Magnetron Mode Transitions

Magnetron Mode Transitions E. C. OKRESS Westinghouse Electric Corporation, Elmira, New York ....................................................

....................................................

11. Instability-Voltage Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Mode-Competition Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Mode-Stability Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Assessment and Ramifications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . American Magnetron Bibliography-(since 1946) . . . . . . . . . . . . . . . . . . . . . Foreign Magnetron Bibliography-(since 1946). . . . . . . . . . . . . . . . . . . . . . .

Page 503 503 513 517 524 531 533 533 536

This chapter deals primarily with the stability of magnetrons in the steady and transient states. Though the treatment relates primarily to the Randall-Sayers (i.e., strapped and symmetrical) type of magnetron, the results are by no means so restricted. The magnetron parameters effecting stability of operation are evaluated in the light of modern knowledge. The development and present status of mode stability theories are critically reviewed and their applicability evaluated. Factors associated with transient loading, frequency stability, magnetic field anomalousness as well as structural aspects associated with the resonant system and interaction space are considered. This chapter begins with a brief review of the pertinent results, of trials and tribulations, leading to our present knowledge of the magnetron. Evaluation of the principal theories of mode behavior are next considered. Criteria are discussed which evolved from these theories, including that of Willshaw-Copley, which was largely superseded by that of Rieke and the latter further developed by Moats. The ramifications and applicability of these criteria to design and performance of magnetrons is assessed.

I. INTRODUCTION The pulsed symmetrical Randall et al. (1) magnetron was strapped by Sayers et al. ( 2 ) in an effort to improve the efficiency and stability of op503

504

E. C. OKRESS

eration in the ?r-mode (S).* Although the subtle effects of this invention were not initially appreciated, it later was apparent that the original (light) strapping (4, 5 ) substantially increased the couplingt between the resonators. Not only did this remove the degeneracy of all$ lower order or doublet modes, but it significantly reduced the contamination of the T-mode field due t o the lower order mode fields in the interaction space of the symmetrical magnetron. This improvement was primarily due to the increase7 of the frequency separations, compared5 with the original unstrapped prototype, and the threshold voltage separations between the lower order modes and the ?r-mode, for given anode and end-cavity lengths (5). Strapping the resonant symmetrical magnetron also increased the characteristic admittance of the original resonators, which resulted in higher electronic efficiency. The cumulative effect of strapping the magnetron was better control of mode selection, and hence a more efficient and stable tube evolved having higher input power capacity and a higher frequency limit than its unstrapped predecessor. Although straps affect the loading (7) and distort the fields of the lower order modes, these aspects were apparently neither recognized nor controlled. Surprisingly, no detrimental effects resulted as a consequence of these factors. I n the light of difficulties encountered in attempting to improve substantially the power capacity and frequency range of the originally strapped symmetrical magnetron, it is indeed remarkable that its originally conceived cathode-to-anode ratio, type of straps, segment tip shape, type of resonators, etc., functioned so effectively together. Soon after straps were introduced, it seemed desirable that the adjacent lower order mode and its associated low-order Hartree harmonics be made more difficult to excite and afford a larger unobstructed range of operation for the ?r-mode. This was done b y increasing the starting voltage of the adjacent lower order modes over that of the ?r-mode as much as possible. As a consequence, a trend of increasingly tighter strapping was started. Tighter strapping also provided a means t o compensate for the adverse effects of larger numbers of resonators. This was a consequence of increased cathode and anode sizes (for greater dissipation and

* Alternates are principal highest order or singlet mode. The singlet 0-mode is of minor consequence in symmetrical compared with asymmetrical magnetrons (for example rising-sun structure). t Coupling through the magnetron end spaces and interaction spaces are not considered. $ Some modes may already be split (6) by the coupled load to one cavity, the degree of which depends upon loading (7). 7 Amounting to 100/loaded Q or of the order of 7%. 8 Amounting to 100/loaded Q or of the order of I %.

MAGNETRON MODE TRANSITIONS

505

higher power, respectively) while maintaining the applied magnetic field within practical levels a t high frequencies. It was soon realized, however, th at such continued extrapolation resulted in designs which failed (8-10) to function as anticipated, owing to limitations other than insulation, emission, etc. Consequently, designs for the highest and lowest frequencies and highest power capacity were limited t o primitive and costly incremental extrapolation techniques. Eventually, considerable understanding was acquired in compromising such factors as resonator admittance (governing the a-mode electronic efficiency) with the ratio of cathode to anode diameter and their individual values which govern a-mode starting characteristics for a given magnetic field and anode voltage. This was especially difficult to achieve at long wavelength and a t high power with good transient and steadystate characteristics. The result of these investigations and the performance of new designs provided important clues for subsequent development of mode change criteria. For example, mode transitions were observed t o create substantial changes in operating parameters as the applied voltage was gradually increased, after the a-mode was satisfactorily initiated. Occasionally, it was not even possible to start in the a-mode. The salient features of these and like phenomena provided the means whereby the transient state involving mode selection, and the steady* state involving mode stability, were eventually resolved into two distinct aspects of magnetron operation. The importance of the rate of rise of the applied voltage with respect to the build-up time of the a-mode, and the influence of modulator characteristics were finally resolved (11). It was then evident that if the open circuit voltage of the modulator were not restrained at the starting voltage of the a-mode, or its rate of rise were not favorable with respect to the rate of build-up of the a-mode, mode skipping or misfiring could result. Under these conditions the voltage could pass through the *-mode starting range too fast and reach the starting range of the adjacent lower order mode or its component. Subsequently, strap breaks were introduced for loading (12) and distortion of the adjacent lower order doublet mode, reducing their adverse effects on the *-mode, and thereby preventing interference with the starting and operation in the *-mode. The limitation of straps and strap breaks soon became apparent necessitating better alternatives. Although these alternatives provided improvements in specific cases, they possessed limitations which hindered their general application. Straps and strap breaks have therefore not disappeared. Increased knowledge ( I S ) of these devices has enabled much more effective utilization of their inherent prop-

* Time long compared

with the period of electromagnetic field of the r-mode.

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E. C. OKRESS

erties toward improving the extent of a-mode operation. I n an attempt to eliminate strap breaks, it has been found th at loading of the lower order mode doublets may be achieved (1.4) by introducing two identical pronounced asymmetries in the strap system. * These asymmetries are two localized points of .strap capacitances, strap inductances, or cavity inductances. One point of introduction is an arbitrary angle with respect to the coupled load, and the other point is a radians from the first. This scheme has met varying degrees of success in practice. The second endeavor t o eliminate strapping evolved into two primary methods; the shaped anode segment tips (15) and the asymmetrical resonator system (16) (for example, rising-sun, porcupine, etc., structures). The shaped segment tips arose from the plausible assumption th a t the effect of the Hartree harmonics in the interaction space are detrimental to the a-mode. Actually, subsequent performances have shown these detrimental effects to be minor. I n any event, it was not possible t o retain the usual rectangular segment tip shape and expect to eliminate all the Hartree harmonics. However, by shaping the segment tips of the resonators of a n unstrapped symmetrical magnetron, according t o a predetermined curve (perpendicular to the lines of the field of the a-mode), only the fundamental space component should be retained with no Hartree harmonics. It was apparent that such shaping of the segment tips produced only moderate improvement in electronic efficiency and substantial improvement in frequency pushing over but a limited region of the performance chart. I n a n attempt to explain the mode transitions and provide a guide for design of magnetrons for the highest powers and widest range of frequencies, mode change criteria were developed. The most important of these were Rieke’s ( I S ) and Willshaw-Copley’s (9). Recently a more plausible mode change criterion than the Willshaw-Copley was formulated by Moats (17) on the basis of Rieke’s (13) original hypothesis and Van der Pol’s (18) nonlinear theory. While the Willshaw-Copley criterion asserts that conditions associated with the competing (13) mode are the sole controlling factors responsible for mode change, the Rieke and Moats criteria maintain that after a strong a-mode is established, it is very difficult for another mode t o build up in its presence, The latter indicates th a t in order for a mode change to occur, the a-mode must either be weak to begin with or must decay of its own accord independently of any influence of other modes. Furthermore, it stipulates conditions necessary for stable a-mode operation and maintains, within the confines of this hypothesis, that apparently no inherent limitation exists u p t o cut-off conditions be-

