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- Permanent Link:
- http://ufdc.ufl.edu/UF00084188/00001
## Material Information- Title:
- On space-charge wave propagation in crossed electric, magnetic, and centrifugal force fields
- Added title page title:
- Space-change wave
- Creator:
- Lear, William Edward, 1918- (
*Dissertant*) Larsen, M. J. (*Thesis advisor*) Chen, Wayne H. (*Reviewer*) Smith, C. B. (*Reviewer*) Fyner, Mack (*Reviewer*) Smith, Alex G. (*Reviewer*) Hanson, Harold P. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1953
- Copyright Date:
- 1953
- Language:
- English
- Physical Description:
- 147 leaves. ; 28 cm.
## Subjects- Subjects / Keywords:
- Anodes ( jstor )
Cathodes ( jstor ) Charge density ( jstor ) Electric fields ( jstor ) Electrons ( jstor ) Magnetic fields ( jstor ) Magnetism ( jstor ) Magnets ( jstor ) Velocity ( jstor ) Waves ( jstor ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Electronics ( lcsh ) Vacuum tubes ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- Abbreviated introduction: Space-charge waves are variations in charge density in a cloud of electric charge, usually an electron beam, which are propagated throughout the cloud. To understand more fully the nature of these waves it is necessary that we examine the charge conditions existing in an electron beam in a vacuum tube.
- Thesis:
- Dissertation (Ph.D.) -- University of Florida, 1953.
- Bibliography:
- Bibliography: leaves 145-146.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000577387 ( ALEPH )
13977482 ( OCLC ) ADA5082 ( NOTIS )
## UFDC Membership |

Full Text |

ON SPACE-CHARGE WAVE PROPAGATION IN CROSSED ELECTRIC, MAGNETIC, AND CENTRIFUGAL FORCE FIELDS By WILLIAM EDWARD LEAR A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1953 ACKNOWLEDGMENT The author wishes to express his sincere gratitude to Dr. W. W. Harman, former chairman of the supervisory committee, for his continued guidance and encouragement, and to the present chairman and members of the committee for many valuable discussions and suggestions. He is also indebted to Mr. R. P. Derrough, whose cooperation made possible the experimental part of this work. LIST OF TABLES Observed Osoillations in Small- Cathode Magnetron . Observed Oscillations in Magnetron With Intermediate Cathode Radius . Observed Oscillations in Outside- Cathode Magnetron . iii Table 5.1 5.2 5.3 Page 95 101 109 LIST OF ILLUSTRATIONS Figure 1.1 2.1 2.2 2.3 3.1 3.2 4.1 4.2 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Page Unstable Space-Charge Condition in a Plasma . .. Klystron Amplifier . . Traveling-Wave Tube . Electric Field of Traveling-Wave in Moving Coordinate System . Outside-Cathode Magnetron . Comparison of Theoretical Cutoff Curves . . Spiral-Beam Traveling-Wave Magnetron Linear Traveling-Wave Magnetron . Block Diagram of Circuit Used to Determine Oscillation Frequencies . Observed Oscillations, Tube No. 1 . High-Frequency Oscillations, Tube No. 1 . . Static Characteristics, Tube No. 1 . Observed Oscillations, Tube No. 2 Frequency vs. r/rc, Tube No. 2 . Static Characteristics, Tube No. 2 . Observed Oscillations, Tube No. 3 . Static Characteristics, Tube No. 3 . 3 12 28 52 59 69 80 83 93 97 98 99 105 S106 . 107 S 115 S 116 LIST OF ILLUSTRATIONS--Continued Figure Page II.1 Angular Velocity in Outside Cathode Magnetron .. 156 11.2 Electron Trajectories in Outside- Cathode Magnetron ..... 137 TABLE OF CONTENTS Page LIST OF TABLES S & & .* & * LIST OF ILLUSTRATIONS . . Chapter I. SPACE-CHARGE WAVES INTRODUCTION II. ANALYSES OF SPACE-CHARGE WAVE TUBES * * The Klystron The Electron Wave Tube The Traveling-Wave Tube The Traveling-Wave Magnetron The Magnetron Oscillator :I. THE CYLINDRICAL DIODE MAGNETRON WITH OUTSIDE CATHODE . V. ANALYSIS OF THE SPIRAL BEAM TRAVELING-WAVE MAGNETRON . V. EXPERIMENTAL RESULTS . . Inside-Cathode Tube With Small Cathode Inside-Cathode.Magnetron With Intermediate Cathode Radius The Outside-Cathode Magnetron Discussion of Experimental Results CONCLUSIONS . . APPENDIX I. DYNAMICS OF THE ELECTRON BEAM - THE VELOCITY POTENTIAL . APPENDIX II. D.C. CONDITIONS IN THE OUTSIDE- CATHODE MAGNETRON . * 58 * 79 . 92 * 0 * 0 * S 122 124 132 iii * iv * * . 11 II I VI. APPENDIX III. TABLE OF CONTENTS--Continued SOLUTION OF THE DIFFERENTIAL EQUATION FOR F ... BIBLIOGRAPHY . . .*. BIOGRAPHY . . . Page 142 145 147 vii CHAPTER I SPACE-CHARGE WAVES INTRODUCTION Space-charge waves are variations in charge density in a cloud of electric charge, usually an electron beam, which are propagated throughout the cloud. To understand more fully the nature of these waves it is necessary that we examine the charge conditions existing in an electron beam in a vacuum tube. An ordinary vacuum tube which may operate at a pres- -6i sure of 10-6 microns, for example, still contains roughly 101 gas molecules per cubic centimeter. Therefore, it is not an accurate picture to conceive of an electron beam moving along its way unhindered in such a tube. As a mat- ter of fact, a one milliampere beam may produce perhaps 1012 gas ions per centimeter of travel.l(a) The positive gas ions thus produced are relatively immobile compared to the electrons, so that a given rate of ion formation will serve to neutralize the space-charge effects of a much higher rate of electron flow. (a) rscrpt number refer to Bibliography. Superscript numerals refer to Bibliography. Now let us assume that such a condition of neutraliza- tion exists in an electron beam. If a portion of the beam should be slowed down, say by a retarding electric field, a dense cloud of electrons would result, and the region ahead of this cloud would be positively charged due to the presence of the relatively stationary positive.ions. This condition is shown in Figure 1.1. However, such a configura- tion of charge is obviously unstable. Electrons from both ahead of and behind the positively charged region will now flow into this region with the result that it soon becomes negatively charged, leaving two new positive regions ahead and behind. These new regions are then filled with elec- trons from both sides, and thus the process continues with two waves of varying charge density being propagated along the beam, one forward and one backward. These waves travel with a velocity which is characteristic of the beam and which is superimposed on the average velocity of the beam. An alternative way of looking at the phenomenon is in terms of fields rather than charges. An axial electric field exists between the regions of high and low electron density, and since this field is time-varying it produces an accompanying magnetic field. Thus two electromagnetic waves are propagated along the beam, one forward and one backward. Tonks and Langmuir2 in their work on discharge tube -..+- -; +. .. .... ... . r -. .... .... i .. ... .......... ........ . pI ---,+- s-4 4 - -- + d + p+ + -. ,- - -+ ... "`ii- I + + - i- --I -- - ... + + 1 + + + '-. -+ + + 1. + -- ., i p + + { , -- +t. S. ^ - ..... + .. . i.-- i--- .- ++ 1Z --- . .. .--. 1 -.. ... ... ... ... ,+ : - . ,, -. .. .. ... .. ; .. .. .. .. + + j + r ... .. + + + + .. .. .. .... .. S:- + .. .. ... +:I .... + ; -.. + _- -- -+ i-- .... .... - I + --j- --- '- -- t ; + -- U 4 I - It I I" -' i . -:--!~~~~~~~~ ~~~~ ~ -, ~ -- i -- -- -- .' *''.<- '*J' -- :-- ---------------------- -. -- -- --- r.---^- ---, + .. II . .. ... ++m *- -:= SI Io o- - + +i P ^ *! ^ 0 *+ :" :... ... .! I o,0 .i... . . : .. . . -'E- 1 % ~ ~ ^ I" -r I 0' i *+ : + = + + i : : , 1 + i - i .+ .. {. + .. .. + - . ... +, I *I' 1 : ._ -: - phenomena have shown that the natural angular frequency of oscillation of the electrons due to a disturbance in the plasma (the name given by them to the region in which the electronic charge is neutralized by the presence of posi- tive gas ions) is given by (1.1) Cp= I vpc where ( is the electronic charge density (a negative num- ber), C is the charge on the electron (a positive number), yv is the mass of the electron, and 6 is the permit- tivity of the medium. MKS rationalized units will be used throughout this work.' Bohm and Gross3 have developed a similar theory for oscillations in electron streams by assuming a solution of the form exp j ( 0() obtaining the eigen-equation relating W and P and solving the equation for w with fixed r The presence of complex values for oW is in- terpreted to mean instability in the beam and to give the frequency of oscillation. However, Twiss4 casts doubt on the validity of this approach and shows that in the case of the two-velocity electron stream where such an analysis would indicate instability, amplification actually occurs rather than oscillation. The conclusion drawn is that there must be two beams traveling in opposite directions to produce the feedback necessary for space-charge wave oscil- lations to occur. Another logical possibility for oscilla- tion, not discussed by Twiss, is the case where a single beam is traveling in a circular path, as is the case in the cylindrical magnetrons to be discussed later. In addition to these natural-period waves which may be excited by some transient disturbance of the plasma, there is, of course, the possibility of the forced excitation of space charge waves of any angular frequency u by the ap- plication of a time-varying electric field of that frequency to a portion of the plasma. The velocity of propagation of such forced vibrations will be of interest and will now be determined. Haeff7 has analyzed the more general case of an electron beam with a continuous velocity distribution, and we shall consider his results when we discuss the electron-wave tube in Chapter II. For the present, however, let us consider a beam in which all electrons travel along the z axis with the same average velocity, vo, and in which the average charge density, Pa is constant. We now apply a longitudinal time-varying electric field of angular frequency, uJ to the beam and determine the velocity of propagation of the space-charge waves which are excited. The analysis is es- sentially the same as that given by Harman in Reference 1. we make the assumptions that solutions will be of the form exp ( W -- P ) and that the alternating components of charge density, P, and of velocity, -It are small compared to the average values. The charge density and velocity in the beam may be written (1.2) P = -o op (1.23) (1.5) r o l + Ir( Three fundamental laws will be employed to produce a relationship which can be solved for P which in turn gives the phase velocity of the waves. These laws are (1) Poisson's equation, (2) the continuity equation, and (3) the force equation. First, applying Poisson's equation V-V = -V - or and, remembering our assumption that the variation of all alternating quantities is as exp ( we have (1.4) = Here only the a.c. component of charge density is used since it is assumed that the average charge density, eo is zero due to the presence of positive ions. Or stated otherwise, the potential in the beam is assumed to have only an a.c. component, so when the a.c. and d.c. terms of Poisson's equation are equated respectively, the result is equation (1.4) plus a second equation, Po = o . The current density, L in the beam is the product of charge density and velocity. This gives, from (1.2) and (1.3), (1.5) L = P o + .