturation range for given AC voltage, but it is not exactlysymmetric aroundVs . Second, at high gain operationG[DPAC/PAC>1, the AC voltageV1 changes considerablywith the DC voltageV0 ; the spoke width at saturation isfixed but the spoke current is not. Third, the hub heightdchanges with applied voltage according to~43!; that breaksthe symmetry, even in low space charge situations, as is evi-dent in Fig. 6.

V. SPOKE CHARGE EFFECTS ON THE GC FLOW

The introduction of the spoke charge has a significanteffect on the magnetron performance. The self-fields of the

FIG. 6. Ratio of the spoke current to the saturation current versus DCvoltage detuning from synchronism, for three different AC amplitudes.



space charge modify the GC flow and the complex gain.While the gain is determined by the GC current, the DC andAC self-fields depend on the actual charge distribution, re-sulting from the finite Larmor radius spread around the GCstreamlines. For simplicity, the actual spoke charge distribu-tion is replaced by charge cloud with effective densityns andwith a trapezoidal shape, of widthp at the base, (12w)p atthe top. That yields a wavelength-averaged~DC! spoke den-sity,

n0~X!51

2nsS 12w

X2d

D2dD . ~61!

Using Eq.~61! inside Eqs.~19!–~20! yields the effective DCpotential, including spoke charge effects,

V 0~X!51

2vp2X2, X,d,

~62!V 0~X!5V 0~d!2E0~d!~X2d!

11

4nvp

2~X2d!2S 12w

3

X2d

D2dD , X.d,

where vp2524pnH , vs

2524pns , and n[vs2/vp

2 is therelative strength of the effective spoke density. The effectivespoke densityns is determined by setting the effective po-tential resulting from Eq.~61! equal to the exact valuedV 0(D) from Eq. ~20!. It follows that

ns524 dV 0~D !

4pw~D2d!2, ~63!

where w512w/3. Since the Larmor radius spreads thecharge, the ration[ns/nH5vs

2/vp2 is less than unity, al-

though the actual GC density is the same in the hub and thespoke.

The spoke charge fields must be incorporated into thesynchronous~Hartree! voltage. By definition, the synchro-nous voltageVs corresponds to synchronous hub top, i.e.,E03B0 velocity equal to phase velocity atX5d,

E0~d!

V5vp51. ~64!

Combining the resonance condition~64! with the DC fieldE0(d) from Eq. ~62!,

E0~X!52vp2X, X,d,

~65!

E0~X!52vp2d2

n

2vp2~X2d!S 12

w

2

X2d

D2dD , X.d,

yields the synchronous hub height

ds5V

vp2 [

1

vD, ~66!

wherevD is the diocotron frequency. While cylindrical ef-fects have so far been ignored, they must be included in thedefinition of the resonance, since the detuning is also a smallparameter. The inclusion of the centrifugal force in the forcebalance modifies the resonant condition~64! to

E0~d!

V512

1

V~r c1ds!, ~67!

where r c is the cathode radius. Hence, the modifiedds isgiven by

ds5V

vp2 S 12

1

V~r c1V/vp2! D . ~68!

Substitutingds in Eq. ~62! yields the synchronous voltage,

Vs5VSD2ds2 D 1

n

4vp2w~D2ds!

2

5Vs1n

4vp2w~D2ds!

2, ~69!

where Vs[V(D2ds/2) is the value without the spokecharge. The spoke charge increases the BH voltage for agiven hub density. In general, it can be shown that the addi-tion of charge above or below the synchronous layer, respec-tively, increases or decreases the BH voltage.

The effect of the AC space charge is accounted for by achange in the AC amplitude to the effective valueLV1 , andby a constant phase shiftC relative to the AC phase~bound-ary condition! at the anode. The GC equations of motionemploying the effective field approximations~62! and ~28!for the DC and AC potentials become

dX

dt5L

V1

V

sinhX

sinhDcos~Y1C!, ~70!

dY

dt52L

V1

V

coshX

sinhDsin~Y1C!2

E0~X!2Es

V. ~71!

The solutions to these equations are again given byH(X,Y)5const, where

H~X,Y!5LV1

sinhX

sinhDsin~Y1C!2V 0~X!2EsX. ~72!

The spoke topology is again determined by the existence andthe number of fixed points, defined by the solutions of

j~X![2E0~X!2Es

LV1 /sinhD5coshX. ~73!

The variation in the detuning across the anode–cathodespace creates more than one fixed point and separatricesshaping the GC flow, depicted in Figs. 7~a!–7~d! obtained fordifferent values of detuning. We will take the detuning at thehub top,

j52E0~d!2Es

LV1 /sinhD, ~74!

as the control parameter for synchronism. The detuning canalso be expressed in terms of the voltage mismatchV02Vs

divided by the AC amplitudeV1 ,

j5V02Vs

LV1

sinhD

D S 12dsD

1nwD2dsD D ~75!

by using~62! and ~69! to express the fields in terms of thevoltages.



The graphic solutions of~73! in Fig. 8 shows the inter-sections of the6coshX curves with the detuning curvej(X)for various applied DC voltages. The cathode detuning re-mains fixed atjc5Es/(LV1/sinhD! because of the space

charge limit E0(0)50. Inside the hub the detuning is astraight line of slopevp

2/(LV1/sinhD! extending up to thehub heightd. The hub heightd, marked by the ‘‘knee’’ in thej(X) curve ~the slopedj/dX drops atds due to the smallereffective charge of the spoke!, increases with DC voltage.For DC voltage below synchronism,j,0, there is alwaysone fixed pointS15(Y152p/2, X1s! with X1s the root of

E0~X!2Es

LV1 /sinhD52coshX, ~76!

and one separatrix curve throughS1 . For large negative de-tuningj,2coshd, S1 is above the hub, Fig. 8 curve~a!, andthe separatrix limits the width of the spoke toD5p/22Y2 ,Fig. 7~a!, where

Y2~d!5sin21S sinhX1s2uju~X1s2d!

sinhd D . ~77!

A low-voltage cutoff is reached and the width drops to zerowhen the sin argument in~77! exceeds 1, meaning that theseparatrix does not intersect the hub. Decreasing the detun-ing to

j.j1[2coshd, ~78!

FIG. 7. Guiding center streamlines in the effective field approximation, Eqs.~62! and ~72!, including the effects of the spoke self-fields. HereLV151.5,nH50.8nB , n50.25, andVs53.830.~a!–~d! correspond to different values of voltage detuning fromVs . See the text for details.

FIG. 8. Graphic solutions of Eq.~73! showing the location of the variousfixed points. The fixed points generated by the intersections of coshX withcurves~a!–~d! correspond to the GC streamlines shown~a!–~d!.



by increasing the DC voltage, shifts the fixed pointS1 insidethe hub, Fig. 8 curve~b!. At this point the spoke attainsmaximum width equal to half-wavelengthY22Y15p, Fig.7~b!. The spoke width remains maximum and equal to half-wavelength as the DC voltage increases from slightly belowto slightly aboveVs and j changes sign, until a further in-crease in voltage causes the curvej(X) to intersect coshX,as shown in Fig. 8, curve~c!. The intersection pointsX2s andX2e are given by the larger and smaller root of

j~X![2~Ed2Es!1~n/2!vp

2~X2d!

LV1 sinhd sinhD5coshX. ~79!

A trapped particle island appears, centered aroundO25(X2e,p/2) and bounded by the separatrix passingthroughS25(X2s,p/2). When it first appears the island liesoutside the spoke and the spoke current is not affected, untilthe separatrix passing fromS2 intersects the hub, Fig. 7~c!,as the island size grows with detuning. The final change inthe streamline topology occurs whenj passes through avalue j2 for which the two separatrix curves coincide;j2 isdefined implicitly by the condition

H~X1s ,2p/2!5H~X2s ,p/2!. ~80!