* Neglecting asymmetry due to the coupled load unless this is excessively reactive (1%

MAGNETRON MODE TRANSITIONS

507

cause of mode stability alone. This assertion has been plausibly demonstrated (17). Yet, it apparently does not rule out other a-mode limitations (19) arising from space charge, transit time, induced current, pulser regulation, debunching (17, do), thermionic (%'I), and enhanced (22) emission. I n relating cathode emission to a-mode current limitation or boundary, i t is necessary to take due cognizance of the fact th a t a change in this boundary with cathode temperature may be due to changes in leakage (23) current. It has been reported (24) that the maximum ?r-mode current boundary (1) attains an optimum as a function of cathode temperature and (2) decreases beyond the optimum as the cathode temperature increases. According to Wilbur et al. (25) item (1) is essential for 1ow-Q operation. However, (2) is open to question, for as Moats suggests, the decrease may be due to changes of alignment of the cathode (resulting from thermal expansion or distortion of the cathode supports) rather than t o emission changes of the cathode. I n contrast t o the Rieke and Moats criteria, the Willshaw-Copley criterion maintains that the governing factor for a mode transition is inherent in the mode which builds up. As will be shown, this mode may suppress the established mode only if the latter is weak to begin with or becomes unstable of its own accord, and not because its instability and threshold voltages have been reached. Weak or strong denotes the relative strength of a mode's ability to persist either against possible competition from another mode with more suitably imposed boundary conditions or against debunching or loading effects. More recent attempts t o establish a better mode change criterion than those cited are wanting in several respects and so cannot be considered as contenders t o the Rieke and Moats criteria. Efforts t o improve the inherent frequency stability of the a-mode in symmetrical magnetrons by investigating subtle factors affecting the frequency stability are not as extensive as those relating to frequency pushing and voltage tuning (26). Nevertheless, investigations for improving these characteristics for specific purposes have yielded information of considerable value relating to inherent frequency stability. One source of instability is characterized by an abrupt minute frequency change at a particular anode current, magnetic field, and loading. Present conjecture, based on limited experimental results, indicates th a t this gauss-line discontinuity may be associated with an instability in the space charge, the precise nature of which is not evident on the basis of the available data (27). A type of transient instability (28) [other than those resulting from the usual resonant line effects (29)l recently resolved is th a t associated with loading of the ?r-mode during its build-up period. The adverse reflections which represent heavy loading of the a-mode are due t o the improper

508

E. C. OKRESS

position of the duplexer (ATR box). I n the course of the build-up of the r-mode, while the duplexer is deionized, the duplexer may create a reflection of such phase and amplitude a t the magnetron so as t o represent a considerable mismatch, in the form of a large reactance in series with the line. Extensive investigation of these phenomena revealed th a t such adverse loading of the magnetron during the starting period can be eliminated by properly locating the duplexer a t an optimum position in the line near the magnetron (28). 11. INSTABILITY-VOLTAGE CRITERION Willshaw-Copley (9) formulated a semi-empirical mode-change criterion based on Bunemann’s (30) small-amplitude hypothesis assuming the Brillouin-Bloch (31) single-stream state. This criterion considers exclusively the influence of another mode on the established mode of the symmetrical magnetron. Twiss (SW),however, has shown that the existence of the single-stream state is impossible in the presence of Maxwellian velocity distribution of electron emission. Also, it is not evident how the normal double-stream state in an operating magnetron could ever shift to the single-stream state. Furthermore, an instability voltage (inherent in the Willshaw-Copley treatment), in the presence of a strong established mode is questionable. Also, it is not clear why the established mode should give way to another as advocated by this criterion. Despite these discrepancies and its restricted applicability, this criterion has been instrumental and has been an impetus in magnetron development. It represents a unique attack of this difficult mode interaction problem, resulting in noteworthy qualitative predictions. It is in order, therefore, t o include a brief discussion here. Bunemann matched the reactive components of the wave admittance a t the surface of the space charge t o the transformed load. Such matching is achieved only with resonators for a given mode or resonant frequency and magnetic field, a t a critical space-charge cloud diameter. With this critical diameter is associated a n instability voltage equal to the anode voltage for which the Bunemann match is feasible. Spontaneous initiation of the nth mode occurs when the anode voltage exceeds not only the threshold (33) voltage but also the instability voltage for this mode. At the instability voltage the uniform single-stream space-charge flow becomes unstable and spokes of space charge (characteristic of the particular mode excited) develop (34). No account is taken of the fact th a t according t o this hypothesis oscillation is supposed to start from the single-stream state when no anode current flows. Actually, mode changes occur during current flow conditions. Neglecting this consideration, maintenance of the excited mode requires that the variation of its reactance

MAGNETRON MODE TRANSITIONS

509

with frequency be a negative quantity. For such inverted reactances, corresponding instability voltages have been evaluated (9, 3 5 ) , neglecting loading of the mode involved. For this excited mode to deliver power t o the load, according to Bunemann, its threshold voltage (33, 36) must also be exceeded, allowing the space-charge spokes t o reach the anode (34). According to the Willshaw-Copley criterion, after the a-mode steady state is attained, the applied anode voltage may be increased until the instability voltage of the adjacent lower order mode is reached. The ?r-mode is then expected to collapse, even though this adjacent lower order mode

FIG.1. Static and dynamic magnetron diagram.

would not be initiated until its own threshold voltage is reached. The range between the a-mode’s threshold voltage and the instability voltage of the adjacent higher voltage lower order mode may be increased by increasing (9) the normal (37) cathode to anode diameter ratio. €lowever, any improvement resulting from such change may be justified on firmer bases. These will be considered later. In this respect, consideration must be given to the influence of this change on the a-mode field in the interaction space ( 3 7 ) . The electronic efficiency is reduced by the increased cathode diameter with the result that compensation (8) in power may be necessary. For Hartree harmonics in excess of 4,the resulting instability voltages cease to correlate with the Hartree or the Bloch (36) threshold voltages. These are in close correspondence with actual starting voltages. I n direct contrast to the performance predictions by the instability voltage criterion, actual results have demonstrated (38)t h a t a strong a-mode can be sustained from its inception until it almost reaches the limiting curve (20, 39) (Fig. 1) without displaying any inherent instability! Thus, as will be shown later, this indicates that once a strong a-mode is initiated, lower order modes cannot start until it becomes inherently unstable, independent of the possible effects of these lower order modes. Although the Willshaw-Copley criterion is not quantitatively