-o + o l = o +* , Here, applying the small-signal approximation, we have neglected the second-order product P, T, Since the cur- rent is due to electron motion alone, the positive ions of the plasma being considered stationary, the value of Po is the electron charge density and is not zero as it was above. Making use of the continuity equation in one dimension, we have or (1.6) W o = r r(,o +-o,;) 8 The force equation in the one-dimensional case is d r aV (1.7) yv d e . The derivative may be expanded and (1.7) rewritten in the form v(1.)- e V (1.8) Tt d )j from which we get (1.9) u.-vr rv If (1.6) is now solved for v1 we obtain (1.10) SPo r Substituting (1.10) into (1.9) yields (1.11) * n II- o, ~ h Or (1.11) may be solved for to give e- pr P V ^ n.f _ ^+Iro1 (1.12) r r' (\4AJrv' We.next substitute this value for the a.c. charge density into (1.4) to obtain an equation which can be solved for r it is e P (1.13) 1 - (w +.d rro) We have seen from (1.1) that the quantity in the numerator is the square of the plasma frequency Op Making this substitution and solving for r we obtain the desired re- sult. It is t* WA)p (1.14) C = j Since the propagation constant, r of an unattenu- ated wave of frequency uw traveling with phase velocity vp is C = J vp we see that equation (1.14) corresponds to two waves-having phase velocities of and L Wp _-ro respectively.. That is, when u >7A p the two space-charge waves are propagated along the beam with respective velocities that are slightly less than and slightly greater than the average velocity of the beam. A number of practical electron tubes make use of the fact that space charge waves may be propagated along an electron beam. Among these are the klystron, magnetron, 10 traveling-wave tube, electron-wave tube, and traveling-wave magnetron. It will be of interest to compare the methods of analysis which have been used on these various' tubes and to consider them as devices belonging to the single class of space-charge wave tubes rather than as a group of unre- lated devices. This will be done in Chapter II. CHAPTER II ANALYSES OF SPACE-CHARGE WAVE TUBES As stated in Chapter I, our purpose in comparing the various methods of analysis of several tubes will be to point up similarities, not differences. Obviously a de- tailed account of each method of analysis, some of which are the subject of lengthy papers or of books, would be out of place here. What we shall attempt to do, though, is to see the physical picture underlying the operation of each tube, the fundamental laws used in the analysis, the sim- plifying assumptions which have been made, and the end re- sults. The analyses, with the exception of the treatment of initial conditions in the wave approach to the klystron, are essentially the same as those given by the authors cited. The Klystron A two-cavity klystron is shown in Figure 2.1. An elec- tron beam is formed in the electron gun and is accelerated by the d.c. anode potential Vo. In passing through the gap of the resonant input cavity, the beam is further acceler- ated or decelerated by the alternating gap voltage V1sinWltl, CAITYR OWTUT CAVITY K\H-a.BNH BEAM FIGURE 2ol .ELBG Su - - SCOLLS~rCTOB where tI is the time at which an electron passes through the input gap. Electron "bunches" appear in the drift space as electrons which have been accelerated catch up with elec- trons which have been decelerated. The bunched electron beam passing through the gap of the output resonator induces a gap voltage which has a large component at the input angu- lar frequency W1. If the output cavity is tuned to this frequency, the result is an output voltage which is an amplified version of the input voltage. Obviously, if a portion of the output is fed back into the input terminals in the proper phase relation, the device will also serve as an oscillator. The most common method of analysis of the klystron9'101 is to consider the .particle mechanics problem of an electron acted on by steady and alternating forces and from the re- sult to obtain the alternating current produced in the beam. The fundamental laws used in such an analysis are conserva- tion of energy, conservation of electric charge, and the equations of classical mechanics. Assumptions which are made in the most elementary theory are (1) small signal (i.e., the a.c. voltage V1 is small compared to the d.c. beam potential Vo), (2) space-charge forces are negligible, and (3) the transit time of electrons across the resonator gaps is negligible. A more elaborate approach in which these as- sumptions are not made gives correction factors which must 14 be applied to the results of the simpler analysis. The energy equation is (2.1) "T ''= V and since V in this case is the sum of the d.c. and a.c. potentials seen by an electron, there results (2.2) t = -;; (Vo+ V, sartW t.) The time that it takes an electron to traverse the S distance S, between resonator gaps is T = 2, which, making O v use of (2.2), results in an approximate form for the transit time (2.3) T= To (I- si- n ) where To is the transit time of an electron at the d.c. beam potential, Vo. A conservation of charge equation is now written. It is (2.4) Io di, = T, t > which says that if the quantity of charge Io dt1 passes through the input gap in time dtl, this same quantity of charge will pass through the output gap in time dt2 and will have the new rate of flow, 12. The time, t2, at which an electron arrives at the out- put gap is (2.5) t, = ,T = t, + To To o51t . Equations (2.5) and (2.4) may be combined to give an expression for the current 12 at the output gap. It is Io (2.6) - where x is a constant called the bunching parameter and is given by (2.7) r = TT N V, N being the number of oscillation cycles corresponding to the average transit time To. Equation (2.6) is the desired current expression except that it is in terms of the departure time t1 instead of ar- rival time t2, and since the relation between t1 and t2, equation (2.5), is a transcendental equation, 12 must be presented graphically as a function of t2. A Fourier series analysis of this curve gives the expression (2.8) I+ = Io[ I +2J.1 () s;J, t- T7N) + 2 J,(X)i S ;(uij-ATTTN) where Jn is the Bessel function of the first kind of order n. The second term is the one of interest in an ampli- fier and gives for the fundamental component of output cur- rent (2.9) Iz, = Io J.() s (.tL r) A transconductance, gm, is defined for the klystron as the ratio of the peak value of fundamental output current to the peak value of input gap voltage. The voltage gain expression may then be written by considering the equiva- lent circuit of the output cavity. This gain equation is (2.10) Voltage gain = -= 9m RR cos . V, R5+RL Here R is the shunt resistance of the output cavity, which includes cavity losses and beam loading effects, RL is the load resistance, and 42 is given by (2.11) tn= s' l)x (R5+ RL))XS where Xs is the shunt reactance of the cavity. As was mentioned previously, the effect of the simplifying assumptions made in this analysis have been the subject of many investigations, but they will not be dis- cussed here, since our object is not a thorough treatment of klystrons but rather an understanding of the general method. Also, we shall not consider the subject of reflex klystrons, although it should be mentioned that the same approach may be used to find the beam admittance, 12/V1' When the conductance component of this admittance is nega- tive and greater in absolute value than the cavity shunt conductance, oscillation will occur. An alternative analysis of klystron operation, and one which is of more interest here since our subject is space- charge waves, is one which considers the effect of the space- charge waves which are produced by the upsetting of the equilibrium conditions in the plasma when the beam is acted on by the external field in the input gap. As we saw in Chapter I, two unattenuated space-charge waves propagate along the beam, one faster than the average beam velocity and one slower, we shall consider only the case where w > >up This condition is not necessary, but it allows us to see the physical picture without a great deal of mathematical embellishment. Using the condition W >j'p in equation (1.14) and combining it with equation (1.10) gives (2.12) Since the positive sign corresponds to the wave which is being propagated with a phase velocity greater than Vo, we see that the density and velocity variations of the fast wave are approximately in phase while those of the slow wave are approximately 180 degrees out of phase. The total velocity and charge density at any distance z from the input gap may then be written (2.13-) '- where the subscripts s and f represent slow and fast waves, respectively. The velocity of an electron passing through the input gap is determined by the sum of the d.c. and a.c. acceler- ating potentials and is given by (2.14) o) = where V1 is, of course, assumed to vary as the real part of exp j W t. But if V1 < Vo, (2.14) may be written in the approximate form (2.15) where S) we- Comparison of (2.15) and (2.13) for z = 0 shows that the a.c. component of velocity is Vf (2.16) VV = v, = +f * Then from (2.12) (2.17) p = PO = P4/ 9 and from (2.12), equation (2.17) may be written (2.18) P'= I-f ^- ( v) The a.c. component of current density, il, is the a.c. part of the product of p and V and, neglecting second- order effects, was found in (1.5) to be , = P, -0r + "; PO _ (2.19) Substitution of (2.16) and (2.18) in (2.19) gives (2.20) = P. z -. r' at z = 0. We now impose the condition 1, = 0 at z = 0 and obtain for the two components of a.c. velocity V, i "If 8Vo'o VC/ (2.21) , and for the components of a.c. charge density J V, But we are considering only the case for which W >>p , so the ratio is small compared to 1 and may be neglected. The velocity and density waves may then be expressed approx- imately as *.), a) (2.23) o-^ b W ^ ^ P P- +(~e)EdV The a.c. current density now