The trapped island migrates to the base of the spoke, cen-tered aroundO2 and bounded by the separatrix passing fromS1 , Fig. 7~d!. That causes an abrupt limitation in the spokecurrent since trapped particles cannot reach the anode. Thespoke width drops toDs5Y21p/2, where

sin Y252S vp2~d22X1s

2 !2Es~d2X1s!

LV1 sinhd/sinhD2

sinhd

sinhX1sD .

~81!

The left spoke boundary is the streamline touching the hub atY152p/2. Current is completely cut off when that stream-line cannot reach the anode. Using the streamline equation~72! to find the maximum height of the spoke boundary, thecutoff limit is j.jc , wherejc is given by

jc5sinh D1sinhd

D2d2

nvp2w

4LV1 /sinhD~D2d!. ~82!

By sweeping through all possible spoke topologies it hasbeen shown that there is a voltage range~‘‘synchronous re-gime’’! corresponding to the maximum widthDs5p at thespoke base. That range is given by

j1, j,j2 , ~83!

with the limitsj1, j2 given by~78! and~80!. In the synchro-nous regime the saturated spoke current is simply propor-tional to the AC amplitudeV1 ,

I x5I s5V1

L0. ~84!

The hybrid impedanceL[V1/I s , relating the AC amplitudeto the DC current assumes a minimum valueL5L0, inde-pendent ofj and given by

1

L052nHb

L

V

sinhd

sinhD. ~85!

Despite the changes in the spoke topology from space chargeeffects, such as the spoke ‘‘tilt,’’ the same saturation currentresults under given AC amplitude as in the absence of spacecharge. What does change is the synchronous voltageVs andthe resonance width aroundVs . In addition, the flat currentregion as well as the cutoff voltages are not symmetricaroundVs .

Outside the synchronous region, the hybrid impedanceincreases quickly from the restriction in the spoke width withincreasing detuningL5L0/J. The functionJ<1 is relatedto the ratio of the spoke width to the saturated widthp,

J[

*Y1

Y2 dY cos Y

*2p/2p/2 dY cos Y

5sin Y2~d!2sin Y1~d!

2, ~86!

and depends on the detuningj. The spoke current decreasesas

I x5V1

L0J 5I sJ. ~87!

The ratioJ of the spoke current to the saturated current isplotted versus the detuningj in Fig. 9, for given, constantvalues of AC amplitudeV1 . On the low-voltage side, thecurrent drop is similar to that in the absence of space chargeeffects, Fig. 6. The extent of the flat-top region on the high-voltage side is, however, smaller than in the charge-free caseunder the same AC amplitude. The sudden decrease in thecurrent is due to the emergence of ‘‘trapped particle islands,’’such as in Figs. 8~c!–8~d!. A current recovery occurs at evenhigher detuning, explained as follows: as the island size in-creases with detuning, the pointS2 shifts above the anodeand trapped particles are now able to contribute to the anodecurrent. The decline and recovery of the current with detun-ing explains the ‘‘S’’-shaped current versus voltage charac-teristics that are sometimes observed in crossed-field devices.The detuning can only increase to the point where the hubheight d, being a function of the applied voltageV0 , riseswithin a Larmor radius distance from the anode. At this point

FIG. 9. Ratio of the spoke current to the saturation current versus appliedvoltage detuning from synchronism, for three different AC amplitudes. Theeffective spoke density is fixed ton5ns/nH50.25.



there is loss of magnetic insulation and the tube shorts out,the cutoff voltage being independent of the AC amplitude.

It remains to self-consistently evaluate the effective-fieldparametersn, L, and C. It will be assumed that the spokecharge effects are small compared to the AC voltagestrength,n/V1;e!1, and consequentlyL;11e, C;e.The evaluation is based on the expansion of the GC equa-tions in powers ofn/V1 around the zero spoke charge limitn50; in effect, the spoke charge effects are treated as pertur-bations around the ‘‘unperturbed’’ GC orbits without spokecharge fields. Since the charge density, and thusn, is alwaysbounded by the Brillouin density limitvp

2<V2, the n/V1expansion improves with increasing magnetron AC power.

The self-consistent spoke boundaries are found from Eq.~72!,

sin Y1,2~X!5sin Y1,2~X!1n

LV1

vp2

4

sinhD

sinhX~X2d!2

3S 12w

3

X2d

D2dD , ~88!

whereY1,2 are the zero spoke charge boundaries obtainedfrom Eq. ~47!. Expanding~88! and letting L51, keepingonly first-order contributions from spoke charge, one findsthat the effect of the space charge is a parallel shiftdY(X) ofthe spoke boundaries,

Y1,2~X!5Y1,2~X!1n

V1dY~X!1O S n

V1D 2, ~89!

dY~X!5vp2

4sinhD

~X2d!2

Asinh2 X2sinh2 d

3S 12w

3

X2d

D2dD . ~90!

Both boundaries are equally shifted, thus the spoke width,Y22Y15Y22Y11O (n/V1)

2, and the total spoke chargeequal that without self-fields. The effective spoke densityfollows from ~20! and ~63!:

ns54

w~D2d!2Ed

D

dXEd

X

dX8nH2p

@Y2~X8!2Y1~X8!#. ~91!

Since the induced shift is zero at the spoke base,dY(d)50,and since the current equals the current drawn at the spokebase,“–J50, the same current flows in the spoke as in theabsence of spoke-charge effectsn50,

I s5bnHLV1

V2sinhd

sinhDJ1O S n

V1D 2. ~92!

In conclusion, the unperturbed spoke boundaries are suffi-cient for the lowest-order computation of the effective spokedensity~91! and the actual spoke current~92!.

To find L and C from ~29!–~30!, one needs the ACcomponent of the spoke self-fielddV 1, given by Eq.~22! interms of the AC spoke charge. The Fourier transform of thespoke charge over the modified boundaries~89! yields

n1~X!5nHp E

Y1

Y2dY sin Y52

nHp

@cos Y22cos Y1#

52nHp

n

V1sin Y2~X!dY~X!, ~93!

n21~X!5nHp E

Y1

Y2dY cos Y5

nHp

@sin Y22sin Y1#

52nHp

sin Y2~X!1O S n

V1D 2.

~94!

The largest component of the AC spoke charge is 90° out ofphase relative to the AC potential at the anode; that is aconsequence of theE13B nature of the flow forming thespoke. The out-of-phase AC charge is approximately thesame as for a perfectly centered spoke without self-field ef-fects. The in-phase AC charge is much smaller and comesfrom the modifications of the GC flow induced by the DCspoke self-field.

Near synchronism,j1, j,j2 , the symmetric spoke pro-file for synchronous operation without spoke charge effects,Eq. ~48!, can be employed for the lowest-order evaluation of~91!–~94!. Applying ~48! inside~91! yields the ration of theeffective spoke density to the hub density,

n54

wp~D2d!2Ed

D

dXEd

X

dX8 sin21S sinhdsinhX8D . ~95!

The AC spoke potential is found by substitution of the ACdensity~93!–~94! inside~22!. For the effective-field compu-tation one needsdV 61 at the base of the spoke,

dV 21~d!522vp

2

p

sinh2 d

sinhD Ed

D

dXsinh@D2X#

sinhX, ~96!

dV 1~d!52n

V1

2vp2

p

sinh2 d

sinhD Ed

D

dXsinh@D2X#

sinhXdY~X!.

~97!

The main component of the spoke AC potential is out ofphase and of the same order as the vacuum AC solution. Thein-phase AC self-field is smaller, of first order inn/V1 .Equations~95!, ~96!, and~97! self-consistently determine thespoke self-fields to ordern/V1 .