510

E. C. OKRESS

reliable, it has qualitatively shown plausible mode-change behavior in restricted cases and under the conditions previously cited. For this reason the mechanics of its application are of interest. The mode-change performance, according t o this criterion, begins with a measurement of the frequency, Y, of all possible modes, n, in the (cold) resonant system. The corresponding instability voltages, V,, for the n modes, are then evaluated (9, 35). The increasing order of these instability voltages qualitatively indicates the order in which modes of operation can occur. For example, if the condition Vu(n+l)

< V,, < Vn( n - 1 )

(1)

prevails, the n mode will start in the range if where V t , denotes the threshold voltage associated with mode n. Consider a n increasing input voltage V , and V t , > V,, with finite impedance source, which is equivalent to an emf in series with a n internal resistance, R. The nth mode will start at V t , with initially no anode current, but will rise linearly a t a specific rate, with increasing V . This rate is defined by the qualitative Slater (40) relation (which may be too small by a factor of 2): vc - Vt, (3) = IdK

(g),

where Iddenotes the Langmuir (40) current or diode current of the same dimensions as the magnetron interaction space for voltage difference Vc. Symbol K denotes a coefficient dependent upon tube performance. Symbol V , denotes the Hull (40) cut-off voltage. Symbol I, denotes the anode current. Therefore, IIanlmin = 0 (4) if Vtn - RI,(,+i) > Vu,. If (as is generally the case), V t , < VUn,then with increasing voltage, for the (n - 1) mode and maximum range of ?r-mode current,

and

MAGNETRON MODE TRANSITIONS

51 1

I n the case when the input voltage is decreasing, with a finite impedance source, the n t h mode once started continues until V < Vt, or V = Vu(n+l), if Vucn+i)> Vtn.Also

if Vu(n+l)> Vt,, which is independent of modulator impedance. Furthermore, V u n - Vtn R[Ia(n-~)lmin IIanImax = (10) (aV/aIa>n R

+

+

I n the case where the input voltage is increasing or decreasing,

and is also independent of the modulator impedance. The range of mode currents will in general be different under increasing voltage conditions from that under decreasing voltage conditions. To illustrate the extent of application of the foregoing instability voltage criterion for predicting operation, a typical heavy double ring strapped symmetrical magnetron is considered. Its salient parameters, together with the results of the Willshaw-Copley treatment and comparison with practice are given in Table I for the following parameters (9) : Anode radius: Anode length: Magnetic field: Anode voltage : Anode current: Number of resonators: Wavelength : Lower mode wavelength:

ra = 1.27 cm 1=2cm H = 1700 oersteds Vz30kv I, = 0 to 60 amp N = 12 A, = 10 cm A 6 = 8.5 cm

The loading, which was hitherto neglected will reduce all lower order mode voltages with respect to that of the ?r-mode. Only the lowest order Hartree harmonic, y (the first reverse mode), is considered. The higher order components of all the lower order fundamental modes are associated with fields in the interaction space which fall off much faster from the anode to

512

E. C. OKRESS

the cathode (41). They are also characterized b y significantly different operating ranges compared with the a-mode. An interesting aspect of Table I is the predicted increase in a-mode current range with increasing cathode diameter consistent with reasonable efficiency. The discrepancies and qualitative features of the Willshaw-Copley criterion are self-evident from the results of Table I. The actual operation shown in the table refers to limitation by mode change. TABLEI. Operating Ranges and Mode Change Characteristics for a Strapped Symmetrical Magnetron According to the Instability-Voltage Theory

I

rln/N (cm) (cm) V,, (kv)* Vtn (kv)* Operation: Start a t V (kv) Start a t I,, (amp)

To

An

* Rieke

-7/5/12

6/6/12

5/5/12 0.450 8.5 33.8 44.3

0.575 8.5 39.3 39.4

0.63: 0.450 8 . 5 10.0 39.8 24.4 36.6 13.2

0.575 10.0 29.2 29.8

0.63! 10.0 30.2 27.8

0.450 8.5 2.5 3.5

0.575 8.5 27.1 30.0

44.3

39.3

39.8

29.8

30.2

no

no

0

66

33.2 0

0

35

0.635 8.5 28.0 28.0 28.0

0

33.8

39.3

39.8

30.2

5 5

116

174

32

0

0

0

5

60

100

(43).

The significance of the instability voltage is practically lost when it is considered that the a-mode can satisfactorily attain steady state and continue t o grow to such large amplitudes that nonlinear effects prevail. The a-mode does not collapse a t or even above the instability and threshold voltages of the higher voltage mode. Indeed, there is no evidence that this other mode could build up in the presence of a strong a-mode. Only conditions in the a-mode determine its instability and allow another mode more suitably matched to the new conditions t o build up. Our present understanding of this competing mode concept is due primarily t o Rieke (4.2). This unique phenomenon actually occurs when the input voltage is in a region where simultaneous initiation of two modes is possible. As these two approach large signal amplitudes a t which non-

MAGNETRON MODE TRANSITIONS

513

linear effects are important, definite mode selection for a given pulse is determined by the fact that the rate of build-up of one mode is affected more by the amplitude of the other than by its own. 111. MODE-COMPETITION CRITERION Generally, mode transitions (mode skips and shifts) occur in which interaction with the space charge is a significant factor. This interaction is due t o the actual mode separations and degree of loading of the lower order modes with respect to the s-mode prevailing in practice. Rieke investigated various mode selection processes (43) displayed b y dynamic ( V , I ) traces, in which interaction or competition and instability in the originally established mode are a primary factor when nonlinear effects are significant. ( V , I ) traces were used extensively b y Rieke because they provide a convenient basis for discriminating between the various mode selection processes. To illustrate the salient features of mode transitions involving interaction or competition between modes and instability of the originally established mode a t high current, the following examples are presented. Assume t ha t the rate of rise of the input voltage pulse is favorable for establishing steady-state operation in a given mode. Then as operation at a given magnetic field is continued toward higher currents, the established mode may eventually become unstable and cease operation. There is as yet no apparent mode competition involved in the process. On this basis, it is found that the dynamic ( V , I ) trace moves along the modulator characteristic, as illustrated (44) by Fig. 2. When the operating region of a higher voltage mode is reached the steady state is presumably established in that mode. This follows provided the mode separation is not too great nor the modulator impedance and magnetic field so low that nonoscillation occurs. This foregoing phenomenon is characterized as mode shift due t o instability in the originally established mode. Another significant example of mode transition in which mode interaction or competition is a primary factor, is mode skip. I n this case such small voltage mode separation exists that the operating regions of the two modes are nearly coincident, as illustrated (43) by Fig. 3. Although this situation involves a nearly equal Hartree harmonic wavelength product, a sufficient condition for mode skip is not necessarily indicated. I n this example, both modes are presumed to be initiated simultaneously under small signal conditions. As the amplitude's of the two modes rise until nonlinear properties of the space charge create mode competition, only one mode prevails at the expense of the other. Such a phenomenon is expected to be rather insensitive t o the rate of rise of the applied pulse. Within limits the relative competition is such that random conditions de-

514

E. C. OKRESS

I

FIG.2. Mode shift as a result of instability of the originally established mode a t the high-current end of operation (44).