becomes (2.24) La 4V0 'V p puJ' Again making use of the condition that w >pp we see that the current becomes largest for the values of z which make sin = 1. In other words, the output gap should be placed at a position J- x s away from the input gap if it is to intercept maximum a.c. current. Equation (2.24) then tells us that the magnitude of the a.c. current density, 12, at the position of an output gap located -5 I- meters from the input gap will be (2.25) L I a We shall carry this analysis no further since it now proceeds 'in the same fashion as the mechanics problem ap- proach. We are now in position, however, to compare the currents obtained by the two methods. To compare equations (2.9) and (2.25) we remember that we have assumed in the wave analysis that V1 o- Vo. If we use this same condition in equation (2.9) we may use the small-argument approxima- tion (2.26) 22 The value of Is in (2.9) then becomes (2.27) 1 Z = o V- . But N is the number of oscillation cycles in the average transit time, which, for the gap spacing I L becomes so (2.28) N /= 4 J - So (2.27) now may be written (2.29) TI = Tr W 4- Vo &p Thus there is a factor of TT difference between the currents predicted by the two analyses. Equation (2.29) was derived with fewer approximations and we should expect it to be more accurate than (2.25). Experimental evidence shows that this is true. However (2.29) cannot be expected to give precise results either since we assumed that a.c. variations of charge density and velocity were small com- pared with the corresponding d.c. quantities, which is def- initely not true in the usual klystron. The thing which we have not yet investigated, and per- haps the most important feature from our space-charge wave point of view, is the physical picture of the bunching process in the wave analysis. We found that two waves of charge density are propagated down the beam. The propa- gation constant of each was purely imaginary, so no ampli- fication takes place, but there is an interference pattern produced. At some distance down the tube the two waves will be in phase and will produce maximum charge density. We found this distance to be M P meters. The Electron Wave Tube This tube follows naturally our space-charge wave approach to the klystron, since the electron wave tube problem is solved in exactly the same npanner as was the klystron problem. The electron wave tube may be con- structed in the same fashion as a two-cavity klystron; i.e., it may have an electron beam passing through the gaps of input and output resonant cavities. While the presence of resonant circuits eliminates one of the advantages of this tube, namely, broad bandwidth, nevertheless such an arrange- ment is perfectly feasible and might perhaps aid in the transition of our thinking from the klystron to the electron wave tube. The fundamental difference between the tubes is the fact that the beam of the electron wave tube is not a single- velocity beam. While the general case would be one of a continuous distribution of velocities in the electron beam, let us, for mathematical simplicity, first consider the case of a beam having electrons of two average velocities, Vo, and voL It will be unnecessary for us to consider the details of this analysis, since it is precisely the same, step by step, as the analysis presented in Chapter I for the propagation constant, F of the space-charge waves propa- gated in the single-velocity beam. The only difference is that equations involving a.c. quantities now have two parts instead of one due to the presence of the second stream of electrons. The resulting equation for r analogous to equation (1.13), is (2.50) + dUo, w'. r) Since the solution for r in (1.13) differed only slightly from ~ we postulate a similar result here and write (2.31) r =-o where vm is the arithmetic mean of to, and Uo ; that is (2.32) V-r. = Vo, = -o, + . We now substitute (2.32) into (2.51) and (2.31) into (2.30) and seek to solve the resulting equation for oC Equation (2.31) tells us that a real part for a means either a gaining or an attenuated wave, since the wave varies as exp (- r ). If d is assumed to be very small compared to either Vro, or urL (i.e., the beam velocities are close together) and if the two plasma frequencies Jp, and Wp, are assumed equal (average charge densities of the two beams the same), an approximate solution results which is (2.33) WP =V. tri 4 The equation for the normalized propagation constant, , has real roots for values of y rJ, which is a measure of the velocity separation of the two beams, between zero and -F2. One of these roots is negative, which means one of the four waves indicated by equation (2.33) will be amplified. Of the other three waves, two are unattenuated and the other is attenuated. When the velocity separation factor -,p is greater than F2, four unattenuated waves are propagated, and while interference of the type found in the klystron may exist, there is no amplification. Within the range where --- is real it has a maximum value of 0.5 when /p = / The maximum electronic voltage gain of the tube is then (2.34) 9b); (d = zo og,9 exp(o.s )P. = 4. 54 * (234 a' I~. r" r r This exponential increase in the wave will not continue indefinitely, however, since saturation will limit the a.c. beam current to a maximum value approximately equal to the d.c. beam current. If the input and output cavities are replaced by non- resonant circuits, say by helices, then amplification should occur over a very wide frequency range. And such is indeed the case, wide bandwidth being one of the outstanding fea- tures of the electron wave tube. We shall not extend the analysis to the case where the two streams of electrons have different average densities, since, as stated previously, our object here is the under- standing of general principles rather than a thorough analy- sis. The case of a continuous distribution of velocities in a beam, rather than only two velocities, will, however, be considered when the plane magnetron is discussed. Here again we shall see electron wave type amplification taking place. Before leaving the subject of the electron wave tube, we must not forget our primary purpose in comparing the analyses of various tubes; i.e., we should get the physical picture of the manner in which amplification occurs from the space-charge wave point of view. Suppose that the, two space-charge waves with which we are now familiar have been set up in each of the two streams of the two-velocity beam. The waves of the faster beam will be moving through those of the slower beam. Let us consider a region of high elec- tron density in the faster beam as it overtakes a similar region in the slower beam. Coulomb forces acting between the two regions will cause a decrease in the velocity of the fast bunch and an increase in the velocity of the slow bunch. The result is an intermingling of the bunches which gives an increased charge density. This in turn produces an increased axial electric field with a corresponding in- crease in energy stored in the field. Thus we see that the energy for the amplification process comes from a decrease in the kinetic energy of the electrons of the faster beam. The Traveling-Wave Tube This tube is similar to the electron wave tube in that interaction between waves takes place along the length of the tube and results in amplification. Here, however, the interaction is not between space-charge waves but between a space-charge wave and a guided electromagnetic wave trav- eling near the beam. The structure used to guide the elec- tromagnetic wave may be one of many types, but the helix shown in Figure 2.2 is a commonly used form. Its purpose is to guide the wave along near the beam with a phase veloc- ity which is near the average beam velocity. For this rea- son it is called a "slow-wave structure". Analyses of the traveling-wave tube have been of two ------ -- .T 4 - -1--- 717-7' I~~ I~ j *~~~~ 4 .-717 K-I *1~~~~ii 71--7j -I L I. ; - .717.: -f--,.. -.;.I-_ -4- -:: 1- -4- f4 ... JI 77 --f 4-4 -7 *. I, L- FL Ei cm f -c -4 -c+ _.. 414$ 7 NsI 412 7144 j WW. I' ____ mi -7I L I--T types; in one13 a circuit approach is used and in the other1415 a wave approach. The object in either method is one with which we are now familiar; i.e., a solution is obtained for the propagation constant r to determine whether amplification is possible. The problem is attacked from the circuit point of view by using a lumped constant equivalent circuit for the slow- wave structure as in ordinary transmission line theory. Transmission line equations are used to determine the volt- age which would be excited in this circuit by the beam cur- rent. Then the force equation and the continuity equation are employed to determine the convection current due to a voltage being propagated along the circuit as exp (jC t-r ). The two resulting equations must be consistent and can be solved for r The assumption is made that a.c. varia- tions of electron velocity are small compared with the aver- age velocity. It is also assumed that the beam and slow- wave structure are in such close proximity that all the dis- placement current due to the beam flows into the equivalent circuit as an impressed current. The solution of the transmission line equations results in the equation (2.35) V = r 30 for the voltage at any position along the line. Here Fo is the propagation constant of the line in the absence of the beam, and Zo is the characteristic impedance of the line. These two line constants are given by the familiar expressions in terms of B, the shunt susceptance per unit length of line, and X, the series reactance per unit length of line. They are ro = j ^ (2.36) X o= T /Z1 I1 in equation (2.35) is the a.c. component of beam current. The second expression relating V and Il is obtained by an analysis similar to that used in obtaining the klystron a.c. beam current by the wave approach. The force equation is (2.37) e e at + V where vo and vI are again the average and a.c. parts of velocity, respectively. The derivative is expanded as in (1.8) and when the magnitude of v, is neglected with respect to vo the resulting equation is _- ' (2.38) -r) The continuity equation is now employed to obtain a relation between a.c. charge density p, and a.c. current density il. With the assumed exp (tJt-Pj) variation with t and z, this becomes (2.39) p. The total convection current density is (2.40) L LO. +. =CL(v. )(po+P.) which, neglecting the product v. gives (2.41) L, = foT + p o Substituting (2.38) and (2.39) into (2.41) yields r4) (2.42) L, = vo(-r L or, writing (2.42) in terms of current I