According to the effective-field parameters depend es-sentially onV and the A–K gapD. Level plots ofn in D2Vspace are shown in Fig. 10~a!. These curves also determinethe minimum AC power for the validity of then/V1 expan-sion. The level plots forL and C are shown in Figs. 10~b!and 10~c!.

VI. MAGNETRON CHARACTERISTIC CURVES

In the preceding section the spoke currentI x has beenobtained from the GC flow as a nonlinear function of the DCvoltage detuning and the AC amplitude. The current curves



shown in Fig. 9 were obtained, assuming constant AC am-plitude along each curve. In fact, at high-power operation,the AC amplitude changes with the applied DC voltage,hence the actualV2I curves do not resemble the flat-topcurves of Fig. 9. In this section the current equation~87! iscoupled with the power balance equation for the self-consistent evaluation ofV1 and the detuningj, closing the

set of equations describing magnetron operation. The satura-tion of the spoke width shows up as a ‘‘linearity’’ of theV– Icurve around the BH voltage.

The circulating magnetron power is expressed in termsof the AC amplitudeV1 as

21

Pc5NV12

ALc /Cc

, ~98!

whereALc /Cc is the anode circuit impedance per segmentandN is the number of anode segments~at p-mode oscilla-tion there are half as many spokes,N52M !. Thus, one candefine the total magnetron impedance ofN segments in par-allelZ5 ALc /Cc/2N, writing

Pc5V12

2Z. ~99!

The total power losses are expressed in terms of the cavityQby

PL51

Q

V12

2Z. ~100!

Of the total losses, only a fractionhc is due to the poweroutput of the tube~the rest being Ohmic dissipation at thevanes!, thus the output power is

PO5hc

1

Q

V12

2Z, ~101!

wherehc is the circuit efficiency. The electronic efficiency isdefined as the ratio~power converted to AC!/~power V0I 0supplied by the external source!, thus the power balance atsteady state yields

1

Q

V12

2Z5hV0I 0 , ~102!

whereI 05MIs . Combining~101! and~102! casts the outputpower asPO5hchV0I 0 .

At steady state, the power loss equals the power transferto AC, expressed in terms ofJ–E1 in Eq. ~37! Equating theRHS of ~37! and ~100! yields expression~40! for electronicefficiency,h[@V 0(D)2V 0(d)]/V0 . Using Eq.~62! for theDC potential V 0(X) to express E0(d) in terms ofV0[V 0(D), one finds

h5S 12d1r

D D 12vp2d2/2V0

12d/D

1nvp2

4V0DrF S 12

d1r

D D S 12w

3

r

D2dD1r

DG .~103!

Formula ~103! has the following interpretation. In the zerospace charge limit, when the DC field has the nominal~vacuum! value E052V0/D, the efficiency from the GCshift D2d2r is h0512(d1r)/D. The hub charge in-creases the DC field above the hub relative to the nominalvalue; that is taken into account by the term multiplying thefirst bracket on the RHS of~103!. The third term propor-tional to n derives from a further increase in the DC field

FIG. 10. ~a! Level curves of the effective spoke density inkD2V/v space.~b! The same for the effective amplitudeL. ~c! The effective phase shiftC.The hub density is half the Brillouin density.



strength above the hub due to the spoke charge. Clearly, thespace charge tends to increase the efficiency under givenoperation voltageV0 .

Expression~103! for the efficiency neglects the changein the GC kinetic energy above the hub. To subtract the frac-tion of potential energy that goes into increasing the GC driftenergy ~AC and DC!, one needs to consider second-ordercorrectionsO (V1

2) in the GC drift velocity, Eq.~11!; thesecorrections result from the curvature drift, causing the GCstreamlines to deviate from the equipotentials.22 Inclusion ofthe second-order corrections in the spoke current densityJand in the power balanceJ–E1 leads to the corrected effi-ciency,

h5h1dh

5h21

2 K @u02~D !1u1

2~D !#2@u02~d!1u1

2~d!#

V0L ,

~104!

whereh is Eq. ~103!. The evaluation ofdh in Appendix Dyields

h5h2nvp2

VV0~D2d!S 12

w

2 D2

L2V12

2V2V0

sinh2 D2sinh2 d

sinh2 D. ~105!

Finally, the frequency shift at steady state is given byevaluating the RHS of~60!, where the ‘‘out-of-phase’’ cur-rent is

I s52nHV1

V

sinhX

sinhD EY1

Y2dY sin Y. ~106!

The spoke boundaries are expanded asY1,25Y1,21(n/V1)dY, whereY1,257sinhd/sinhX are the boundariesof a centered spoke with no space charge, anddY(X) is thespoke charge-induced shift. Substitution inside~38! yields

2 dvV12

2Z52nHMb

V1

V sinhD Ed

DdX E0~X!

sinhX

sinhD

3EY1

Y2dYS sin Y1cosY

n

V1dY~X! D .

~107!

It follows that the frequency shift has two contributions:

dv5MbZ

V1

vp2

2pV S sgn~j!

sinh2 DP~j!

1sinhd

sinh2 D

n

V1Ed

DdX E0~X!dY~X! D . ~108!

The first bracketed term carries the contribution of the un-perturbed~zero space charge! streamlines, while the secondbracketed term in~107! yields the effect of the streamlineshifting due to space charge effects. Near synchronism, thespoke is centered aroundY50 and the phase integral* dY sinY is nearly zero, yieldingP~j!50. Away from syn-

chronism, the streamlines that cannot reach the anode foldback into the hub and shift the center of charge nearY56p/2, contributing to a finite ‘‘reactive’’ current. The foldedstreamline contribution yields

P~j!50, uju,coshd,~109!

P~j!5Ed

cosh21 jdX E0~X!

3Asinh2 X2@Aj2212uju~cosh21 j2X!#2,

uju.coshd.

The shiftdv is plotted in Fig. 11 as a function of the detun-ing parameter. Notice that the frequency shift would be zeroat synchronismj.0, if one neglected the streamline shiftdY(X) due to the spoke charge effects.

Equations~62! for the DC potential,~87! for the current,~100! for the output power,~102! for the power balance, and~108! for the frequency shift, constitute the five main mag-netron equations with seven unknowns:V1 , P0 , I 0 , V0 , dv,the hub heightd, and the hub densityvp

2. There are also thesix auxiliary equations:~29!, ~30!, ~95!, ~96!, ~97!, and~105!that define, respectively,L, C, n, dV 1, dV 21, andh. Elimi-nating any four of the unknowns among the five principalequations yields a relation between the remaining three. Wecan always leave the hub density as an unknown, and obtaina relation between any two of the quantities that can be di-rectly measured experimentally, with the hub density as theparameter to be determined. Of particular importance are theDC voltage versus DC currentV2I and the output powerversus DC currentP2I characteristics, respectively. Theelectronic efficiencyh that is easily measurable is also aconvenient quantity for comparisons with experiments.

As an example, the characteristicV2I curve is obtainedby expressing the AC anode potentialV1 on the left-handside of the power balance equation~102! in terms of thespoke current from Eq.~92!, yielding

V052p2V2 sinh2 D

hQZb2M2L2vp4 sinh2 d

1

J2 I 0 . ~110!

FIG. 11. Frequency shift versus detuning without~dashed! and with~solid!spoke-charge effects.



