ENGINEERING RESEARCH INSTITUTE UWIVERSITY OF MICHIGAN ANN ARBOR ON THE PROPERTIES OF TUBES IN A CONSTANT MAGNETIC FIELD by O. DOEHLER with J. BROSSART and G. MOURIER Technical Report No. 9 Electron Tube Laboratory Department of Electrical Engineering A Translation from: Approved by: ANNAIES DE RADIOELECTRICITE H. W. WELCH, J.. by GEOR1GE R. BRPEWER W. G. DOW Project 2009 CONTRACT NO. DA-36-039 sc-15450 SIGNAL CORPS, DEPARTMENT OF TEE ARMY DEPARTMENT OF ARTY PROJECT NO. 3-99-13-022 SIGNAL CORPS PROJECT NO. 27-112B-0 February 1952

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MAJOR REPORTS ISSUED TO DATE Contract No. W-36-039 sc-32245. Subject: Theoretical Study, Design and Construction of C-W Magnetrons for Frequency Modulation. Technical Report No. 1 H. W. Welch, Jr., "Space-Charge Effects and Frequency Characteristics of C-W Magnetrons Relative to the Problem of Frequency Modulation," November 15, 1948. Technical Report No. 2 H. W. Welch, Jr., G. R. Brewer, "Operation of Interdigital Magnetrons in the Zero-Order Mode," May 23, 1949. Technical Report No. 3 H. W. Welch, Jr., J. R. Black, G. R. Brewer, G. Hok, "Final Report," May 27, 1949. Contract No. W-36-039 sc-35561. Subject: Theoretical Study, Design and Construction of C-W Magnetrons for Frequency Modulation. Technical Report No. 4 E. W. Welch, Jr., "Effects of Space Charge on Frequency Characteristics of Magnetrons," Proc. I.R.E., 38, 1434-1449, December 1950. Technical Report No. 5 H. W. Welch, Jr., S. Ruthberg, H. W. Batten, W. Peterson, "Analysis of Dynamic Characteristics of the Magnetron Space Charge, Preliminary Results," January 1951. Technical Report No. 6 J. S. Needle, G. Hok, "A New Single-Cavity Resonator for a Multianode Magnetron," January 8, 1951. Technical Report No. 7 J. R. Black, H. W. Welch, Jr., G. R. Brewer, J. S. Needle, W. Peterson, "Theoretical Study, Design, and Construction of C-W Magnetrons for Frequency Modulation," Final Report, February 1951. i

Contract No. DA-36-039 sc-5423. Subject: Theoretical Studyg Design and Construction of C-W Magnetrons for Frequency Modulation. Technical Report No. 8 G. R. Brewer, "The Propagation of Electromagnetic Waves in a Magnetron-Type Space Charge," July 1951. Technical Report No. 10 G. Hok, "Space-Charge Equilibrium in a Magnetron: A Statistical Approach," July 13, 1951. Technical Report No. 11 J. S. Needle, "The Insertion Magunetron: A New External-Cavity Magnetron for Low-Power Electronically-Tunable Operation in the 10 to 20-cm Wavelength Range," August 1951. Technical Report No. 12 H. W. Welch, Jr.,'Dynamic Frequency Characteristics of the Magnetron Space Charge; Frequency Pushing and Voltage Tuning," November 1951. Technical Report No. 13 J. R. Black, J. A. Boyd, G. R. Brewer, G. Hok, J. S. Needle, W. Peterson, S. Ruthberg, R. F. Steiner, H. W. Welch, "Theoretical Studyr, Design and Construction of C-W Magnetrons for Frequency Modulation," Final Report, January 1952. ii

tEREFACE During the past three years, several articles presentinz theoretical considersations on the subject of the magnetron and the prop1csed magnetron travelling-wave tube hlave appeared in the Annales de Radicelectric it. A number of these articles have been translated from the trench fior use in this laboratory. A series of four articles by 0. Doehler et al. were of considerable interest to the personnel of the E.,lectron Tube Laboratory, and it is thought that they wTiould be of greater interest to other workers in this field if available in an English translation. It is to be noted that for consistency with the coordinate system usually used in the United States, the x- and y-axes have been reversed in the course of this translation. In the translation, all errors have been corrected according to "Errata," Vol. 3, P. 183. This writer would like to express his appreciation for the assistance received from Dr. B. A. Ihlendorf, Editor of the Engineering Research Institute Publications, who carefully edited the translation. Ann Arbor, Michigan July, 1951 George R. Brewer iii