V

I

FIG.3. Mode skip as a result of competition between modes (43).

termine the outcome. As first pointed out by Rieke, a t large mode amplitudes the nonlinearity of the space charge makes simultaneous operation of two modes inherently unstable. This confirms experience th a t simultaneous operation does not exist in the steady state. An equivalent circuit (Fig. 4) may be constructed t o represent simultaneous excitation of two fundamental modes at different frequencies and

MAGNETRON MODE TRANSITIONS

515

field configurations. Interaction is exclusively by means of the space charge. Each mode is represented by a parallel resonant circuit whose excitation is characterized by an alternating voltage, 8, and an electronic admittance, Ye. The small electronic susceptance, Be, involved is neglected. However, the property of the steady-state modulator with an output voltage, V = V , - V ( I ) ,and output current, I , is included for a step-function open-circuit voltages, V,. The nonlinearity of the space charge for two modes is characterized by the dependence of their respective electronic conductances on (1) the alternating voltages of the two modes and (2) the input voltage. The input current is dependent on both

w

(3

a

1 0

w

0

L

rn

FIG.4. Network representing simultaneous excitation of two modes in a magnetron. alternating voltages as well as upon the input voltage. Since the amplitudes of the alternating voltages are under consideration, only the real parts of the circuit and electron conductance need be equated for each mode. This is needed to obtain the characteristic relations for the transient state (45)’ including the approximate effect of the steady-state modulator characteristic. The transient state relations for each mode have to be solved in conjunction with the dependence of electronic conductance on the alternating voltages and input conditions. The result in terms of variables C: N - (ge 4- g ~ ) / 2 Cand # N In p/po for the a-mode is expressed as Gr = 4r(#r, #%)and 9%= 4n(#s, #,J for the lower order mode. The system’s transient behavior is then described by the locus

JlrlGn

=

C:r(G,#n>ltn(+r,

#7J

(12)

Its solution, containing integration constants representing the noise voltages from which each mode is initiated, may be represented by a Nordsieck (46) interaction diagram. I n Fig. 5 such a diagram is illustrated for two interacting modes with their conductance contours resolved. The in-

516

E. C. OKRESS

tersection of the contours, f r = f n = 0 represents a possible state of simultaneous operation in two modes. The starting ranges of the modes are near the locus of the modulator characteristic. When the two modes overlap, the interaction diagram can preserve its general symmetry for all modulator characteristic loci. The modes’ noise voltages are represented as point-density distributions, (J.r, J.n)o, near the origin of the interaction diagram. The probability that the system’s locus lies between any two other loci is directly proportional to the point density between them.

FIG.5.’Nordsieck-Rieke interaction diagram.

There is zero probability for a locus to proceed to the simultaneous oscillation state, represented by the intersection of the mode contours, ( f r = fn = 0). This intersection or saddle point, as it is referred to, possesses inadequate conditions for stability. Under small signal conditions in Gn both f T and tnare independent of Jln. The dependence of f w on J.,, is then as if the n mode were absent and vice versa. Rieke further surmised that under large signal conditions where f r and f n are both large, nonlinearity makes the dependence of f n on Jlr stronger than its dependence on and vice versa. I n other words, the rate of build-up in either of two modes is affected more b y the amplitude of the other than by its own amplitude. It will be subse-

+,,

MAGNETRON MODE TRANSITIONS

517

quently shown that this judicious assumption of Rieke’s is valid in the light of present knowledge. This is so despite the fact that the (-contours actually move during the transient, and so build-up does not occur along a stationary modulator characteristic as assumed. Furthermore, mode, selection and therefore E, is effected by the magnetron parameters (for example, magnetic field, loading of modes, open-circuit voltages, modulator impedance, and mode separation) and the manner in which these parameters affect the interaction field and noise distribution. The fact that 5 is a time function means that mode interaction during the transient may occur. The r-mode build-up may also be affected even though the n-mode may not suppress the r-mode. A particularly significant example substantiating Rieke’s mode change criterion will be deferred t o a subsequent section in which his hypothesis is further developed.

CRITERION IV. MODE-STABILITY Rieke’s mode-competition criterion has received significant confirmation since its inception. Moats (1‘7) has recently advanced convincing arguments, based upon results of Van der Pol’s (18) nonlinear theory for the magnetron, which substantiates Rieke’s views. He also presented experimental evidence supporting the qualitative results, relating to loading effects of a n established mode on another smaller amplitude (externally maintained) mode. I n the application of the Van der Pol theory, the magnetron is treated as a feedback oscillator in which the feedback replaces the grid circuit of the triode. The electromagnetic fields of the various modes in the interaction space are superimposed upon the passive system and interact only through the space charge. The feedback is distributed throughout the interaction space, for which the transit time is of the order of cycles. This is in contrast to the triode case of a lumped circuit having fractional transit time. A constant feedback factor is assumed, and this is related to the resonant system and interaction space geometries, particularly the ratio of cathode to anode diameters. The gain factor without feedback is related to the amplitude of the modes and loading of the resonant system. A high-Q resonator system, normally inherent in the Randall magnetron is assumed. Also a frequency mode separation is assumed such that the impedance of one equivalent resonant circuit (Fig. 6) representing a mode, isnegligible a t the resonant frequency of the other. I n this respect it is pertinent to mention that Van der Pol considered significant coupling and resolvable modes. The space charge is represented by a nonlinear current function which is proportional t o the feedback voltage. Both modes are characterized by the same feedback factor. This restriction, however, may be eliminated by introducing a correction, equivalent to changing the ratio of inductance to capacitance,

518

E.

C. OKRESS

without altering the resonant frequency. The effect of variations in the phase of the feedback factor is not considered. I n spite of these and other simplifying assumptions, considerable insight into magnetron mode interaction has been gained. This hypothesis, to a significant degree, has been confirmed experimentally. The following . c I w

v,

= IV, I cot w,t I-

F

MODE

LOWER ORDER n MODE

-

vn = IVn lcos w,t

FIG.6. Magnetron equivalent circuit for two modes.

procedure, based upon the foregoing assumptions, outlines the salient features of Moats (17) analysis for mode interaction during build-up. An alternating voltage

across a resonant circuit (Fig. 6), representing a particular T- and y-mode may be related t o the associated alternating current as

f

=

L(lRl,I P Y l ) f

=

+

fY(l~*l,

#(P)

IP,I)

(14) (15)

It is assumed that the angular frequency, o,for each mode be harmonically related. The ?r-mode component of current may be derived by finding the fundamental Fourier component of in phase with the voltage. T ha t is,

#(v)

Although frequency and phase shifts will occur as a result of loading the active system, the effect of these shifts on the rate of build-up of the

519

MAGNETRON MODE TRANSITIONS

modes is assumed to be negligible. For any arbitrary phase, 4, of o,t there will be n phases of 4. Furthermore, as n + 00 , then

r,(lv,l, Ivy])= &

/"i2=$(P) 0

cos w,t

0

*

d(w,t)

. d(w,t)

(17)

0

For a n evaluation of build-up characteristics of modes, this relation requires the fulfillment of the following conditions:

where the rate of build-up of

T- and

y-modes is expressed in terms of *

and

Results of numerical integration of the current relation for a particular nonlinear odd function (because of a high-Q resonator),

$(v)

=

a tanh b p

are illustrated by Fig. 7. This function is such th a t a Taylor expansion for small values reduces it t o the original Van der Pol (18) expression. Also, since -+ - I for large - 7 $(P) -+ + I for large + P $(PI = +ab for small 7 a study of magnetron build-up is feasible. With respect t o Fig. 7, the electronic conductances for both modes are

and

* The symbol u denotes the respective real part of the complex frequency in e(a+fo). Build-up or decay of the sinusoidal wave is then expressed by e*u'. The subscript T or y denotes the mode involved.