The loaded cavity impedanceZ can be taken equal to thecold cavityZ; for fixed AC amplitudeV1 at the anode, achange in the AC energy inside the A–K gap due to thespace charge does not alter significantly the total energystored in the magnetron cavity, since the energy in the A–Kgap remains a small fraction of the energy stored in the an-ode structure. Expression~110! looks like Ohm’s law,

I 05V0

R, ~111!

except that the ‘‘radiation resistance,’’ defined by

R52p2 sinh2 DV2

hQZb2M2L2vp4 sinh2 d

1

J2 , ~112!

depends implicitly onV0 throughd, L, h, andJ. The exactsolution for theV2I characteristic requires the simultaneoussolution of all relevant equations to express the above vari-ables in terms ofV0 and I 0 only. This task, in general, canonly be carried out numerically, and a code is currently underdevelopment.

On the other hand, it is well known that during usualoperation, the magnetron characteristic curves are relatively‘‘flat.’’ For operation near synchronism, we have seen thatthe spoke width is saturated,J51, while the other factors inthe denominator of~110! are slowly varying functions of thevoltage. One can therefore expand the characteristic equa-tions around the synchronous operation point, obtained atsynchronous~BH! voltageVs . Any characteristic curve inthe synchronous operation range is then determined by thevaluesVs , I s , ds , hs , Ps , and their local derivatives. Thefull set of equations has to be solved only once in the pro-cess, to determine the synchronous parameters. This methodyields ‘‘linearized’’ characteristics appropriate for compari-sons with experiments.

The linearizedV2I curve is obtained by expanding 1/Rin V02Vs , yielding

V0F11bSV0

Vs21D G5RsI 0 , ~113!

where

Rs52p2V2 sinh2 D

hsQZb2M2vp4Ls

2 sinh2 ds,

~114!

b5Vs

d

dV S ln 1

RDs

.

The analytic formula forb is evaluated in Appendix E. Thedynamic impedance is defined by the slope of theV2Icurve. From~113!, it follows that

R5dV0dI0

5Rs

11b. ~115!

The lower-current cutoff voltage, given byVc5(121/b)Vs ,falls well below the BH voltage. Notice that the precise valueof Vs includes the spoke charge contribution and is higherthan the BH value based merely on the hub charge. Thepower-current (P2I ) characteristic results in a similar man-ner from ~99!–~102!,

P5PsF11aS II s21D G , ~116!

where

Ps5hsVsI s5hsRsI s2, I s5

Vs

Rs,

~117!

a5I sd

dI~ ln P!s .

Expressions~113!–~116! fail outside the synchronousrange. Returning to the nonlinear equations, the exact volt-age limitsVr1 ,r2

for the validity of the ‘‘linearized’’ expres-sions are

Vr1 ,r25Vs7j1,2LV1

D

sinhD S 12dsD

1nwD2dsD D 21

,

~118!

wherej1,2 are given by~78! and~80!. HereVr1andVr2

markthe edges of the flat-top current regions, as those shown inFig. 9 for constant AC amplitude. The AC amplitudeV1 isnot the same for each edge and must be computed from thepower balance equation. Since the AC powerP increaseswith the DC voltage and the distanceuVr2Vsu in ~118! isproportional toV15A2PZ, the resonance is quite broaderabove the BH voltage than below, i.e.,uVr2

2Vsu.uVr12Vsu. Outside the synchronous region, the resistanceR, Eq.~111!–~112!, increases by 1/J2. The main effect of the de-tuning is carried by the spoke width factorJ, plotted in Fig.9. HereJ depends strongly onV02Vs , going fromJ51 atthe edges of the synchronous regionVr1, Vr2 to J50 ~infi-nite resistance! at the current cutoff voltages,

Vc1,25Vs7jc1,2LV1

D

sinhD S 12dsD

1nwD2dsD D 21

.

~119!

Here one must take the limit (jc1,2V1)→0, where the valuesjc1,2 at cutoffs have been defined in Eqs.~77! and ~82!. Thetwo cutoff voltages are located above and below the synchro-nous voltage range. The lower cutoff voltageVc1 lies abovethe cutoff value obtained from the linearized characteristic~113!. There exists some parameter regime where there is noupper cutoff; the magnetron operation is extended toward theHall voltage, until loss of magnetic insulation and abruptmagnetron short-out occur.

VII. COMPARISON WITH EXPERIMENTS

The normalized equations in the previous section areconverted back to natural units for comparisons with experi-mental results. The characteristic equations~113! and ~116!remain the same after being converted in natural units, withV given in volts, currentI in amperes,R in ohms, andP inwatts. The radiation resistance is obtained in ohms from

Rs5S 931011

vpD 2 2p2V2v2 sinh2~kD!

ZhQ~kb!2vp4M2L2 sinh2~kds!

,

~120!



whereZ is in ohms, all dimensions are in cms,v52p f ,where f is the p-mode frequency in Hz, and thep-modewave numberk52p/2a, with a the vane period. All sub-scripted quantities (s) are computed at the BH voltageVs ,Eq. ~69!, given by

Vs5V0F2V

v S kD2kds2 D 1

n

2

vp2

v2 w~kD2kds!2G ,

~121!

wherevp5v/k is the phase velocity in cm/s, the synchro-nous kinetic energyV0 is given in volts by

V05mevp

2/2

1.6022310212 ergs/V, ~122!

and ds is

ds5vpV

vp2 S 12

v

V~krc1Vv/vp2! D . ~123!

Neglecting cylindrical effects inds , ~121! becomes

Vs5V0F2S VD

vp2

vp2

2V2D 1n

2

vp2

v2 wS kD2vp2

V2D 2G . ~124!

In the Brillouin density limit vp25V2, one obtains from

~124!,

Vs5V0F S 2 B

B021D1

n

2

V2

v2 w~kD21!2G , ~125!

whereB0[mecvp/eD G. The first bracketed term on theRHS is the familiar expression3 for Vs , while the secondcarries the spoke charge contribution.

The theoretical results from~113!–~117! are comparedwith theV2I curves for three tubes, Litton 4J50 atQ5208,Fig. 12~a!, and Litton 4J52 atQ5282, Fig. 12~b!. The ex-perimental points, taken, respectively, from Figs. 6.42 and6.44 in Ref. 2, are marked by X’s. The various solid curvescorrespond to different hub densities~normalized to the Bril-louin density!. The theory assumes constant hub densityalong each characteristic. Though a different density yieldsthe best match for each tube, the optimum densities generallyfall between 0.45nB and 0.55nB . This is taken to imply thatthe hub density is not close to the Brillouin density, a fre-quently made assumption, but, in fact, quite smaller. Thecomparison between theory and the experimentalP2Icurves is given in Figs. 13~a!, and 13~b! for the same tubes asin Figs. 12. Again, the experimental points are taken, respec-tively, from Figs. 6.27 and 6.28 in Ref. 2. The hub densityyielding the bestV2I fit also yields the bestP2I fit.

The fact that a particular density choice yields goodagreement between theoretical predictions and experimentalresults for various characteristic curves, strengthens the va-lidity of the present approach. It is further encouraging thatthe optimum density choices for three tubes of different de-signs fall close to half the~nominal! Brillouin density. Thatobservation agrees with numerical results from simulationsinvolving magnetron operation as an amplifier~CFA!, usingthe PIC codeMASK.23 A typical plot of the rationH/nB overthe interaction space of a crossed-field amplifier with distrib-uted emission cathode is shown in Fig. 14. That ratio stays

nearly fixed and close to 0.5, although the AC power in-creases more than ten times from input to output. Furthercomparisons with a larger variety of experimental data willdetermine whether the new equations can become a usefulanalytic tool, providing zeroth-order scaling laws for magne-tron design. The applicability of previously introduced ‘‘scal-ing laws’’ for magnetron operation is limited between similartube designs: one can use the characteristics of an existingtube to extrapolate the performance of a new tube belongingto the same ‘‘family.’’ In essence, one reproduces variationsof an existing design. The present approach allows perfor-mance evaluation from ‘‘scratch,’’ without any reference toexisting tube performance.