rather than cur- rent density i, we have Io p. r v (2.43) I. = J v -r Here To is the d.c. beam current, Vo the d.c. beam potential, and .o = * Now equations (2.43) and (2.35) may be combined to obtain the desired expression for r It is .i. A. r, r (2.44) V(r -re(.r)I = 1 . To solve this equation for f we use the same technique that was used in the case of the electron wave tube; i.e., we look for waves which are traveling with velocities near the average beam velocity vo and write (2.45) r= + + = o + where oL is assumed to be small compared with o Since we are looking for a wave which travels at a speed near that of the electrons, we will consider the case where the d.c. beam velocity is equal to the velocity of the wave in the absence of the beam. If (2.45) is then substituted in (2.44) an equation in oc results which when solved shows that three waves are propagated along the tube. One is unatten- uated, one is attenuated, and the other, the one of interest, is amplified. A fourth root for the equation in r which was lost in the mathematical approximations above, shows that there is an unattenuated wave propagated in the -s direction. The negative real part of the root of oC which shows amplification is T P ) 4 The voltage gain in db is then 9,.t (ab) = A + zo l0,5 exp v P4 X ,v J . The factor A is an attenuation term which is due to the fact that the input voltage is divided equally among the three waves which travel in the +z direction. Its value in this case can be shown to be -9.54 db, if N is large enough so that only the gaining wave is significant at the output. In terms of the number of wavelengths N in the interaction space the gain is (2.46) ja.n (db)= 9. 4 + 47,3 CN, where C = ( ) 13 This parameter may be put into a more general form which does not include the characteristic 3 EL impedance of the line. It can be shown that C g-p sVo where E is the peak magnitude of the electric field acting on the electrons, and P is the power flow. The field-theory approach to the traveling-wave tube has been carried out by Chu and Jacksonl4 for the particu- lar case of a cylindrical helix with a cylindrical electron beam along the axis of the helix. The general method will be outlined before any of the details are presented. The tube is divided into three regions. These are (1) the region inside the beam, (2) the region between the beam and the helix, and (3) the region outside the helix. The wave equations for the axial components of the electric and magnetic fields are solved in the three regions. This is easily done for the two charge free regions and for Hz (the axis of the helix is the z axis in cylindrical coordinates) inside the beam, since the equations are homogeneous. The equation for Ez inside the beam is inhomogeneous, however, since it includes the axial current density iz. Here, as in preceding analyses, we once again call upon the continu- ity equation and the force equation with the small-signal assumption to give us an equation relating is and Es. The wave equation is then solved for Ez. Maxwell's curl equa- tions are now employed to give the other field components in the three regions. The helix is idealized by assuming it to be a lossless, infinitely thin cylindrical sheet, but conducting only in a direction which makes the angle f with a normal to the axis. This imposes boundary conditions at the radius of the helix which allow evaluation of the constants in the wave solutions for the two charge-free regions. To match the solutions at the radius of the beam a method employed by Stratton16 is used in which a radial wave admittance looking toward the axis is defined for the two regions, and the values of admittance at the beam radius are equated. This results in an equation for the propagation constant which can be solved and the nature of the roots examined to determine whether a gaining wave is possible. For convenience in handling the mathematics the wave solutions are divided into TE and TM waves. It is shown that the TM wave is the one primarily responsible for inter- action between the waves and the electron beam, since only the TM wave has the axial component of electric field neces- sary to produce space-charge waves of the type we have dis- cussed. We shall omit the steps involved in determining all the field components, since these steps are quite straightfor- ward, and start here with the radial admittance functions for the TM wave inside and outside the beam. The normalized radial admittance for the TM wave inside the beam is defined by (2.47) = * Substitution of the values found for Hp and Ez gives for Yri SK I, (n o) (2.48) YrZ = 7 (n r)1 where K p (r L+ K1) I 0 and Il are the wChere 0 Bessel functions of the first kind of order zero and one, respectively, and n is given by I I (2.49) nL Pil + 7TrTb6 Lar.3 LJr - Here b is the radius of the electron beam and Io and vo are the d.c. beam current and velocity, respectively. A corresponding, but considerably more complicated, expression for the radial admittance outside the beam is k" je pcLL,(p aL) Ta(p aj k atpG.) ., (pr k, (p) L PPL--) (2.50) Yo [K, C)ot4, ,rJK -rp )Ka) *(P 1.(prp + K.(p j ( Equations (2.50) and (2.48) may now be equated for r a b, the beam radius. The resulting equation is quite formidable, however, since the desired unknown r is con- tained in p and n, which in turn are in the arguments of the Bessel functions. This difficulty is circumvented by writing a simplified approximate expression for both (2.48) and (2.50). Equation (2.48) is simplified by assuming that nb << 1. This allows the replacement of the Bessel functions by their small-argument approximations and results in an expression for Yri which is K n1- b (2.51) /' = J p * The assumption nb << 1 means essentially that the beam is thin and that all electrons are acted on by the same axial electric field. This same assumption was made in the cir- cuit approach when the gain parameter C in equation (2.46) was written 1I ) '- ( p. \P 8 V. * The radial admittance function Yro, when plotted for real values of p, shows a pole at p = p2 and zeros at p : 0 and p = p1. This curve of Yro vs. p may be approxi- mated in the region of pl and p2, which turns out to be the region of interest, by the expression a Yr.] P-P. E PPI- (2.52) Yr= -(P -P P) P-. (P'P P-FP' Remembering now that p2 (CL+ K) and considering that the phase velocity of the waves is small compared to c, the velocity of light, we may approximately replace jp by F This results in the expression for the radial admittance, (2.53) = c where r. = f and r, = PL Equations (2.51) and (2.53) may now be equated to give the desired expression for r It is (2.54) X, \ This is a cubic equation in r A study of the roots of the equation for a particular set of tube dimensions and d.c. beam current reveals that there is a range of d.c. beam velocity for which one of the three waves is amplified. But for velocities above or below this range, all three of the roots of r are imaginary; i.e., the waves are neither amplified nor attenuated. The velocity at which maximum gain occurs is the synchronous velocity, which means that the velocity of the electrons is the same as the wave ve- locity on the helix in the absence of the beam. The theoretical gain obtained from the wave approach agrees fairly well with that obtained by the simpler cir- cuit approach. As in previous analyses, we shall not go further into the wave approach, although there are still many points of interest which we have not investigated. Instead we shall look for.the physical picture of the amplification process in the traveling wave tube. The theory shows that for an electron beam traveling with a d.c. velocity equal to the phase velocity of the free wave, the phase velocity of the forced wave will be lower than the electron velocity. This proves to be true even for a beam velocity somewhat lower than the synchro- nous velocity. So if an observer werq to station himself in a coordinate system which is moving with the velocity of the gaining wave, he would see the electric field of the wave as a static field and the beam drifting slowly by in the positive z direction. Electrons would see alternately a retarding field, then an accelerating field, as they drifted through the wave. They would move faster in the accelerating field regions and slower in the retarding field regions. The net effect would be a bunching of the beam by the wave. But while the wave is bunching the beam it is also extracting energy from the beam as it slows down the bunches in the retarding field regions. If the process were allowed to continue, complete bunching would result and electrons would then be slowed down below the wave ve- locity by the strong retarding field. This would result in a loss of synchronism and amplification would cease. The normal traveling-wave tube operates at a much lower level of bunching than this, however. 40 It is apparent that the traveling-wave tube has rather low efficiency, since the only energy that can be trans- ferred from beam to wave is the kinetic energy correspond- ing to the difference between beam velocity and wave velocity. The outstanding feature of the tube, as in the case of the electron wave tube, is the broad bandwidth pos- sible if the slow-wave structure is one (such as the helix) for which the phase velocity of a wave is relatively inde- pendent of frequency. The Traveling-Wave Magnetron This tube is similar to the traveling-wave tube in that there is a wave guided along a slow-wave structure and being amplified by extracting energy from an electron beam. There is a fundamental difference in the energy transfer in the two tubes, however. In the traveling-wave tube we have seen that the beam, in being slowed down by the wave, gives up kinetic energy to the wave. In the magnetron amplifier, electrons in a retarding field region move closer to the anode, giving up potential energy to the wave, with essen- tially no change taking place in their kinetic energy. This change in potential energy can be quite large compared to the change in kinetic energy which takes place in the traveling-wave tube. The traveling-wave magnetron, there- fore, has considerably higher efficiency than the traveling- wave tube. Analyses of this tube have again been of two forms, the circuit approach and the wave approach. In the former method, which we shall consider first, Pierce1 has extended his circuit analysis of the traveling-wave tube to include transverse motion of the electrons and the presence of the static electric and magnetic fields. The voltage equation for the transmission line in the traveling-wave tube case was