ON THE PROPERTIES OF TUBES IN A CONSTANT MAGNETIC FIELD Page PART I CHARACTERISTICS AND TRAJECTORIES OF TBE ELECTRONS IN TEE MAGNETRON, by O. Doehler........... 3 PART II THE OSCILLATIONS OF RESONANCE, by O. Doehler... 31 PART III THE TRAVELLING-WAVE TUBE IN A MAGNETIC FIELD, by J. Brossart and O. Doehler ~ ~ * * * 67 PART IV THE TRAVELLING-WAVE TUBE WITH A MAGNETIC FIELDn by 0. Doehler, J. Brossart, and G. Mourier....... 95 iv

PART I ON THE PROPERTIES OF TUBES IN A CONSTANT MAGNETIC FIELD by O. Doehler Annales de Radioelectricite Vol. 3, No. 11, Jan., 1948, pp.29-39 Summary This article treats the static and dynamic properties of the magnetron. Using the theories of L. Brillouin, the author gives the relations defining the space charge, the distribution of potential, and the characteristics for the magnetic field smaller than the critical magnetic field. The author then studies the oscillation frequencies of the multicavity magnetron and showrs that the results of his calculations agree well with the measured values. Introduction The present work treats the static and dynamic behavior of the magnetron. Although much has been written on this question, this article seems justified, for it brings together the different ideas on certain points idhich have been published heretofore. The first part treats the space charge, the distribution of potential, the trajectories of the electrons, and their characteristics; the

second part studies the regions of oscillation by taking as a basis the preceding results. These results are the point of departure of the theory which considers the rmagnetron as a tube for the propagation of waves. One begins with the idea of L. Brillouin in order to arrive at the relations concerning the space charge, the distribution of potential, and the characteristics for the magnetic field smaller than the critical magnetic field. However, in the use of the magnetron, these quantities are most important for the field larger than the critical field. Certain experimental results make one think that the usual ideas should be conmplemented in this domain from the ideas borrowed fronl other phenomena. One should study the motion of the electrons, not from the kinematic point of view, but from the statistical point of view; in fact, because of the very large current density, the interactions between the electrons are strong. It is therefore no longer permissible to regard the electronic current as a continuum but as having the properties corresponding to a discharge in a gas. There does not exist an exact statistical theory of the electrons in the magnetron analogous to that of the discharge in the gas. This theory will not be developed here. Nevertheless, we can derive certain conclusions from the measurements of the distribution of potential and base the calculations on these. The measurements we conduct presuppose an approximately constant density of electrons, independent of the dimensions of the system in the static state. TWre can then calculate the trajectories of the electrons for a magnetic field 1.2 to 1.5 times greater than the critical magnetic field. In the second part we will show that the oscillation frequencies of the multicavity magnetron, calculated according to this hypothesis, are in good agreement with the measured values. It is the same for the efficiency and the admittances of the oscillating magnetron.

3 Finally, the results of this treatment will give quantitative indications useful relative to a new type of amplifier tube: the magnetron travelling-wave tube, which will be discussed later. PART I: CHARACTERISTICS AND TRAJECTORIES OF THE ELECTRONS IN THE MAGNETRON When we study the dynamic operation of the magnetron, it is neeessary to understand the static operation and in particular the characteristics and the electron trajectories in this type of behavior. This problem has already been treated by L. Brillouin,l who has shown the importance of the space charge; the distribution of potential is not the same as that in a saturated diode with space charge. Wte must distinguish three regions: 1. In the region below the critical point, the magnetic field is sufficiently weak so that the electrons reach the anode. This region is practically without interest. 2. In the region around the critical point, the magnetic field being equal or slightly greater than the critical field, we find ourselves between the knee above and the knee below on the curve of plate current as a function of the magnetic field. We know that in this region the electronic oscillations arise in the magnetron with or without gaps. 3. In the region above the critical point, the magnetic field being 1.2 to 1.5 times greater than the critical field, the plate current is small. In this region arise the oscillations of resonance in the magnetron with gaps and oscillations in the form of spirals in the magnetron without gaps.