VIII. SUMMARY AND CONCLUSIONS

The GC fluid model treats the cathode electron layer~hub! as a ‘‘passive’’ source of the spoke charge. The hub

FIG. 12. TheoreticalV2I curves~solid! versus experimental data~X’s! for~a! Litton 4J50 atQ5208 and~b! Litton 4J52 atQ5282. Different curvesin each plot correspond to different choices of the hub densitynH/nB , asmarked.



density is prescribed and the hub height results from thedensity and the A–K voltage subject to zero cathode field~space-charge-limited emission!. Energy exchange with thecavity modes takes place through the GC motion in thespokes, while cyclotron rotation effects average out. It wasshown that the power transfer rate to the AC fields is givenby the product of the spoke current times the DC field inte-grated over the spoke height, (1/Q)(V1

2/2Z!5I 0*d

D2r dX E0 . Thus, the electronic efficiency5~power to

AC/I 0V0! is approximately equal to the ratio of the potentialdifference between the spoke top and spoke base, divided bythe total DC potential in the A–K gap,

h5V 0~D2r!2V 0~d!

V01dh. ~126!

Changes in the kinetic energy associated with the AC driftE13B0 yield the efficiency correctiondh. Finite Larmor ra-dius effects enter indirectly through the average distanceD2d2 r traveled by a GC before an electron hits the anode.Here r is the average GC distance from the anode at themoment of impact; because of the AC drift,r is smaller thanthe actual Larmor radiusr. In Ref. 6,r has been computed asa function of the AC amplitudeV1 and shown thatr/r→0 asV1→1. To the lowest order the efficiencyh is independentof the AC potential or the anode current, depending only onthe space charge distribution. Magnetrons and CFAs convertmainly potential energy into wave energy, in contrast withmost of the other microwave devices.

The synchronous~BH! voltageVs was redefined in Eqs.~124!–~125! by including the contribution from the spokecharge. It was proven that there exist a synchronous voltagerange of width,

UV0

Vs21U< V1

Vs

sinhD

D, ~127!

aroundVs . In that range the spoke width remains constantand equal to half-wavelength, despite the changes in the de-tails of the streamline topology. The spoke current in thatrange is proportional to the AC amplitudeI s5V1/L0, whereL0 given in Eq.~85! depends on the hub density and the hubheight. Combination of the power balance at steady state,

~1/Q!~V12/2Z!5hV0I 0 , ~128!

with the efficiency Eq.~126! and use ofI 05MIs for Mnumber of spokes, yields aI2V characteristic of the form

I 051

RV0 , ~129!

where the radiation resistanceR is given by

R5~L0 /M !2

2hQZ. ~130!

Notice thatR is proportional to the square of the impedanceL0/M from M spokes connected in parallel, divided by theinteraction impedanceZ. Spoke charge effects enter the ef-ficiency h, the BH voltage, and the effective AC fieldL. Itturns out that the effective spoke densityn can be computedfrom the profile of an unperturbed~zero self-field!, symmet-ric spoke in perfect synchronism. The spoke self-field param-etersL, C, andn are then obtained by expanding the per-turbed by the space charge flow around the symmetric spokeprofile in powers ofn/V1 . Here L and n are tabulated asfunctions of the cyclotron-to-RF frequencyV/v and theA–K gap D; the level curves corresponding tonH50.5nBare plotted in Figs. 10.

Expression~130! is a nonlinear Ohm’s law becauseRdepends, among other factors, on the A–K voltageV0 . In the

FIG. 13. The same as in Fig. 12 for the magnetron output power.

FIG. 14. Typical evolution of the hub density^nH&/nB around the interac-tion space during numerical PIC simulation of magnetron operating as anamplifier ~CFA!. The interaction length is measured in degreesu around thecircular cathode andnH& is the average density between the cathode andthe nominal Brillouin heightdB5eVDC/meV

2D2.



synchronous range the dependence ofR onV0 is weak, com-ing from the dependence of the hub heightd on the A–Kvoltage, Eq.~62!, and the dependence ofh, L, andn on thehub height. Expansion of~130! aroundVs , by expandingL0

around the synchronous hub heightds51/vD , results in the‘‘linearized’’ characteristics~113! and~116! and the dynamicimpedanceR from the slope of theV2I curve.

It is interesting, at this point, to compare the form of thenonlinear magnetron characteristics with thelow gain char-acteristics for crossed-field devices. In recent experiments,7,8

with a sheet beam CFA, the gain versus voltage was foundnearlysymmetricwith respect to the voltage~frequency! de-tuning from synchronism, with a maximum gain at synchro-nism. That is, a fundamental difference between crossed-fielddevices and the rest of the ‘‘unbound electron’’ microwavedevices that are characterized by anantisymmetricsmall sig-nal gain versus detuning curve. The present treatmentshowed that, under fixed AC amplitude and hub height, thespoke current is symmetric, relative to voltage detuning, Eqs.~54!–~56!. In low gain amplifier situations,DP!Pin , wherePin is the input power, one may take the AC amplitude ap-proximately constant, equal to the input value, and indepen-dent of the applied detuning. However, at high-power, highgain magnetron operation, a change in the applied voltagehas a significant effect in the AC power; one cannot changethe detuning without changing the AC amplitudeV1 . Thechange in V1 acts back to change the spoke currentI 0}V1/V, despite the constant spoke width. Magnetron gainis not symmetric around the synchronous~BH! voltage, but amonotonically increasing function of the applied voltage.Yet, a leftover of the low-gain symmetry remains: the currentcutoff voltages are located around the synchronous voltage.The saturation of the spoke width manifests itself in the flatshape of the characteristic curves near synchronism.

Concerning the applicability to relativistic magnetronoperation, the present treatment is correct to ordervp/c andneeds modifications when (vp/c)

2 effects are important. No-tice that the form of the GC equations~10!–~14! remainsinvariant: theE3B drift does not depend on relativistic masseffects @the cyclotron frequency appearing in~10!–~14! re-sulted from normalizing theE/B ratio using the rest mass#.What is affected is the magnetic field strength in the synchro-nous frame,B0→B0[12(vp/c)

2]. In addition, the magneticfield of the slow waveB1;(vp/c)

2E1 cannot be neglectedfor mildly relativistic phase velocities. It is worth observingthat there is no relativistic bunching over the cyclotron angle:since no cyclotron resonance is involved the cyclotron rota-tion energy,g' is invariant in the synchronous frame.

The discussion is now turned to the limitations of thepresent model. First, the hub density has been treated as con-stant along theV2I curve, with a prescribed value, to beconfirmed from the agreement with experiment. It has alsobeen assumed that the DC field is zero at the cathode. Thetheoretical evaluation ofnH requires the introduction of abalance equation between current drawn at the spoke baseand current emitted by the cathode. Secondary emission de-pends on the impact energy, which, in turn, depends on theDC field at the cathode. The impact energy and the currentbalance at the hub provide the two additional equations for

the self-consistent determination of the unknownsnH andEc . The task will be taken up in future work. Numericalsimulations indicate thatEc at the cathode is small, but notnegligible. A finiteEc increases the synchronous voltageVs

under given hub heightd, causing an upward shift of theV2I curves without affecting the slopedV/dI. It is intrigu-ing to observe that in Figs. 1 and 2 the curves correspondingto lower hub density yield better slope agreement with theexperiment, while the voltage falls short. It is plausible that atheory with finiteEc and smaller hub density yields betteroverall agreement with the experiment.

The second limitation concerns the effective-field ap-proximations. Though the choice ofL and C according to~30!–~31! minimizes the error between the actual potential~27! and its approximation by~28!, there is no guarantee thatthe error is small enough. Our experience is that theeffective-field solution tends to underestimate the space-charge potential and the error increases with increasing ratioof self-field to vacuum field. At high AC power, as in thecases examined, the self-fields are sufficiently small for thevalidity of the approximation. In cases involvinglow ACpower and relatively high self-fields~the self-fields dependonlyon the spoke charge, which is practically independent ofthe AC power! a better formula may be needed for theeffective-field representation.