520

E. C. OKRESS

With the aid of Fig. 7, the following observation is made for any value of parameter blP,( within the limits 2 > blPyl > 0 (which excludes blP,l = 2), the variable blP.1 is seen to cause a reduction of g,(]P,l, IP,l) from unity. When the values of the parameter and variable are interchanged, however, it is found that blPrl reduces g r ( l P , ) , (P,()even more. Hence, the rate of build-up in one mode is not affected as much by its own magnitude as by the magnitude of the other mode, in confirmation with Rieke’s hypothesis. An interaction diagram may be derived from Fig. 7. That is, for a particular g.(lP,I, IF,() in Fig. 7, the value of the

Y

0 h

-0.4a/b

blV,. 1.2

I

I

I

I

FIG.7. Results of numerical integration of build-up, Eq. (17). This shows electronic conductances as a function of the amplitudes of the alternating voltages in each of two modes with the other as parameter (17).

variable blV.l corresponding to each bl P,I parameter is transformed to Fig. 8 t o form the u. = 0 contour. By interchanging 18,1 and lVyl and repeating the previous process, the formation of the u7 = 0 contour is achieved. Such an interaction diagram illustrates the transient and stability characteristics of one mode with respect t o another for specific circuit parameters, g. = gr. The four possible solutions for 1f.l = lfrl = 0 of Van der Pol’s equation are represented in Fig. 8. As indicated, two solutions represent stable operating points. One of these is the ordinate intercept of the u-,= 0 contour, for which u. < 0. The y-mode is stationary in magnitude, since ur = 0, and stable, since any departure of operation away from this point will change conditions so as to cause it to return. Any tendency for the ?r-mode to build u p is counteracted. The other stable point is the abscissa intercept of the u. = 0 contour, for which u7 < 0. The ?r-mode is also stationary in magnitude, since us = 0, and stable, since any departure of operation away from this point will

MAGNETRON MODE TRANSITIONS

52 1

change conditions so as to cause it to return. Any tendency for the y-mode t o build up is also counteracted. Of the two remaining solutions, both represent unstable operating points. One of these is located a t the origin and is unstable because u, > 0 and uy > 0. The other unstable point is located a t the intersection of the u, = 0 and b y = 0 contours, the saddle point. At this saddle point, if both the r-mode and the y-mode are disturbed, where u, = uy = 0, so as to cause operation t o enter region 11, the r-mode will continue building up while the y-mode will continue t o decay. If the operating point enters region 111, it will likewise STAELE OPER.

SADDLE POINT

LJNSTABLE OPER. d%I FIG.8. An interaction diagram derived from Fig. 7, illustrating transient characteristics of two modes under conditions when either mode may be stable.

continue t o depart from the saddle point. If the operating point begins to enter regions I or IV, conditions will induce it to go t o region I1 or I11 instead, and i t will continue departing from the saddle point. I n region I, where ur > 0 and uy > 0, both modes are building up. I n region 11, where u, > 0 and uy < 0, the r-mode is building up, b u t the y-mode is decaying. I n region 111, where us < 0 and uy > 0 the r-mode is decaying but the y-mode is building up. Finally, in region IV, where uu < 0 and uy < 0, both the r- and y-modes are decaying. At the origin a perturbation can cause build-up of both the r- and y-modes. When the originally established mode is relatively weak, then conditions represented by Fig. 9 prevail. Here g, is as in Fig. 8. However, gr has been increased by virtue of heavier loading. The abscissa intercept for the u, = 0 contour is the only stable operating point. There are two

522

E. C. OKRESS

other points which are unstable; one of these is a t the origin and has been considered, and the other is an ordinate intercept of the u? = 0 contour, for which ua > 0. Hence, any tendency to build-up by perturbation would result in operation moving t o the region in which the y-mode decays. Hence, the requirement for a mode to build u p in the presence of a n established mode is that the latter first weakens or be weak initially. For the case of a n initially established weak mode, denoted by the a,' = 0 contour of Fig. 9, and a subsequent condition denoted b y the ur = 0 contour, the s-mode would build u p and suppress the y-mode. This is in accord with Rieke's criterion. T o the extent that a weak mode is established initially, the Willshaw-Copley criterion is plausible.

I/

UNSTABLE

\UNSTABLE

OPER.

OPER.

FOR:gy

)ow

blvrl

FIG. 9. Another interaction diagram derived from Fig. 7 illustrating transient characteristics of two modes under conditions when but one mode may be stable (ignore ux' = 0 contour) and when either mode may be stable (ignore U , = 0 contour).

T o illustrate, the persistence of the established mode referred to as hysteresis by Van der Pol, mode changes which may occur when operation in the s-mode is initiated under conditions illustrated by Fig. 9, will be considered. As the parameters are gradually altered t o the conditions illustrated by Fig. 8, the originally established s-mode persists until the abscissa intercept, for the uy = 0 contour, exceeds th a t for the ur = 0 contour. The y-mode will then build up and suppress the ?r-mode with the change occurring at (blF,l)., = (bIF,J),. With the y-mode established and the s-mode nonexistent, a gradual reversal of the previous procedure indicates persistence of the established y-mode until a mode change occurs at (blFTl)y = (blF&. He did show experimentally (Fig. 10) that strong s-mode operation can suppress other modes and thereby allow satisfactory operation through higher threshold voltage modes u p t o the limit curve of Fig. 1.

MAGNETRON MODE TRANSITIONS

523

No discontinuities were reported over the range of a-mode operation illustrated in Fig. 10. This performance also agrees with Rieke’s criterion. Moats (17) has experimentally shown that the conditions for mode change are determined exclusively by the strength of the established mode. Precautions were taken in his experiment to alter conditions in only one mode by selective loading. Referring t o Fig. 11, when the loading of only mode A was changed, the voltage boundary between modes A and B was altered as indicated. No other change occurred. A similar

12

-

10 -

ACTUAL M MODE OPERATION

THEORETICAL STARTING VOLTAGE FOR YY8 MODE

THEORETICAL STARTING VOLTAGE FOR 4/4/8 8OR II-MODE

z 2 6 8

-

HULL CUT-OFF

9 W

2 4 -

n - ~ OPERATION ~ w STARTS

2-

I

0

.

I

I

1000 .I500 2000 MAGNETIC FIELD IN GAUSS

500

FIG. 10. Illustrating satisfactory r-mode operation of a particular magnetron (2539) from satisfactory starting to the limit (Fig. 1) without encount6ring *-mode instability (17).

result was obtained for each of the other modes. Each of the mode change (voltage) boundaries is associated with an abrupt reduction in current during the mode transition. This indicates a decay of the original mode and build-up of the subsequent mode. To determine the effect both of the a-mode space charge on the quiescent lower order mode and of decreasing the loading on a lower order mode, a series of experiments were undertaken (17).The results of these experiments tend to confirm the Moats’ hypothesis of mode change. I n particular, it was found that the quiescent mode was loaded by the s-mode space charge, and this loading increased with increase in the amplitude of the established mode. As a consequence, it was concluded in confirmation

524

E. C. OKRESS

with Moats’ hypothesis, that a large amplitude of one mode tends to suppress build-up in another. Furthermore, the loading of one mode by another is a function of the relative amplitudes. This is not directly proportional to the ampIitudes but saturates at high levels.