rP1. ar, (2.35) V= r where the assumed variation exp (dt -P ) is understood but omitted for brevity. In the same analysis the linear charge density f was found to be r uJ (2.39) -7d Combination of these two equations gives _,- ,.,, r o fO (2.55) V = 7 - We might intuitively expect an equation of the same form in the present case with the replacement of the linear charge density, by some term which takes account of the fact that there are variations in charge density in the transverse plane and that the displacement current into the line is now a function of the distance of the charges from the line. Such an equation does result, and we shall use it as a starting point here without going through the prelim- inary steps in its derivation. It is (2.56) V = -& - Here 4 is a function of x and y (beam and wave are travel- ing in the z direction) which, when multiplied by the line potential V, gives the potential in the vicinity of the line, and '= The function 4 is assumed to be given by the equation (2.57) C4 = r, t - By making use of this assumed variation of we may rewrite the potential expression of (2.56). It becomes IT (2.58) V =_____ (7L41L r where As in the case of the traveling-wave tube analysis, the force equation and the continuity equation are used to obtain another expression relating V and F which can be combined with (2.58) to eliminate V. The method of pro- cedure is the same, the only difference being that in the present case there are the added complications of motion in two dimensions and of the presence of the static magnetic field B. The equation which results is ^ V r fo^V (dQ -f)td.&"J roo Here M L where Uo) is the cyclotron radian frequency, If (2.59) and an equation for r derived from the con- tinuity equation are now substituted in (2.58) the desired equation for P results. It is (2.6o) ^r- r where H = L.) O It will be remembered that Po is the propagation constant of the wave on the slow-wave structure in the absence of electrons. Again we look for solutions where the propaga- tion constant of the waves in the presence of electrons is not greatly different from Po and let (2.61) r= r(( ot). With the assumption p4< equation (2.60) may be reduced to a fourth degree equation in p We obtain four wave solutions, one of which increases in amplitude with z. This amplification occurs for two ranges of electron velocity, but the one of interest, as in the traveling-wave tube, is that for which the electron velocity is near the circuit phase velocity. One of the values of p shows the possi- bility of a wave which increases in the -z direction, but this is not generally utilized. The gain in this tube varies with the quantity H much the same as the traveling-wave tube gain was dependent on the factor C. But we see that H is -proportional to ( whereas C was proportional to Since is a factor less than unity, the gain of the traveling-wave magnetron will be somewhat less than that of a traveling- wave tube having the same circuit, beam current, and beam potential. However, as was mentioned earlier, the nature of the energy conversion makes the efficiency of this tube much higher than that of the traveling-wave tube. The wave solution for the traveling-wave magnetron has been carried out by Brossart and Doehler22, for the plane magnetron, which is a limiting case of the cylindrical mag- netron with large anode and cathode radii and small anode- cathode spacing. Two types of tube construction are considered. -In one the electrons are emitted from a cathode which is external to the interaction space and enter this space parallel to the electrodes and with a velocity determined by the balance of static magnetic and electric field forces. Space-charge forces are neglected in this case. In the other system the inner electrode is the cathode. Here the electron trajec- tories are epicycloids, and space-charge is neglected only for the case of small currents. The analysis shows that the gain expression is of the same form in the two cases, the only difference being in the magnitude of one of the param- eters in the final equation. The slow-wave structure is a flat helix with its axis along the y-coordinate axis. Wave solutions are then as- sumed to be of the form exp (?)J Pi) The system is assumed to be infinite in the z direction. The first step in the analysis is the solution of the wave equation for the transverse and longitudinal compo- nents of electric field. These components are (.2o6r) _' -r (2x2 ,,i 1 h 'b)% rd) (2.62) E -. ii-sink (4 Pd) Here AV defines the amplitude of the wave and depends on the initial conditions, and t is the transit time given by 2= i to where to is the time at which an elec- tron enters the interaction space; d is the spacing between electrodes. Next, the electron trajectories under the influence of the time-varying fields are assumed to be small perturba- tions, dw and Jdt on the d.c. trajectories. The force equation is then used in combination with equation (2.62) to obtain expressions for dS and For the linear trajectory case these are dxr (2.63) P;- ra v W f. J) ef S sC ( I d) where Lo is the cyclotron frequency, -o is the equilib- rium position of the electron, and f = "'j'r7 o being the d.c. beam velocity. In arriving at (2.65) the approximation )J1< 4 o has been made. For the case of cycloidal trajectories the expressions are more involved. They are (2.64) dwi. J ' ,. -- I Equations (2.64) are deceivingly simple, because d. v r I AV., r and e T 8, s (' Pd) A with Q and T given by 0.Q J. (I[ t Ita ^ ' ( Xr J1 ( x j,) and Tr J. (^ [ tr.r + X + rx,))s where Xr is the amplitude of the cycloid. The next step in the analysis is to make use of the continuity equation in conjunction with the equations for efX and cf to obtain an expression for the longitudinal component of a.c. current, iy. The small-signal approximation is made. The result is (2.65) Z : r I rd J co(r x) 6 r xi.r Vo if .Ki4P rd) for the linear trajectory case. Here Vo and I are the d.c. potential between electrodes and the d.c. beam cur- rent, respectively. For the cycloidal trajectories X.- X. (2.66) L Jd d r .4- dX if, t? Xr Vaf S Aig rPd) X:o The final step of the solution is to write a conserva- tion of energy equation (2.67) dP t dP, + d P = o . Here -dP is the apparent power given up by the electronic current along a small path dy, dP, is the apparent power taken by the helix along the path dy, and dcP& is the in- crease along dy of the apparent power which is propagated in the direction of the wave. These three quantities are found respectively from -dld (2.68) dP = P + .)= do. ~dPL H e r e.- h p r te i e ph e e curet log sal pthdy f s heaparntpoe where the asterisks indicate conjugate quantities. Y and 0 are the attenuation and phase constants of the wave, with the zero subscripts being used to denote these quanti- ties in the absence of the electron stream. The coupling resistance R must be determined from the wave equation solutions on the particular slow-wave structure and is given in this case by (2.69) R; @(d V 6 where o and 4 are factors which depend on the dimensions of the helix. If now the proper substitutions are made in equation (2.67) and the real and imaginary parts are equated, the result is (2.70) ( >- ) + D , The factor D is a constant for a particular tube type and marks the only difference in the result for the two cases considered here. The quantity v in (2.70) is the phase velocity of the wave in the presence of electrons. A study of equation (2.70) shows that there are two waves propagated in the forward direction, one of which is strongly attenuated and the other strongly amplified. Each is propagated with a phase velocity v equal to the arith- metic mean j (V-r + I) of the electron velocity and the free wave velocity. The value of Y the real part of the propagation constant, is given by k, V JaX I s ') y wd6P 6'd (i -+ (2.71) + 4 X s4d) ca- zoM 4 ; where q9 is the pitch angle of the helix, and LeooXr) z [cos.jr -1 J ) 1-t .Oxr) +jx 3.c -I )[ .+ X r) + ( X. + l2 0B ,) -" + (,. x In particular, for the case of electron velocity and free wave velocity the same, and for a lossless helix, the gain is shown to be (2.72) gao.t = + 8.7 r. db. with r given by (2.71). For a physical picture of the amplification process let us examine the case of electron velocity equal to wave velocity (we remember that this case produced no amplifica- tion in the traveling-wave tube). The interactions which take place are more readily observed if we place ourselves in a frame of reference which is moving along in the y direction with the velocity of the wave and electrons. Then the traveling-wave appears as a static electric field as shown in Figure 2.3. Electron motion is now influenced only by the r.f. field, which appears to be stationary, and the static magnetic field, since there is no static electric field in our moving frame of reference. Electrons are initially stationary with respect to the fields, and, therefore, there is no magnetic field force, S=- e(ix ) However, as soon as an electron begins to move under the influence of the electric field, there will also be a magnetic field force. For example,.an elec- tron at A in Figure 2.3 will be forced to the right by the electric field (arrows on the field lines represent lines of force on an electron and are opposite to the electric field direction). As it moves toward the right, there is a mag- netic field force which is directed downward, so the net ef- fect is a drift to the right and downward, away from the anode. An electron at B will move up and to the right, one at C will move down and to the left, etc. The net effect of the action of the fields will be a bunching of the electrons ---. 4 - -I I. Iw Ki- r-- -.