A third limitation concerns the impact of cylindrical ef-fects. Presently, the effect of curvature was included only inthe evaluation of the synchronous velocity at the hub top, Eq.~67!. In general, the cylindrical corrections to the profiles ofthe AC and DC potentials, as well as the GC orbits, scale asthe inverse aspect ratioD/Ra , whereD5r a2r c is the A–Kgap andr a , r c the anode and cathode radii. For the tubesunder consideration the inverse aspect ratio is of the order of10%. Hence, a model based on rectangular geometry is notexpected to predict the experimental results within betterthan 10% accuracy.

Finally, the wave–particle energy exchange in the spacebetweenthe vanes has been neglected, by assuming particleabsorption at the vane-tip heightD. It has been argued24 that,in certain cases, the electron interactions with the AC fieldsbetween the vanes~‘‘slots’’ ! may have a measurable effecton the magnetron gain and the energy dissipation at the an-ode.

ACKNOWLEDGMENTS

The author wants to thank David Chernin for many use-ful and stimulating discussions.

This work is supported by U.S. NRL Contract No.N00014-92-C-2030.

APPENDIX A: EFFECT OF CYCLOTRON CURRENT

The interaction near the pure drift resonancev2ku50affects the GC motion, but not the cyclotron rotation. Onlythe part of the distribution involving the GC coordinates isrearranged by the RF interaction; the part containing the dis-tribution over the cyclotron coordinates still evolves alongthe unperturbed cyclotron orbits. It is shown here that when



the electrons are initially uniformly distributed around eachGC location, the energy exchange between the wave and thecyclotron part of the electron motion is zero.

The total electron velocity isv5U1vc , whereU givenin Eq. ~14! includes both the DC and the AC drifts, andvc isthe gyration~cyclotron! velocity. Note that sinceU dependsonly on the local fields at the guiding center location, anythermal spreads appear in the cyclotron velocity. The phasespace can be completely specified by the guiding center co-ordinates (X,Y), the cyclotron velocityvc[uvcu, and the gy-roangleu, in place of the usual coordinates and velocitiesx,y, vx , vy . The electron distribution in phase spaceF(x,y,vx ,vy) is thus expressed as

F~X,Y,vc ,u!5n~X,Y! f ~vc ,u!. ~A1!

The GC velocityU does not enter the distribution as anindependent variable, since it is completely specified by theparticle’s GC position. The two expressions forJ–E1,

n~x,y!E2`

`

dvxE2`

`

dvy f ~vx ,vy!v–E~x,y,t ! ~A2!

and

n~X,Y!E0

`

dvc vcE0

2p

du f ~vc ,u!~U1vc!–E~x,y,t !, ~A3!

are fully equivalent. For a uniform distribution around eachGC,

f ~vc ,u!51

2pf 0~vc!, ~A4!

the cyclotron current contribution to the energy exchange is

Jc•ERF5n~X,Y!E0

`

dvc vcE0

2p

du1

2pf 0~vc!vc–E~x,y,t !.

~A5!

One could now expand x5X1(vc/V)sinu y5Y1(vc/V)cosu in the argument of the electric field, and takethe u average term by term. A direct and more elegant ap-proach is to notice that

E0

2p

du vc5V R dl, ~A6!

wherer is the circumference of a circle of radiusr5vc/V.Then, according to Stoke’s identity,

E0

2p

du vc–E5V R dl–E5VE ~“3E!–ds, ~A7!

where the surface integral is taken over the area of a circle ofradiusr. The last term is zero for the electrostatic approxi-mation“3E50, thus for uniformu distribution,

Jc–E50. ~A8!

This result holds for any time or space dependence ofE. Inparticular, it is true for any wavelength-to-gyroradius ratioL/r. A uniform distribution in u is the statistically mostprobable distribution, in the absence of cyclotron resonancesand relativistic bunching effects.

APPENDIX B: NONLINEAR MODE STRUCTURE ANDHIGHER HARMONICS

The solutions of the space charge potential have beenobtained by imposing a monochromatic boundary conditionthe AC potential at the anode. Yet, higher harmonics areintroduced in the mode structure from two effects.~a! Theperiodic anode geometry introduces amplitude modulationon the vacuum mode structure.~b! The bunched spoke dis-tribution produces higher harmonics. In what follows we of-fer a justification for maintaining the monochromatic waveboundary condition at the vane tips during nonlinear opera-tion.

Because of the periodic anode geometry, the vacuum po-tential solutions have the form of Bloch waves,

V ~x,y,t !52V1S sinh~kx!

sinh~kD!sin~ky2vt !

1(lÞ0

sinh~klx!

sinh~klD !sin@kly2vt# D . ~B1!

Thus, single-frequency eigenmodes have a wide wave num-ber content, wherekl5k1 l2p/a carries the perioda of theanode vanes. The phase velocitiesv/kl are far belowv/k,given thatk<kl . Since theE03B0 velocity is near resonancewith v/k, the lÞ0 terms are out of synchronism and do notcontribute to the AC energy exchange with the electrons.Thus, harmonics do not enter the right-hand sideJ–E1 of theenergy balance equation~37!, and are neglected from themode structure used to obtain the GC orbits and the spoketopology. The full harmonic effects are retained on the left-hand side of~37! through the interaction impedanceZ thatrelates the total power flow in the cavity with the AC ampli-tude of the fundamentalV1 at the anode. The energy ex-tracted from the synchronous interaction with the fundamen-tal goes into building all components of the Bloch wave inEq. ~B1!. Harmonic effects are thus included in the powerflow and the AC dispersion relation without complicating theGC orbit calculations.

The Fourier transform of the bunched space charge inthe spoke produces AC harmonics. These become thesources of harmonics in the AC potential of the ‘‘loadedcavity’’ modes. In the synchronous frame, where the spacecharge distribution is ‘‘frozen,’’ these harmonics are solu-tions of the ES Poisson’s equation~17!. The formal solutionis written as

V ~X,Y!52V1 sinhX sin Y1dV 1 sin Y1dV 21 cosY

1(l>2

@dV l sin~ lY!1dV 2 l cos~ lY!#, ~B2!

where

dV 6 l~X!54pE0

D

dX8 n6 l~X8!Gl~X,X8!. ~B3!

In the lab frameY5y2vpt, the AC potential~B2! becomes~going to natural units!,



V ~x,y,t !52V1 sinh~kx!sin~ky2vt !1dV 1

3sin~ky2vt !1dV 21 cos~ky2vt !

1(l>1

@dV l sin~ lky2 lvt !1dV 2 l

3cos~ lky2 lvt !#. ~B4!

The AC potential~B4! is of the form V (x,y2vpt). Thisfollows from the electrostatic approximation: a spoke travel-ing at the synchronous velocityvp!c drags its electrostaticfield with it. All nonlinear harmonics (lk,lv) propagate atthe same phase velocity with the fundamental, and representa nonlinear waveform steepening in the A–K space. It willbe argued now that the anode circuit does not support har-monics. Hence, the amplitudedV l of a harmonic, which canbe large in the A–K space, drops to zero at the vane tips,dV l(D)50.