BEFORE CHANGING LOAD AFTER

CHANGING

LOAD

FIG. 11.. Representation of oscilloscope traces of a magnetron illustrating mode shifts due to changing load of mode A only; restoring original load, changing load of mode B only; restoring original load and changing load of mode C (17).

V. ASSESSMENT AND RAMIFICATIONS I n the course of the development of the foregoing major mode transition criteria, considerable information was gained toward the development of the present performance status of the Randall magnetron. The qualitative Rieke and Moats hypotheses correlate in general with magnetron performance. In essence the further developed Moats’ hypothesis asserts that (1) a strong established a-mode tends to suppress other quiescent lower order modes, (2) the adverse effect of one mode on another is not a continuously increasing function with amplitude, (3) once a strong mode has been established, other modes cannot build up in its presence, (4) the cessation of the established mode is a function of its own ultimate instability (for example, loss of synchronism due t o excessive input voltage, etc.) apart from the influence of any other mode, ( 5 ) the dependence of the rate of build-up in one mode upon the amplitude in the other is greater than upon its own amplitude, (6) mode competition prevails as soon as the amplitudes of two modes are simultaneously initiated, the nonlinear effects then promoting the stronger mode t o suppress the weaker one, and (7) a weak mode may have a n adverse effect on a substantially stronger mode.

MAGNETRON MODE TRANSITIONS

525

No better mode stability criterion has been developed since that of Rieke and Moats. Recently, however, Welch (47) and his coworkers have shed considerable light on this difficult problem in attempting to formulate an improved criterion particularly for the maximum stable a-mode current (in the absence of an adequate criterion for the maximum stable .Ir-mode voltage). I n contrast to the Rieke and Moats criterion, that of Willshaw-Copley is basically formulated on the effect of one mode on an already established mode. It asserts that, as soon as the established mode attains the instability voltage of another mode, it collapses. This other mode is initiated when its threshold voltage is attained. Although this criterion has been shown to be questionable, i t nevertheless agreed with practice in restricted instances. The lack of general validity has already been discussed in Section 11. I n spite of its qualitative and restrictive character, it was indirectly instrumental in improving magnetrons (9, 35, 48). One outstanding result of the Willshaw-Copley semiempirical criterion was the prediction that to increase the ?r-mode current the ratio of cathode t o anode diameters must be increased beyond the conventional Slater (20) optimum. Changes in the cathode size affect the transient behavior more than the steady-state behavior. Increasing the ratio of cathode to anode diameters increases the rate of build-up and preoscillation noise for all modes and hence may improve a-mode stability. However, the field intensity of all the lower order modes are greater than that of the a-mode a t the cathode. Therefore, in order to discourage mode competition and take advantage of the higher preoscillation noise and faster rate of build-up in the a-mode due to the increased ratio of cathode to anode diameters, i t is necessary adequately to distort and properly t o load the lower order modes. Thereby substantially greater power capacity may be realized. It may also be necessary to provide adequate voltage mode separation between the a-mode and the lower order mode and in proper voltage sequence, so that the lower order mode threshold voltage is higher than that of the a-mode. Thereby the lower order mode cannot build up ahead of the a-mode and delay it. Considerations regarding the loading of the lower order modes indicate, however, that this may not be necessary and it has proved not to be in practice. This occurred when the lower order mode loading was adequate, so that the a-mode could be strongly established (e.g. Fig. 10). Nevertheless, this suggestion is a safe practice whenever feasible. Merely increasing the cathode to anode diameter ratio without considering factors such as loading, distortion, etc., of the lower order modes may lead to delay in build-up of the a-mode and even a-mode skip. The fact that the efficiency decreases as the ratio increases beyond

526

E. C . OKRESS

Slater’s value (because the transit time decreased, etc.) indicates that a compromise in other tube parameters is necessary. Using a different basis, Hagstrum et al. (8),also increased the cathode to anode ratio beyond the Slater value. He altered the magnetron parameters to compensate in part for the resultant drop in efficiency. A reasonable efficiency with significantly greater a-mode current and power output capacity thereby resulted. The method of approach in this instance is unique and warrants elaboration. Hagstrum made an empirical study of a-mode failure of pulsed symmetrical magnetrons with respect to magnetic field, frequency, ratio of cathode to anode diameters, load conductance, and rate of rise of the input voltage pulse. A limitation in the input power was encountered as a function of magnetic field, frequency, and modulator characteristics. Beyond this operation in the a-mode ceased, regardless of the power capacity of the particular magnetron involved. The A modulator regulation is a factor in this instance. If it is poor, the applied voltage is B reduced as the a-mode is initiated, thereby interfering with its starting. As the anode voltage is continuously increased, a-mode C failure occurs when the input current pulse starts to narrow from its leading edge (Fig. 12b). The current a t which this takes place D was referred to as the critical current, I,. Continuous increase of the anode voltage E causes the original current pulse to disappear (Fig. 12c) in a manner dependent upon modFIG.12. Sequence of cur- ulator regulation. Instead, a jittering pulse rent pulses during mode shift and instability. Nar- may appear (Fig. 12c’). With further increase rowing of pulse shown by in input voltage the normal current pulse form trace B (8). is restored, but in another mode (Fig. 12d and e). This process sometimes repeats with subsequent increase in input voltage. This phenomenon of mode skip and instability is associated with the rate of rise of the input pulse and modulator regulation. In any event, Hagstrum’s work indicated that the critical current increased with magnetic field and frequency. The electronic efficiency also decreased with increased cathode to anode diameter ratio, beyond the Slater value. The critical current and electronic efficiency were found to vary inversely as a function of load conductance (Fig. 13). A compromise was therefore necessary with respect to the desired electronic efficiency and maximum a-mode or critical current. The last parameter investigated was the effect of the rate of rise of

?

527

MAGNETRON MODE TRANSITIONS

the input pulse on ?r-mode build-up. He found th a t the rate of build-up increases with cathode size. Increasing the resonator admittance and/or load conductance counteract in part the decrease in efficiency produced by the increase in cathode size. The resonator admittance, however, is a direct function of the product of loading (loaded Q) and electronic conductance, whereas the electronic efficiency is a monotone function of the latter up t o a maximum. These relationships are particularly important a t high power levels, for increasing the resonator admittance and loading (decrease loaded Q) reduces the alternating voltage between the resonator elements. I n a n effort to determine whether this inverse relationship of critical current (i.e., power output) and electronic efficiency is not completely nullifying, Hagstrum formulated a technique utilizing empirical .-6

-5 ..4

y r fC

-3 ..2

-.I

FIG.13. Average characteristics of a number of magnetrons of various cathode-toanode diameter ratios and loading. Subscript 0 denotes characteristic (Slater-Allis) variable (8).

data. This technique relates critical current and electronic efficiency as a function of loading for a number of magnetrons as illustrated b y Fig. 13. These results do not imply, however, that magnetrons (within the range considered) are equivalent when their ratios of electronic to characteristic conductances are identical. Nevertheless, the empirical relation represented by Fig. 13 has been found by others to be surprisingly reliable in practice even after many fold complete scaling of magnet,ron parameters. The unusually large cathode to anode diameter ratio and resonator characteristic admittance are salient features of the Hagstrum technique. This is particularly of value in solving the mode skip and instability problem encountered a t long wavelengths. It thereby avoids the build-up time consuming a significant portion of the pulse or unnecessarily complicating the modulator design. Although the increase in the ratio of cathode