- so t I -I-- -- -4 -~- m -d I -+A U4 4~ K --4-I-- Th~wmvimIL7'7IIlir in the region of the retarding field between B and C. And in this region the magnetic field force carries the elec- trons closer to the anode causing them to lose potential energy which they give up to the electric field. In space-charge wave terminology, and returning to a stationary frame of reference, we have a traveling wave and a space-charge wave moving along together and mutually aug- menting each other as they travel down the tube. In addition to the high efficiency (around 40% in ex- perimental tubes) already mentioned as a desirable feature of the traveling wave magnetron, the tube also has the broad bandwidth feature of the traveling-wave tube and of the electron-wave tube. The Magnetron Oscillator The most familiar form of the magnetron oscillator, and the one least susceptible to a complete analytical solution, is the cavity magnetron. In basic principle it is the same as the traveling-wave magnetron amplifier just discussed with the structure closed on itself to provide the feedback necessary for sustained oscillations. And since the device is to function as an oscillator, the slow- wave structure is a resonant one rather than broad-band. Here again we have waves and electrons traveling along in synchronism, this time in a closed cylindrical path. And as before, electrons in the retarding phase of the traveling wave move toward the anode, giving up potential energy as they go, while electrons emitted in the opposite phase are sent back to the cathode. The result is the almost complete bunching and very high efficiency characteristic of a cavity magnetron oscillator. The situation in the magnetron oscillator is obviously one of large-signal, rather than small-signal, behavior, and it is also apparent that space-charge effects can no longer be neglected. This produces a problem of such complexity that a completely analytical solution is too difficult and other methods must be used. One approach23'24 is the method of self-consistent fields in which a potential distribution is assumed, the motions of electrons in this potential field are calculated by numerical methods, a charge density dis- tribution is determined from a sufficient number of trajec- tory calculations, and the potential due to this charge dis- tribution is then calculated. If the resulting potential agrees with the assumed potential, the problem is solved; if not, a new assumption must be made and the process repeated. This approach has provided considerable insight into magne- tron oscillator operation under particular conditions and has confirmed the fact that the revolving space-charge is in the form of spokes which extend to the anode in the regions of retarding electric field. An analysis for the frequencies of oscillation of a cylindrical magnetron with a smooth anode has been carried out by Harris.5 Full account has been taken of the space- charge, but the small-signal approximation has been used. The dynamics of the electron beam is handled by means of a velocity potential. Since this method will be used in the I analysis of the outside-cathode magnetron to be presented in Chapter III, it is carried out in some detail in Ap- pendix I. The analysis by Harris shows that oscillations should be possible in the smooth anode tube at integral multiples of the Larmor frequency, . The mechanism for amplification (and oscillation) in the smooth-anode tube is not so apparent since there is no slow-wave structure. However, since there is a continuous distribution of velocities in the electrons rotating around the cathode, we see that we have the conditions necessary for electron-wave amplification, and if the structure is closed on itself, for oscillation. The problem of amplification in a plane magnetron with smooth anode has been solved using the velocity potential approach by Macfarlane and Hay25 and by Bohm.26 The func- tion of the crossed static electric and magnetic fields in this case is to provide a continuous distribution of veloci- ties in the electron stream, so it might perhaps have been more appropriate to discuss this work in the section on the electron wave tube. However, a tube of this sort qualifies I as a magnetron due to the presence of the crossed fields, and if feedback of the proper phase were arranged, could presumably operate as an oscillator. The analysis by Macfarlane and Hay is similar except for geometry to that carried out in Chapter III for the outside-cathode magnetron, so only the results will be pre- sented here. It is found that amplifying waves can travel along the beam having continuous velocity distribution, called a "slipping stream" by the authors, for all fre- quencies. If the electron velocity varies linearly across the beam from V_. to V1, a fractional velocity aL is defined by (2.73) = It is found that for o4< 0.42 the tube behaves pri- marily like a two-beam tube and has a maximum gain of 2.1 ~ decibels per unit length, where Pp is the plasma frequency and VTo is the average beam velocity. This is compared with a gain of 4.35 decibels per unit length for the two beam tube. This two-beam tube behavior contin- ues up to a frequency Ir Above this frequency the tube has a low gain of a traveling-wave tube nature. For oC 0.42 the tube behaves primarily like a traveling-wave tube having a gain of about 0.53 decibels ' per unit length for all frequencies above the plasma fre- quency. This is considerably less than the gain of 6 ( p) j decibels per unit length exhibited by the traveling wave tube, but the slipping stream tube achieves its amplification without the use of a slow-wave structure. This traveling-wave tube action, as explained by Macfarlane and Hay, is due to.the presence in the stream of resonance layers which act as highly reactive impedance sheets and can guide waves of slow phase velocity in the same manner as a helix or other slow-wave structure. The case analyzed by Bohm is a limiting case of the slipping-stream tube where IV- =--0 and it is shown for this case that no amplification will result. In addition to the traveling-wave type of amplifica- tion and oscillation in a magnetron, there are other types of oscillation possible in which the magnetron is able to sustain oscillations in an external resonant circuit. Two types27 of oscillators, other than the traveling-wave type, are the negative resistance oscillator and the cyclotron frequency oscillator. Since our primary concern is with traveling space-charge waves, we shall not discuss these oscillators here. However, an analysis for the negative resistance characteristics of the outside-cathode magnetron will be carried out in Chapter III. J -*. CHAPTER III THE CYLINDRICAL DIODE MAGNETRON WITH OUTSIDE CATHODE The cylindrical smooth-anode magnetron with the con- ventional arrangement of the inner electrode as the cathode has been analyzed by Harris as discussed in the preceding chapter. The opposite physical construction, a tube with the outer electrode as the cathode as shown in Figure 5.1, will now be considered. As our analysis will show, such a configuration produces some unique results. The velocity potential approach to the dynamics of the electron beam will be used. The differential equation for F1, the a.c. part of the velocity potential, has been de- rived by Harris and applies equally well whether the cathode is the inner or outer electrode. This derivation is shown in Appendix I, and the result, equation (1.26), will be used as a starting point in our analysis. Thus we have -F,7( + ,-L = -'-- 1. V7 ---- ------- -- -. .-T- -j :_ | .. .[ ...-.- l Z ~~ : ,~' ~ ......-.-h-- .. ....- -.....h..... -. ... , 1 .~.1 n~n ^^rn^^__-^^___ i:_ ^^_4--^ --i- ...7 4 Ir tt' S I -' ' -- --___ .-_4-K .-4--_ __ .._- __ __ _ -- -+ -- ... , .... I ^ ^ x- !-- -- -.- m-^" I I'~ -- --- *--- --- .------ -- lI -' i -. .I ........... ... .-; -- ........ .... i - -' ...., r 'E- S: '---' r- --- -.---- t I .: ~ " - .. .. .. .. _.. ...1.. ~. ... .. -.. ... . ___ -iiI S- -.1 _ . -_ I I - , -,- "t i-"---- ---- I'~ I I 4 -~-1 I : I : - __~~ I. Ir Il_ _ Si. j II i I I. ':I Here = I k L m r ' where vo is the d.c. beam velocity and h is the wave number in the assumed variation of F1 as exp j(wk-4e ). We must have the values of vo and of po the steady- state space charge density, in order to evaluate Wp and , which appear in (3.1). An analysis of the static character- istics of the outside cathode magnetron is given in Appen- dix II. The results of this analysis are, from equations (11.9) and (II21), (3.2) Vr. = = 1 -( ./) and (3.3) z eo e k r4 e. where Uo is the cyclotron frequency. Substituting (3.2) into the expression for v yields (3.4) and substituting (3.3) into (1.1) we have LWP r+-- 61 Performing the indicated operations in (3.1) and simplify- ing the equation gives 0 CL LC L (3.5) + *--) -a- Cj = 0 7 where W L 9~L bo and AuJ. 4 A) It is apparent that some approximation must be made which will make equation (3.6) more tractable. Let us as- sume that r>?>r This means physically that we shall restrict our solution for F1 to the region near the inner radius of the space charge when the tube is operating in a cutoff condition. With this assumption only the higher order terms in the coefficients are retained and (3.5) be- comes 62 wd) e -, F', e, (3.6) rduFr dr s r W which reduces to (3.7) where dC F 3 d F, L dr- F ,- F- o d r r dr r' L L VL L-40 The solution to (3.7) is --c, r or, letting Z- + 4*L = -f=4+ (3.8) F C. r + c rt- . Applying the boundary conditions FI = 0 when r : rc gives (3.9) F, = C, r r(r The inward radial admittance, that is the admittance at the inner radius of the beam looking in the direction of the charge free region, will be the same except for sign as that found by Harris5 for the conventional smooth-anode z ++fT magnetron. The admittance expression is dF, 0dr (3.10) y, - From (3.9), dr and Sdf, F; di- Atv-f dr 4 d dr re 4*; dr may be found. 1- r X rz which may be written (3.11) F: dr F r where and , the positive sign being taken. Substitution of (3.2), (3.3), and (3.11) into (3.10) yields It is I r-rc d Law-t',o ^<~ <44 Wo J (3.12) y, = -- * ,/ r -_ - We now direct our attention to the charge-free region in order that we might determine a radial admittance for that region at the edge of the space-charge cloud. The two values of admittance will then be equated. A proper solution to the wave equation in cylindrical geometry gives the following expression for the axial com- ponent of magnetic field: (5.13) H= J( + Nr) where Jh and Nh are Bessel functions of the first and second kind, respectively, of order h, and K = -. C The solution of (3.13) cannot become infinite on the axis in the case of a hollow cylinder, and we reason that the same sort of solution will hold here even though there is a small anode cylinder on the axis. We then retain only the first part of the solution, setting AL = o. We next make use of the Maxwell equation vC)- - remembering that all field quantities are assumed to vary as exp (wt e) and obtain ")A& J .llr), where Au is the permeability of the medium, and J (kr) is the derivative with respect to r of JC(Kr). Then the radial admittance may be expressed as ,__ JL Cr) (3.14) Y= Ee A Cr) We will now make the assumption that rr <'.1 This means physically that the phase velocity of the wave is small compared with the speed of light. This assumption seems justified since the wave, if any, will be traveling in synchronism with a rotating wave of space charge, and we expect the space-charge wave velocity to