First, one must say that the dispersion relation is deter-mined by the anode structure. It is little affected by the fieldsin the A–K space since most of the circulating AC energy isstored in the anode structure~slots!; the AC potential in theA–K space is, in effect, the fringe field between the vane tipsand the cathode. The intersections of the linev5kvp withthe various Brillouin zones of the dispersion relation are notat equal distances, thus only one of the frequency–wave

number pairs (lk,lv) can satisfy the dispersion relation. Ifthe fundamental (k,v) propagates in the anode circuit, itsharmonics do not and their amplitude must go to zero at thevane tips. This is enforced by constructing the Green’s func-tion with the propertyGl(0,X8)5Gl(D,X8)50,

Gl~X,X8!5sinh~ lX,!sinh~ lD2 lX.!

sinh~ lD !, ~B5!

where X,[min(X,X8), X.[max(X,X8). From ~B3! and~B5!, it follows thatdV l(0)5dV 2 l(D)50. The monochro-matic boundary condition at the anode is thus justified, evenin the presence of space charge. The harmonic amplitude issignificant inside the A–K gap, scaling asdV l.dV 1/l

2. It isworth noting that the nonlinear harmonics contribute in thewave–particle energy exchange, since they are also synchro-nous. Their contribution to the power gain has been ne-glected for simplicity.

APPENDIX C: EFFECTIVE FIELD PARAMETERS

It has been shown~Ref. 6, Appendix D! that the mini-mization of Eq. ~29! in respect toC and L leads to thefollowing solutions:

tan C5S ^x21 sinh2 X1x218 cosh2 X&

^~11x1!sinh2 X1~11x18!cosh2 X& D , ~C1!

L5@^x21 sinh

2 X1x218 cosh2 X&21^~11x1!sinh2 X1~11x18!cosh2 X&2#1/2

^sinh2 X1cosh2 X&, ~C2!

where (8)[d/dX, ^•••& stands for*dD dX,...., and thenonlinear susceptibility elementsx61 have been defined in terms of the

space-charge self-fieldsdV 61,

x61~X!52dV 61~X!

V1 sinhX/sinhD, x618 ~X!52

~d/dX!dV 61~X!

V1 coshX/sinhD~C3!

~the notationx is used here in place ofe in Ref. 6!. Substituting~C3! inside ~C1!–~C2! and rearranging terms yields

tan C52^dV 21 sinhX1dV 218 coshX&

V1^sinh2 X1cosh2 X&/sinhD2^dV 1 sinhX1dV 18 coshX&

, ~C4!

L5F S 12^dV 21 sinhX1dV 218 coshX&V1^sinh

2 X1cosh2 X&/sinhD D 21S ^dV 1 sinhX1dV 18 coshX&V1^sinh

2 X1cosh2 X&/sinhD D 2G1/2. ~C5!

UsingdV 61 sinhX1 dV 618 coshX5 @dV 61 coshX#8, and performing theX integration implied by•••& yields

tan C52dV 21 coshXud

D

AV12dV 1 coshXudD , ~C6!

L5F S 12dV 21 coshXud

D

V1AD 21S dV 1 coshXud

D

V1AD 2G1/2, ~C7!

where A5*dD dX~sinh2 X1cosh2 X!/sinhD5@sinh(2D)

2sinh(2d)#/2 sinhD coshd. Inserting the boundary condi-tion dV (X)6150 atX5D in ~C6!–~C7! yields Eqs.~30!–~31!, respectively.

APPENDIX D: HIGHER-ORDER EFFECTS IN GCMOTION

The GC equations in the synchronous frame of referenceare



dX

dt5

1

V

]H

]Y,

dY

dt52

1

V

]H

]X. ~D1!

The GC Hamiltonian, obtained by time averaging over thefast ~cyclotron! time scale,22 is ordered in powers of the RFamplitude as

H5H11H21••• . ~D2!

The first-order Hamiltonian, used in Eq.~20!,

H1~X,Y!5LV1

sinhX

sinhDsin~Y1C!2@ V 0~X!1EsX#,

~D3!

does not include the kinetic energy of the guiding center. Thefirst-order GC motion according to~D1! and ~D3! followsthe equipotentials of the combined DC and AC potentialviewed in the synchronous frame.

The GC kinetic energy contribution, being of second or-der inV1 , is recovered by going to the second-order averag-ing,

H2~X,Y!5L2V1

2

2V2 @sinh2 X1sin2~Y1C!#

2LV1

2V2

dE0dX

sinhX sin~Y1C! ~D4!

~the electric shear above the hubdE0/dX is also treated as asmall quantity, of orderV1!. In the limit dE0/dX50, H2 isnothing, but the kinetic energy associated with thefirst-orderdrift, H25u1

2/2, where, from ~14!, u15V1/V~sinhX3cosYx2coshX sinYy!. While first-order motion H15const corresponds to constant potential energy, the second-order motionH11H25const corresponds to constant total~potential plus GC kinetic! energy, thus the GC orbits deviatefrom the equipotentials whenV1 is significant.

The combined GC velocity from~D1!–~D3! is given by

U5u01u11u2 , ~D5!

whereu0 andu1 are, respectively, the DC and AC drift andhave been computed earlier, Eq.~14!. The second-order de-parture from the parapotential flowu2 is given by

Y251

V

]H2

]X5

L2V12

V3 coshX sinhX

2LV1

V3

dE0dX

coshX sin~Y1C!, ~D6a!

X2521

V

]H2

]Y

52L2V1

2

V3 sin~Y1C!cos~Y1C!

1LV1

V3

dE0dX

sinhX cos~Y1C!, ~D6b!

and can be explained as follows. A particle drifting along theequipotential surfaces experiences an accelerationg due to

the curvature of the field lines. The accelerationg is found bytaking the time derivative ofu01u1, using Eqs.~10!,

g5F V12

V2 coshX sinhXx,2S V12

V2 sin Y cosY

1LV1

V2

dE0dX

sinhX cos~Y1C! D yG . ~D7!

The applied DC magnetic field then imposes an additionalcurvature driftu252~g3z!/V, which is just Eqs.~D6!.

When the second-order driftu2 is added to the spokecurrent, Eq.~33!, and then applied toJ–E1 on the RHS of thepower balance equation~37!, it can be shown9 that

MbE0

DdxE

2p

p

dy J–E1

5I 0S 2Ed

DdX E02

1

2û0

2~D !2u02~d!&

21

2û1

2~D !2u12~d!& D . ~D8!

The power transfered to AC equals the current~i.e., particlesper unit time! times the sum of the changes in the DC po-tential energy~E0 includes the spoke charge effects! and theaverage DC and AC drift kinetic energies per particle, be-tween the hub topd and the effective anode heightD. Thepotential energy decreases toward the anode while the GCdrift velocity generally increases, because the DC and ACfields get stronger toward the anode. Hence, a small fractionof the potential energy extracted from the GC motion acrossthe A–K voltage is converted to kinetic energy, while mostof it is transferred to the wave energy.

By definition, the efficiency is the left-hand side of~D8!divided by the power supplied by the DC sourceI 0V0 , yield-ing

h5V 0~D !2V 0~d!

V02

û02~D !&2û0

2~d!&2V0

2û1

2~D !&2û12~d!&

2V0. ~D9!

The first term on the RHS is evaluated combining Eq.~62!for V 0(X), the boundary conditionV 0(D)5V0 , and Eq.~65! for E0(X) to expressE0(d) in terms of the A–K voltageV0 , resulting in

V 0~D !2V 0~d!

V05S 12

d1r

D D 12vp2d2/2V0

12d/D1n

vp2

4V0

3DrF S 12d1r

D D S 12w

3

r

D2dD1r

DG .~D10!

The change in the DC drift energy above the hub is the samefor all electrons, thus the second term is found by using Eq.~14! for u0 and Eq.~62! for E0,



u02~D !2u0

2~d!