528

E. C. OKRESS

to anode diameters increases the rate of build-up of the a-mode, the increase in characteristic admittance of the resonant system, tends to counteract the effect of the change in cathode size. However, because the electronic conductance during a-mode build-up is the crucial factor under consideration, a definite net gain results. It is appropriate t o mention th at the Willshaw-Copley treatment in addition t o predicting larger cathode size for higher power output also arrived a t concepts of an optimum mode separation which in certain respects are in confirmation with Rieke’s (49) results. This implies in essence that reduction of mode interaction depends upon mode separation; therefore, i t is desirable to consider the (Hartree) harmonic wavelength product for the a-mode and competing mode. This is done in such a manner that overlapping starting ranges (Fig. 3) are avoided, and the starting voltage of the competing mode component is above th a t of the a-mode. It was also predicted by Willshaw-Copley (9, 48) th a t for the highest power, satisfactory operation can be realized with the fewest number of resonators. I n addition, they pointed out that it was advantageous to increase anode and cathode dissipation density rather than increase the working areas. The deduction with regard to the sequence of starting voltages, in the light of Moats’iand others’ results (Fig. lo), is not a necessary requirement. It may be desirable at long wavelength and high power t o resort to as few number of resonators as possible. I n addition t o the rate of build-up of the a-mode and its relative strength, the effects of preoscillation noise, loading and distortion of the lower order modes, and optimum voltage mode separation are considerations of adequate cathode (total) emission and uniformity of the transverse alternating and axial magnetic fields throughout the interaction space. The foregoing theoretical aspects of this treatment have ignored effects of various aberrations because of the complexity they introduce. However, deviations, including transverse and longitudinal cathode eccentricities (50), cathode and protuberances ( 5 1 ) , etc., have important practical consequences on the transient and steady state performance of magnetrons. Historically, the Willshaw-Copley criterion promoted operational improvement in the strapped symmetrical magnetron by indicating how much the ratio of the cathode to anode radii and/or the number of resonators may be altered. T o a limited extent, this guide resulted generally in some improvement in operation, but not for the reasons indicated by the criterion. For the majority of types to which this criterion has been applied, the a-mode instability voltage occurs a t a higher value than the threshold voltage, with the magnetron starting a t or near the threshold

MAGNETRON MODE TRANSITIONS

529

voltage. It is now recognized by the proponents of the criterion that this has greater significance than the instability voltage, both from the initiation of oscillation and the mode transition aspects. I n any event, since the associated theory is based on the inapplicable single-stream state, it is severely limited in its applications. No attempt has been made, however, to formulate plausible instability theory based on the double-stream state. Rieke’s criterion, based on nonlinear arguments, is sound and has been largely confirmed. Furthermore, Moats has presented plausible theoretical arguments in favor of Rieke’s criterion and extended his results. In fact, he has plausibly demonstrated that once a strong r-mode is established, no apparent limitation exists which can be attributed t o adverse effects of lower order modes. The paramount considerations relating to the stable transient aspect of magnetron operation are inseparably associated with relative loading and relative threshold voltages of the a-mode and undesirable lower order mode components. If the voltage mode separation between the component and the r-mode is such that this component’s threshold voltage is greater than t ha t of the r-mode, then the loading of primarily the r-mode is important. It should be such that its rate of build-up is greater than that of the applied pulse. If this condition is not satisfied, the r-mode may not start and will therefore fail to load down the modulator, with the result that the applied voltage may rise to the starting range of the subject component and excite it if its loading is such that the corresponding rate of build-up is greater than th at of t,he applied pulse. T h e voltage mode separation may also be such that the threshold voltage of the component i s less than that of the a-mode. Then stable r-mode starting will depend on the relative loading of the r-mode and this component. T h a t is, not only must the rate of buildup of the r-mode be greater than th a t of the applied pulse, but the component must be so loaded th a t it does not build up ahead of the a-mode; otherwise, mode skipping will result. The unstable starting of a magnetron due to a n excessive reactive load impedance presented by an unfired, improperly positioned ATR switch may be a serious consideration. For this reason, the path length from the magnetron t o the reflection and back during the transient ought not t o be less than the product of the duration of the transient and the velocity of propagation of the wave in the transmission line. This simple solution conflicts with another consideration, however, namely, the bandwidth of a tunable magnetron. As the physical length of the line is increased, the tuning range for stable starting is correspondingly restricted. This latter consideration implies a location of the reflecting element near the magnetron. An optimum position which compromises these two conflicting conditions must be determined.

530

E. C. OKRESS

With regard t o the steady state aspect of magnetron operation, it is also necessary t ha t the relative loading be in favor of the ?r-mode, but for reasons other than those discussed for starting. Favorable loading may be achieved by a sufficiently wide bandwidth transducer and proper orientation of strap or resonator discontinuities in the case of a symmetrical magnetron. Failure of the established a-mode may occur because of limitations imposed by the space charge irrespective of the influence of any other mode resonance. Such limitations include the current through and charge density of the space charge which can be tolerated. Although various aspects of this complex space-charge stability problem have received attention, notably b y Welch and his coworkers, and plausible descriptions made of recognizable attributes thereof, the status of the subject is still qualitative. However, some estimates have been made of transit time and phase focussing effects associated with the space charge. With regard t o the cathode, end hat emission is still a serious problem which can and frequently does limit the maximum power capacity of a magnetron. The interaction space ought to have uniform axial magnetic field from the mode stability point of view. Excessive deviations, also with regard t o the applied electric field, may cause superinduced spectra, especially if the cathode emission extends the length of the anode. Although considerable progress has been made in understanding magnetron operation, there is still severe lack of quantitative results with regard t o principal factors including space charge stability, transient loading, maximum current boundary, and back bombardment. The design of magnetrons is still largely restricted t o empirical methods. I n the foregoing treatment, methods for determining the input impedance, a-mode loading and lower order mode relative loading during normal operation of the magnetron have been assumed (52, 53),* on the part of the reader. Incidentally, with respect to the input impedance of the tube during ?r-mode c-w operation, the common heterodyne method, using the tube as a local oscillator, is applicable.

ACKNOWLEDGMENT Several figures used in this article have been reproduced, with permission, from other sources. This permission is gratefully acknowledged from McGraw-Hill Book Company Inc., (Q),Figs. 2 and 3. M. I. T. Research Laboratory of Electronics ( l 7 ) , Figs. 7, 10 and 11. Bell Telephone Laboratories (8), Figs. 12 and 13.

* Reference (62) includes a method for determining operating r-mode loaded Q (relation (11)in subject reference) from Rieke diagram or equivalent information. Reference (63)includes a method for determining the loading at a lower order mode resonance, while the magnetron is operating stably in the T-mode.