be not too differ- ent from the d.c. beam velocity. With this assumption we may now use the small argument approximations for Jt and JA and (3.14) becomes Kr (3.15) Y1 = -d ;L We may now equate the two values of radial admittance, equations (3.12) and (3.15), and obtain (3.16) -- = A, Er ^ ~ Recalling that KL= -- u-= LA we may simplify (3.16) to give the desired equation in w This is (3.17) Lao -( ) -4+ ^- = " Since we have assumed solutions of the form exp j(wd-t e ), it is apparent that W must be complex with a negative imaginary part if oscillation is to occur. Application of the quadratic formula to (3.17) shows that will be complex if the following inequality is satisfied: (3.18) 4( -) '- For the assumed case of rl>>~ we see that and the inequality becomes (3.19) 8 -4 )(4-- ^) " Inspection of (3.19) shows that for mr the first term in parentheses is negative and the second term is positive. Therefore, the product is negative and can never be greater than h2. And since h must be a positive integer, an investigation of the relationship between h and m shows that the only allowed value of i< 2 is for h = 2, in which ease 0 = 0. Substitution of 0 = 0 and h = 2 in (3.19) re- veals that the inequality is again not satisfied. So the conclusion may be drawn that, within the limits of the assumptions made, the cylindrical smooth-anode mag- netron with outside-cathode will not oscillate. This is a fascinating possibility because, if, as has so often been postulated, the marked deviation of the cutoff curve of a magnetron from the theoretically predicted curve is due to oscillations, then it should be possible to check experi- mentally the cutoff voltage expression derived in Appen- dix II, equation (11.13), for the outside-cathode tube. In any event, equation (11.13) deserves some special comment. It is repeated here for convenience. e f v f 4 1 (3.20) VC -8. An appreciation of the magnitudes involved here may be obtained by considering a particular tube for which r : 0.655 cm. and ra = 0.0191 cm. For an anode voltage of 1000 volts, equation (3.20) predicts that a magnetic field of only 10-3 webers/meter2 (10 gauss) is required for cutoff. Contrast this with the value of 35 x 10"- webers/meter2 required for cutoff with the same anode po- tential on an inside-cathode tube of the same dimensions. Theoretical cutoff curves for the two types of construc- tion with the same dimensions are shown for comparison in Figure 3.2. Equation (11.21), the expression for the static space- charge density in the outside-cathode tube, is also quite interesting when compared with the corresponding expression for the conventional tube. The two expressions are exactly alike, term for term, each being given by -WO^m e. /, ( -'V (3.21) ao = e + But in the outside-cathode tube I < rc so the charge density is greatest at the edge of the space charge cloud, farthest from the cathode. This should be important from the standpoint of efficiency in the inverted multicavity magnetron, since it puts the greatest number of electrons in the region where interaction with fields takes place. In Chapter II we saw that there are other types of oscillation possible in magnetrons beside the traveling- wave type; this type, we have shown, cannot exist in the ..... I_: .. .. .... ..' .-J = -= := ... :f +' -. . ____ :L-- 2-L | 4 -....I 4- ~ - S -- -" L- T-"" vi __" i. ; :; 1' *" -_- -- __'_ -- ,-**** -* _ S-- ..... .... -+ -L ..... I'-ti. -_^__^_^ _-___^_^-- __ I. ,.- i S -: -. -~~~~ -H --+ j II t .. .1 t sg ._- _. ... i *1 ii 7tt- -j jT T 7i777 t TI _, i i i- -+ ___i .4 I3 : .- ; .. i ----; -~ J--- I i ',' : :11 " I I : t I I-- 11f--' --,- IA - -4------ I $- -- I ------------ 4 -------I--- -----'-+ ---- f---+--- ---,--- i- I i Til I ----q--- - 1 1 : i r-- ... i - -----1-----+-----'--- .... 1 -i i i i i 1 I i- outside-cathode tube with smooth anode. We should, there- fore, investigate the possibility that this tube can present a negative resistance to an external tuned circuit. We shall follow here the approach used by Brillouin21 for the inside-cathode tube. We assume that the potential, the radial current, and the radial position of an electron have d.c. and a.c. components given by V= Vo(r) + V,(r,t) (3.22) I- Io I1 10 ^ =r fo) + 'r,) The radial force equation is derived in Appendix II, equation (11.8), and is (3.23) r = e -+ rie e Br6 Jf and when the value of e from equation (11.9) is substi- tuted in (3.23), the force equation becomes (3.24) nr l"=. t' r r- w t -f ,J dr 4 (& ;Lr We may now define an apparent potential P such that SP (3.25) Yv r = e r Then from (3.24) we see that P V V., where V. is the contribution of the magnetic field to the apparent poten- tial and is given by (3.26) Vc ---- r The force equation now becomes d CG d_ V eSV . (3.27) L d- *-S+ rr r dfL d* c r / r adr But Vo.) = V.) + r, ,vo and Vc(r) = Vc(r.) r, j 1 J r Therefore (3.27) becomes (3,28) 4 d'r -e /d(r) d V(r.) p ~ V, d~ V e V, (5.28) t- (r 7rF /- -;;-r 5W r Equating the a.c. terms of (3.28) gives (3.29) drr -_ _^ e\. ( 3 H<9- { -- d-t *' dr The radial current per unit length, including displace- .ment current, is I =-27~rrr( p- + )=--27rrr-2T- , dt/ it where D is the electric displacement and vr is the radial component of electron velocity. We now note that (5.31) (r (D)- () + (r ), di,)t d and from the divergence theorem in one dimension (3.32) ( ) = We see that (3.31) may be written (3.33) dLrt ) = (r) + r . Comparison of (3.33) and (3.30) gives (3.34) (3.35) r T di ; TT t dV1 (jAt (3.50) where the substitution be JV b=-E 5- has been made. We may now substitute the expressions of (3.22) in (3.35) and obtain t (3.36) (ro+r) V. +r rA)= r-f(rL*rI,)d . If we now neglect the products of a.c. terms (small signal approximation) and equate the a.c. terms of (3.36), the result is (3.37) d t 5 rd L T 1 We first solve the homogeneous equation, i.e., equation (3.37) with the right-hand member set equal to zero, and obtain the natural vibration frequency uj. of r, The solution to the homogeneous equation is of the form (3.58) r,= A 6'' B 6i" where W%. is given by (3.39) 4)V =_ d Le dro/ I d r J- drl The value of Vc is given in equation (3.26), and Vo is given by equation (11.19) in Appendix II. These two poten- tial expressions are equal, so when the differentiations of (3.39) are performed, the natural radian frequency becomes (3.40) W0. - V 7 T In arriving at this result the approximation ->> has been made. We now look for forced oscillations and assume that I1 is of the form (3.41) I, = I E and that rl is of the form (3.42) r, = r4 L . Then equation (3.37) becomes (5.45) ) ( -z)=_ T- ( L r where Z'= o is the transit time of an electron from the cathode to the radius ro. From (5.29), under cutoff conditions, we have (3.44) C = ~ r, and substituting the value of rl from (3.43) this becomes (3.45) ( (A e- ) Equation (3.45) indicates that the radial electric field becomes infinite when uJ-~= We recognize that this is not a physically realizable situation and add a damping factor S to the equation. Thus (3.46) = -- / : E- dr z 716 r. w uw } \ws The d.c. components of equation (3.36) may be equated to give (3.47) = T 2. But Vo as given by equation (II.19) in Appendix II is (3.48) Vo(r) -= -~- '-) where a is the cathode radius. This expression was derived for cutoff conditions, but should hold approximately for . __ small values of Io. Then (3.49) d V / -. ) - = -;- r I I dr 4e. Substituting (3.49) into (3.47) and solving for T we have 7T i'^ I aSo \ (3.50) r= 6 e I r- Now substituting (3.50) and (3.40) into (3.46) we obtain -d X l- E ) -rrC r" (e -t- ) w,'f+- 1" (3.51) = dr The internal impedance of the magnetron is given by (3.52) where ra is the anode radius. This becomes rc. ;s -.dr (3.53) o , / -"d The integration of the impedance expression for the inside-cathode magnetron, which is quite similar to V, tr I; equation (3.53), is the subject of a lengthy analysis by Brillouin. We shall adapt his result to the present case and obtain the approximate result (3.54) T I O1 o " Analysis of (3.54) shows that the condition of physical significance for which the real part of 2 may be negative heresmall. Then the internal resistance is and se e M 5 n b n ir, a. Analysis of ((.54) shows that the condition of physical sigenifica since for which the real part of neg may be negative alueis that ) r large, and c negligible small. Then the internal resistance is (3.55) R We see that equation (3.55.) can be negative for all values of Wx which make co.f( 5. r.j < 0 ii 110" r, 36 r __ I_ ) < -- This means that for values of a) which make R negative the tube is capable of sustaining oscillations in an external tuned circuit. Equation (3.55) shows that the value of R will be greatest when = or when S= ( 7r The first of these conditions gives (3.56) U) The second condition yields (3.57) 1) -, . When the anode radius is quite small compared to the cathode radius, equation (3.57) reduces to (3.56). Thus we see that the outside-cathode magnetron is capable of sustaining oscillations in an external resonant circuit. Oscillations are possible for frequencies which make coS fo r-- ~ '-) but will be strongest for W in the vicinity of 'o/. CHAPTER IV ANALYSIS OF THE SPIRAL BEAM TRAVELING-WAVE MAGNETRON A variation of the traveling-wave magnetron is shown in Figure 4.1. This tube,first suggested by Harris during the course of his work on hollow cylindrical electron beams, operates in the following manner: A hollow cylindrical electron beam is formed in the electron gun. This beam is passed through a radial magnetic field which gives the beam some tangential velocity. The beam then enters the space between two concentric cylinders between which there is a difference of potential as shown in Figure 4.1. This potential is adjusted so that the in- ward electric field force acting on the beam balances the outward centrifugal and space-charge forces. The result is a hollow beam spiraling along the axis of the tube. A slow-wave structure is now wound on the inner con- centric cylinder with a pitch which is equal to the pitch of the spiraling electrons. The physical dimensions of this guiding structure are such that a radio-frequency wave ap- plied to the input will travel down the slow-wave structure 6vw WMBAIw.UAK !RAVLMINGLVAUX KUM"= IPG JI'.1 6-d with a phase velocity which is approximately equal to the average linear velocity of the beam. This gives a wave and an electron stream moving along in synchronism in a spiral path. This appears at first glance to be just a traveling- wave tube wrapped into a spiral, since there is no magnetic field present in the interaction space. However, the