2V05n

vp2

VV0~D2d!S 12

w

2 D1n2

vp4

4V2V0~D2d!2S 12

w

2 D 2.~D11!

The change in the DC drift energy is due to the spoke chargeand goes to zero forn50. We approximate the averagechange in the AC drift kinetic energy with that for a particlestarting at the spoke centerX5d, Y50. The particle locationat impact isX5D.D, Y5f, wheref is the shift from theequipotential surface caused by the curvature drift. From thesecond-order equation of motionH11H25const, one finds,using ~D1!–~D3!, that

LV1

sinhX

sinhDsin f5

L2V12

2V2 ~sinh2 X2sin2 f!

2LV1

2V2 sinh2 d, ~D12!

where the sheardE0/dX above the hub is neglected for sim-plicity. The change in the AC drift kinetic on the right-handside equals the change in the AC potential energy on the left.Equation~D12! is solved as a quadratic equation for sinf,

sin2 f22V2 sinhD

LV1sin f1~sinh2 D2sinh2 d!50.

~D13!

For real solutions the discriminant must be positive, imply-ing the condition V1<V2 sinh2D/LAsinh2D2sinh2 d.When the AC amplitude is large enough to violate this con-dition, the curvature drift and the deviation from the equipo-tentials are so severe that a particle starting at the center ofthe spoke never reaches the anode: current cutoff can alsoresult from very large AC field. Under usual circumstancesV1;1, one finds

sin f.LV1

2V2

sinh2 D2sinh2 d

sinh2 D. ~D14!

Substituting on the LHS of~D12!, which is the change in theAC drift energy, one obtains

^u12~D !&2^u1

2~d!&2V0

5L2V1

2

2V2V0

sinh2 D2sinh2 d

sinh2 D. ~D15!

The total efficiency~D9! is the sum of~D10! minus ~D11!minus ~D15!.

APPENDIX E: LINEARIZED COEFFICIENTS

Within the synchronous operation regime one can ex-pand aroundV05Vs . The termsh, n, andL depend onV0both directly and indirectly throughd(V0). Thus, in general,

d

dV05

]

]V01

dd

dV0

]

]d, ~E1!

where from~62!,

dd

dV05S vp

2@D2ds#2n

2wvp

2@D2ds# D 21

. ~E2!

From the definition~114! follows that R depends onV0throughd. Thusb, also defined in~114!, is given by

b5F 1L2 S ]L2

]d Ds

11

h S ]h

]d Ds

12 coth dsG3

V~D2ds/2!1~n/4!wvp2~D2ds!

2

vp2@D2ds#2~n/2!wvp

2@D2ds#. ~E3!

The efficiencyh depends onV0 both explicitly and implic-itly. Applying ~E1! to Eqs.~103!–~105! yields

S ]h

]d Ds

5H 21/D1~V/2Vs!@12~r12ds!/D#

12ds /D112~r1ds!/D

D

12V/2Vs

~12ds /D !21

nvp2

4VsrDSw3 r2

~D2ds!22

1

D D11

n S ]n

]dDs

nvp2

4VsrDF S 12

ds1r

D D S 12w

3

r

D2dD 1r

DG J S ]d

]V0Ds

11

VsH 12~r1ds!/D

~12ds /D !2Vds2Vs

2nvp

2

4VsrD

3F S 12ds1r

D D S 12w

3

r

D2dD 1r

DG J . ~E4!

The derivative of the effective spoke density is found from Eq.~95!,

S ]n

]dDs

5F 2

D2ds1coth dsGn2

4~D2ds!

wp~D2ds!2 . ~E5!

Finally, from Eq.~31!,

S ]L2

]d Ds

52H 2S 12dV 1

AV1 D 1

AV1 S ]dV 1

]d Ds

1]V 21

AV1

1

AV1 S ]dV 21

]d Ds

1F2S 12dV 1

AV1 D dV 1

AV11S dV 21

AV1 D 2G S 21

A

]A

]d Ds

J ,~E6!



where the derivatives ofV 61 follow from ~96!–~97!,

S ]

]ddV 1D

s

52n

V1

2vp2

p Eds

D

dXsinh~D2X!

sinhX S ]

]ddYD1coth ds dV 1 , ~E7!

S ]

]ddV 21D

s

522vp

2

p

sinh~D2ds!

sinhD1coth ds dV 21 , ~E8!

and

]A

]d52

1

Atanhd1

coshd

sinhD. ~E9!

In the limit A@1, ~E6!–~E8! combine into

S ]L2

]d Ds

52F2dV 1

AV1S tanh ds2 1

A

coshdssinhD D 2

dV 1

AV1S coth ds1 *ds

D dX@sinh~D2X!/sinhX#@~]/]d!dY#

*dsD dX@sinh~D2X!/sinhX# D G . ~E10!

1A. M. Clogston, inMicrowave Magnetrons, edited by G. B. Collins~McGraw-Hill, New York, 1948!, pp 401–459.

2J. F. Hull, Doctoral dissertation, Department of Electrical Engineering,Polytechnic Institute of Brooklyn, 1958.

3J. F. Hull, inCross Field Microwave Devices, edited by E. Okress~Aca-demic, New York, 1961!, pp. 496–527.

4S. Riyopoulos, Phys. Fluids B3, 3505~1991!.5S. Riyopoulos, D. Chernin, and A. Drobot, IEEE Trans. Electron. DevicesED-39, 1529~1992!.

6S. Riyopoulos, Phys. Rev. E47, 2839~1993!.7J. Browning, C. Chan, J. Z. Ye, and T. E. Ruden, IEEE Trans. Plasma Sci.PS-19, 598 ~1991!.

8R. McGregor, C. Chan, J. Z. Ye, and T. E. Ruden, IEEE Trans. Electron.DevicesED-41, 1456~1994!.

9S. Riyopoulos, IEEE Trans. Plasma Sci.PS-22, 626 ~1994!.10S. Riyopoulos, IEEE J. Quantum Electron.PS-31, 1579~1995!.11O. Buneman, in Ref. 3, p. 367.12O. Buneman, R. H. Levy, and L. M. Linson, J. Appl. Phys.37, 3203

~1966!.13R. C. Davidson, K. T. Tsang, and J. A. Swegle, Phys. Fluids27, 2332

~1984!.14R. C. Davidson and K. T. Tsang, Phys. Fluids28, 1169~1985!.15S. Riyopoulos, Phys. Plasmas2, 935 ~1995!.

16L. Brillouin, Adv. Electron.3, 85 ~1951!.17E. Ott and R. V. Lovelace, Appl. Phys. Lett.27, 378 ~1975!.18J. C. Slater, inMicrowave Electronics~Van Nostrand, New York, 1950!,pp. 333–345.

19A. Ron, A. A. Mondelli, and N. Rostoker, IEEE Trans. Plasma Sci.PS-1,85 ~1973!.

20One may compare Poisson’s equation with Schro¨dinger’s equation for aparticle-in-a-box2(\2/2m)¹2V 5EnV . The set of solutionsV n corre-sponding to all non-negative energiesEn does indeed form a complete set.However, Poisson’s equation¹2V 50 represents only thezero energyso-lutionsE50, thus the vacuum potential solutions are not a complete setfor other equations in general. To includeEnÞ0, one must extend Pois-son’s equation to2¹2V 5kn

2V , kn25(2mEn/\

2).0; the resulting waveequation does possess a complete set of solutions!

21In earlier literature, such as Refs. 2–3, the circulating power is expressedusing the peak-to-peak AC voltageVpp52V1 inside ~98!. Hence, the cir-cuit impedanceALc /Cc reported in Refs. 2–3 is four times larger than theimpedance associated with the power definition~98! based on the ACamplitudeV1 .

22S. Riyopoulos, J. Plasma Phys.46, 473 ~1991!.23A. Palevsky, Ph.D. thesis, Massachusetts Institute of Technology, 1980.24H. McDowell ~private communication, 1994!.



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