MAGNETRON MODE TRANSITIONS

531

REFERENCES 1. Randall, J. T., Boot, H. A. H., and Wright, C. S., British Patent 588,185 (May 16, 1947). 2. Sayers, J., and Wright, C. S., British Patent 588,916 (June 6, 1947). 3. Clogston, A. M., Forced Response of a Cavity Electromagnetic Resonator, Ph. D. Thesis, Mass. Inst. of Technol., Cambridge, 1941. 4. Collins, G. B., ed., “Microwave Magnetrons,” 1st ed. p. 19. McGraw-Hill, New York, 1948. 6. Slater, J. C., Mass. Inst. Technol. Research Lab. Electronics Tech. Rept. No. 182 (1942). 6. Okress, E. C., J. Appl. Phys. 18, 1102 (1947). 7. Walker, L. R., in “Microwave Magnetrons’’ (G. B. Collins, ed.), 1st ed. p. 141. McGraw-Hill, New York, 1948. 8. Hagstrum, H. D., Hebenstreit, W. B., and Whitcomb, A. E. Bell Telephone Labs. Memo. MM-45-2940-2. 9. Copley, D. T., and Willshaw, W. E., General Electric Research Laboratory, Wembley, England, C. V. D. Rept. No. 8490 (Aug. 1944). 10. Walker, L. R., in “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., p. 135. McGraw-Hill, New York, 1948. 11. Rieke, F. F., in “Microwave Magnetrons’’ (G. B. Collins, ed.), 1st ed., p. 355. McGraw-Hill, New York, 1948. 1.8. Walker, L. R., in “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., p. 147. McGraw-Hill, New York, 1948. IS. Rieke, F. F., in “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., p. 380. McGraw-Hill, New York, 1948. 14. Brown, W. C., A Report on the Analysis of the Introduction of Dissymmetry for the Purpose of Equalizing the External Loading on the Adjacent Mode Doublets, Raytheon, Waltham, Mass. (July 22, 1947). 16. British Patent 666,689, Campagne Generale De Telegraphic Sans Fil (Feb. 20, 1952); American Patent No. 2,610,309, Campagne General De Telegraphic Sans Fil (Sept. 9, 1952); Hall, J. F., Analysis of a Magnetron Interaction Space for Elimination of Hartree Harmonics, Evans Signal Laboratory, Thermionics Branch, Belmar, N. J. Memo. Rept. No. SCCSCL-RTB2, Project 322A (Jan. 30, 1951). 16. Kroll, N., i n “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., p. 83. McGraw-Hill, New York, 1948. 17. Moats, R. R., Mass. Inst. Technol. Research Lab. Electronics Tech. Rept. 171 (1951).:Caption for Fig. 12 of T.R. 171 should refer to Eq. 45 and not Eqs. 43 and 44. Mode Stability in a Resonant-Cavity Magnetron. Master’s Thesis. Mass. Inst. Technol., Cambridge, 1947. 18. Van der Pol, B., Phil.Mag. 43,700 (1922); Edson, W. A., “Vacuum Tube Oscillators,” 1st ed., Chapter 4. Wiley, New York, 1953; Hayashi, C., “Forced Oscillations in Non-Linear Systems,” 1st ed., p. 106. Neppon, Osaka, Japan, 1953. 19. Welch, H. W., Jr., Ruthberg, S., Batten, 11. W., and Peterson, W., Univ. Mich. Electron Tube Lab. Tech. Rept. 6 (1951); see also Proc. I.R.E. 41, 1631 (1953). 20. Slater, J. C., Mass. Inst. Technol. Research Lab. Electronics Tech. Rept. 118 (1941); Hok, G., “Very High Frequency Techniques,” p. 508. McGraw-Hill, New York, 1947. 21. Rieke, F. F., in “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., pp. 351, 379. McGraw-Hill, New York, 1948.

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29. Jepsen, R. L., Enhanced Emission from Magnetron Cathodes. Columbia Radiation Laboratory, New York, May 1, 1951. 23. Forsberg, P. W., in “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., p. 379. McGraw-Hill, New York, 1948. 94. Welch, H. W., Jr., reference 19, Figure 8.6. 25. Wilbur, D. A., Nelson, R. B., Peters, P. H., King, A. J., and Kollar, L. R., CW Magnetron Research, Final Rept. Contract No. W-36-039 sc32279, General Electric Research Laboratories (April 1, 1950). 26. Welch, H. W., Jr., Proc. I . R . E . 38, 1434 (1950). 27. Lesensky, L., Observations of Space Charge Discontinuities in a Pulsed Magnetron, Raytheon, Waltham, Mass. (1955). 28. Rappaport, H., and Kosai, G., Bandwith Limitations Caused by Magnetrons Unstable Starting from Greensmann, J. W. E. An Investigation of Broadbanding Limits of Microwave Components, Report R-403-54, P.1.B.-336, Appendix V. p. A25, Microwave Research Institute, Brooklyn, New York (Dec. 1954). Early work credited to Kumpfer, B. D., of U. S. Signal Corps, Engineering Labs. and Molnar, J. P., of Bell Telephone Labs. 29. Rieke, F. F., in “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., p. 321. McGraw-Hill, New York, 1948. SO. Walker, L. R., in “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., p. 253. McGraw-Hill, New York, 1948. S1. Brillouin, L., and Bloch, F., Advances in Electronics 3, 145 (1951); Page, L., and Adams, N. J., Phys. Rev. 69, 494 (1946); Bloch, F., Space Charge Limited Single Stream Solutions in a Cylindrical Magnetron with Small Currents, Report 411-175, Radio Research Laboratory, Harvard Univ., Cambridge (May 25, 1945); Hok, G., Univ. Mich., Electron Tube Lab. Tech. Rept. 10 (1951). 32. l b i s s , R. Q., Mass. Inst. Technol. Research Lab. Electronics Tech. Repts. 116 and 117 (1949). 33. Hartree, D. R., Mode Selection in a Magnetron by a Modified Resonance Criterion, p. 10. Dept. Admiralty (British), London. 34. Tibbs, S. R., and Wright, F. I., Temperature and Space Charge Limited Emission in Magnetrons, C. V. D. Rept. Mag. 38, Fig. 2. Dept. of Scientific Research and Experiment, Dept. Admiralty (British), London. 35. Copley, D. T., and Buller, R., Assessment of Mode Change Performance of Magnetrons-Additional Values of Instability Voltage. Extension of Rept. No. 8490. General Electric, Wembley, England (Sept. 1947). 36. Bloch, F., A Tunable Squirrel Cage Magnetron by F. H. Crawford and M. D. Hare, R.R.L. Rept. No. 411-252, p. 14-15. Radio Research Laboratory, Harvard Univ., Cambridge (Oct. 1, 1945); Walker, L. R., in “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., p. 243. McGraw-Hill, New York, 1948. 37. Collins, G. B., ed., “Microwave Magnetrons,” 1st ed., pp. 23 and 440. McGrawHill, New York, 1948. 38. David, E. E., Jr., Mass. Inst. Technol. Research Lab. Electronics Tech. Rept. No. 168 (1950). S9. Allis, W. P., Mass. Znst. Technol. Research Lab. Electronics Tech. Rept. No. 176 (1942). 40. Slater, J. C., Mass. Inst. Technol. Radiation Lab. Tech. Rept. No. 22 (1943). 41. Kroll, N., in “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., p. 74. McGraw-Hill, New York, 1948; Healea, M., Mass. Znst. Technol. Research Lab. Tech. Rept. No. 686 (1944).

MAGNETRON MODE TRANSITIONS

533

4.8. Rieke, F. F., in “Microwave Magnetrons” (G. B. Collins, ed.), 1st ed., p. 344.

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