sig- nificant difference is that here there is a force field, the centrifugal force, which acts at right angles to the path of the electrons just as the magnetic field force does in the traveling-wave magnetron described in Chapter II. The result is traveling-wave magnetron action. Electrons give up potential energy to the wave by falling toward the center conductor, rather than giving up kinetic energy as in the traveling-wave tube. The advantages of this arrangement over the conven- tional traveling-wave magnetron are threefold. First, the interaction space can be made long,as in the traveling wave tube,without making the tube physically long. Second, no static magnetic field is required in the interaction space; true, a magnetic field is required to give the required tangential velocity, but this radial field is relatively small. And third, analysis8 of the space-charge conditions in the beam shows that the charge density varies as , which means that the majority of the electrons will be \ concentrated near the wave being amplified, where they will do the most good. The tube will be analyzed using the velocity-potential approach as in Chapter III, but in this case both the geom- etry and the space-charge conditions will be different from those of the cylindrical magnetron. We shall consider the particular case in which the slow-wave structure is a helix of rectangular cross-section. It will also be assumed that the actual structure of the interaction portion of the tube can be approximated by a linear structure as shown in Fig- ure 4.2. The problem then becomes one in rectangular geom- etry, but one in which the charge density and electron velocity of the actual spiral structure will be used. Since there is no centrifugal force acting on the elec- trons in the linear system, a static magnetic field, B is added to the system. This field is assumed to vary with y in such a manner that it produces a force on the electrons equivalent to the actual centrifugal force. The derivation of the differential equation for Fl, the a.c. part of the velocity potential, proceeds in a man- ner exactly analogous to that given in Appendix I. The only difference is in the geometry. In the present case, Fl is a function of y and is assumed to vary as exp j( w P" ). Since this differential equation has been derived by Mac- farlane and Hay25 and by Bohm,26 it will be used as the -- i7I I~ ~~~ t -a- -H - I I*-------t------1------4 -7- IA I. -- -. iT t-r 1 :: --.-- -:- i--- : -- 2 L -- : AL I' 444. 1 TIM -~1ciL. I I .1. 7. ------ IL 1 F 7 I I .T :i .-~i -~- -irV -- N. L:4JI LI_ t~ I., .j . 4i 4 --l-l-- -H-- _ - ----: -i ,.- -- t ---ti -5 rj7E41{ _- ___ S-- ---- -~- -- _-- ~- _i _ starting point for our analysis. Thus, Here up is the plasma resonance frequency, and is given by (4.2) V , where vo is the d.c. electron velocity. From equation (1.1), the value of UJ; is (4.3) u- * Thus we see that the velocity potential is dependent upon vo, the d.c. electron velocity, and upon 9, the d.c. charge density. All electrons are accelerated to the same velocity in the electron gun, and since the passage of the electrons through the radial magnetic field does not alter the linear velocity, vo will be a constant. As mentioned previously, the equilibrium conditions established by Harris for the beam show that the charge density varies inversely with the fourth power of the radial position. If this re- sult is translated to our rectangular system, the charge density variation becomes (4.4) P = V Here, Y) is the magnetic flux linked by the electrons at the cathode. The expressions for the potential of the two cylin- 8 drical electrodes are also derived by Harris. These are necessary in the design of the tube, but since they are not used in the gain analysis, they will not be given here. The complexity of the differential equation which re- sults when (4.3) and (4.4) are substituted into (4.1) makes it desirable to seek some simplifying assumptions. We re- call from the discussion of the traveling-wave magnetron in Chapter II that maximum gain occurred when wave velocity was equal to electron velocity. Under these conditions F is equal to I plus a small imaginary part. From (4.2) we see that is quite small for r near and that under such conditions it is permissible to neglect 1 in E-7/' This results in considerable simplification of the differ- ential equation, so we shall consider only the case where the phase velocity of the wave is equal to the electron velocity. For this condition, substitution of (4.2) and (4.3) into (4.1) results in the differential equation (4.5) --- d. J. d The dnd result which we are seeking is a solution for r from which we can determine the gain of the tube. The approach will be the same as that used for the outside- cathode magnetron in Chapter III. An admittance, defined as is found at ys, the lower surface of the beam Ex (see Figure 4.2). This admittance is calculated in two ways; one expression is derived from a consideration of the dynam- ics of the beam, and the other from a solution of the wave equation on the slow-wave structure. The two admittance ex- pressions are equated, and the result is an equation in which r is the only unknown. The admittance, Y1, derived from the beam dynamics has 25 26 been determined by Macfarlane and Hay and by Bor6 for the general case of a linear beam. We shall make use of the result obtained by these authors. It is (4.6) /:F, d j .t Fa- o r-- -M An exact solution to the wave equation in which the boundary is a helix of rectangular cross-section is pro- hibitive. However, this particular problem has been solved 22 approximately by Brossart and Doehler in their analysis of the linear traveling-wave magnetron, which was discussed in Chapter II. These authors simplified the problem by as- suming that the high-frequency fields in the interaction space are due entirely to high-frequency current on the face of the helix nearest the beam, the influence of the farther face and the lateral faces being negligible. The values of E and Hz obtained in this manner are (4.7) EA= -s.' d. d-^^ and (4.8) H = co'" I, Al coAk(d-) e In these equations A1 is an arbitrary constant, d is the spacing between the two electrodes bounding the interaction space, 9l is the pitch angle of the helix, and oC is given by (4.9) 0( z 4- ) The factors p and s are, respectively, the pitch of the helix and the length of the portion of a turn lying in the face of the helix nearest the beam. The admittance, Y2, determined from (4.7) and (4.8) is (4.10) YZ Exco- jY) The two admittance expressions, equations (4.6) and (4.10) are now matched at y,, the lower surface of the beam, to give the desired equation for r It is (4.11) C5 r 5 C- [i Ca3 c +CrL C,,r P Co =o 0 where the coefficients are given by IM-M )coJt-' (4.12) CC = k 4~) ed-)\et) (4.15) C,= I d jdo'- 3S 6 o i. A c t o (4.16) C j = y- 27 ^ e oW0 3 a o. a ar and (4.17) Co '= e/ w' + uJ4 . The factor R which appears in C2, C3, C4, and 05, i dF is the ratio, ., evaluated at y ys, Tt is apparent, therefore, that the differential enuation for F1. erua- tion (4-5), must be solved before r, can be evaluated, The solution of this equation is found in Apprndix TII, The fifth degree equation in f, e-uation (&.11), must now be solved. Let us denote rbv (4.1) r : a +' b From the assumed vpripti)n of the wave as pxi i(wt-rx) we see that a is the phase constant And r the attenuation constant. A positive value for b will indicate a wave which is increasing with x. To solve equation (I.11) we try the value r'= This proves to be a solution to the e-'iation and indicates an unattenuated wave traveling in the -x direction with phase velocity, c When the root,r= = is fsctorpd out of equation (4.11), the result is f.19) r4 +B,Pr ?B r + 3 r + B,=o . . r The cieffi~ rt ntsq PrP p ?,n bv (L.20) 8, ro w w R's 1f (4.21) B- y- o-'cfcod4(^-) ' 3-o (Z .22) 03 Y- v s+o tr, a ckK i/ coit%0Cd- ) and (..23) Bo = ~ / co-V cot a(d-y) "e Ise thp metmhoe of ePsumTfle, uadratio factors to solve the quartic fauation, (,.19). That is, we express equation (4,19) in the form (4.24) (r aL, r + )(r -b, r + b.)= For a wave traveling in the +x direction with Dhase velo- city, ITo i either aI or bI nust ePual Y- Tf w choose a,= and eauete coefficients of like powers of f in enuations (4.19) and (6.24), we find that b, This inAicatrs a Dair of waves traveling in the -x direct- ion with a phase velocity enual to onr-half the electron velocity I The four values of r obtained from the solution of equation (4.24) are 2 t (+ ZJ e) ,o (4.25) w,,4 Ur U b wcot.^coi4iCtW-yf) and ^Z(L) 4j '4-ww-6. p. e-) P., " (4.26) '3 =o 7 ;"coi'-eo/4( -fJ . 9, is the value of interest, since it indicates an in- creasing wave traveling in the +x direction. For this wave, the value of the ettenuation constant, $, in e~uat- ion (4.18) is (h.27) b '" 4 - The value of b can thus be celculpted for a particular tube, mh ,gaining wavP is amnlified at the rate of 6 nepers Drr meter, or 8.7 b decibels Der meter. .It should be remembered that the small-signal approximation was made in the derivation of the differ- ential equation for,Fl, Therefore, if the interaction space of a tube is made long enough to allow high-level bunching, equation (4.27) may renutre modification. CHAPTER V EXPERIMENTAL RESULTS Three cylindrical smooth-anode magnetrons were con- structed and data were taken on each to determine the fre- quency of observed oscillations as a function of magnetic field strength and anode potential. The tubes tested were (1) an inside-cathode magnetron with large anode to cathode radius ratio, (2) an inside-cathode magnetron with inter- mediate anode to cathode radius ratio, and (3) an outside cathode magnetron with large cathode to anode radius ratio. Inside-Cathode Tube With Small Cathode This tube was designed to check the theoretical pre- dictions made by Harris for a tube with "vanishingly small" cathode. The method of testing is shown in the block diagram of Figure 5.1. The d.c. anode voltage supply was modulated at a 60 cycle rate by the output of the transformer, T. Oscil- lations which occurred were detected by the crystal, X, and the amplified r.f. envelope was applied to the vertical de- flection plates of the cathode-ray oscilloscope. The hori- zontal deflection voltage was a 60 cycle sine wave which was /I w - :: ( I r '1 I... .1. .. -: I~i.. -- -- - ____ .1 -4. -- -- .- -- t- L i I .I. -t`! . f 1 -I-- -.i I-I:. ;. . -7~ tt --- 77 A- iV- Op~ 7 K K ---Th2-- ~- -----I -r _- * -i --- -i _i- - i-~~ -- . __LL'I *1Ii __II'-- __ __ L i w |

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