4 1. Reg;iorn Below. the Critical Point The characteristic in the region below the critical point has been calculated many times.2-9 Here is the basis of the calculations: (a) The equation of motion, m v = e E -+ e( xB, (1) (b) Poisson's equation, 2 V 0 = and (2) (c) the equation of conservation of charges, v (Pr) = 0. (3) If one writes equation (1) in cylindrical coordinates (r, a, z), the magnetic field. being in the direction Oz and the electric field radial, one obtains:'6- r o2 = 7E(r) +7B (4) b + 2, = - M B, (..5) where q = e/m, X = & (instantaneous angular velocity). Equations (4) and (5) give after integration: r2 + r2w2 = 2779(r) (6) X = [ -%E f l (7) Equations (6) and (7) assume that the electrons leave the cathode (0 = 0, r = rc) with zero velocity.

For the plane magnetron, we find in the same manner, for a magnetic field in the direction Oz and an electric field in the direction y Y (Y) x - B. (9) The initegration of these equations, assuming that the electrons leave the cathode without initial velocity, gives +2 (,c2 y2 = 2 70(y) (o10) x - o Y c (11) where coc = eB/m. Equations (6) and (7) with (10) and (11) give the well-known value for the critical magnetic field Bcr for the cylindrical magnetron / / 6.72 / Bcr -, (12) rp 1 - ) rp 12rpr r7 where Bcr = critical magnetic field in gauss rp = radius of the anode in centimeters rc = radius of the cathode in centimeters Up = plate voltage in volts, and for the plane magnetron _ _ _ (13) d d

where d is equal to the distance between the anode and the cathode in centirameters. If B < B3cr., all the electrons which are present at the same distance from the cathode have the same velocity, and equation (3) gives, for the plane magnet ron ia = v, (14) ia being the current density, and for the cylindrical magnetron Ia 22rrpi, (15) Ia representing the plate current (for unit height of the plate); y and r are determined by equations (10) and (6), respectively. If the current is sufficiently large for the space charge to limit the emission of the cathode, the electric field at the surface of the cathode is zero. Age can in such a case, with the above equations, calculate the current as a function of plate voltage and the magnetic field. Fig. 1 represents the curve obtained by Bethenod2 for a plane magnetron. We have placedalong the abscissa the ratio of the magnetic field to the critical magnetic field, on the ordinate the ratio of the plate current Ia to the plate current Iao corresponding to B = 0. Iao is given by the law of Langmuir-Child: Iao = 2.33 x 10-6 U 2 (16) Iao is in amperes, Up in volts, d distance between cathode and anode in centimeters, and S the surface area of the plate in square centimeters. Fig. 1 shows that the plate current diminishes by 28 per cent when B is increased from zero to Bcr.

1.0 0,9 0.8 B 0,7 _C 02 0.4 o6 0,8 1 Fig. 1 Plate Current as Function of Magnetic Field. Plane Magnetron For the cylindrical magnetron with a cathode of small diameter, we can calculate the plate current as a function of B from a development in series which Moller3 has made use of for calculation of the distribution of potential. Curve 1 (Fig. 2) represents the characteristic with the same coordinates as in Fig. 1. We can see that the plate current diminishes by 12 per cent when the magnetic field is increased from zero to Bcr. A method due to Pidduck7 permits the calculation of the plate current of a magnetron saturated with space charge for the critical magnetic field. This method gives the same results as the calculation carried out according to the method of Moller. Page and Adams9 have likewise developed the calculations for the cathode of large diameter. These authors find that, for rO/rc infinite, the plate current diminishes by 14 per cent when B varies from zero to Bcr for rp/rc = 125, by 10 per cent. In a very exact theory by L. Brillouin,10 the diminution is by 20 per cent for the ratio rp/rc very large. Curve 2 (Fig. 2) represents the results of the measurements of Hull, Mulert,12 and my personal measurements. All the latter give a diminution of current larger than that which we can expect according to the

8 calculations. By itself, the theory of L. Brillouin appears to be in good accord with experience. 0,9 o MEAS. OF HULL x ~ II to MULERT X X 0,8 x " I' DOEHLER x 0,7.0.6. 0' c',6 5Bcr 0 02 0,4 0,6 0,8 1 Fig. 2 Plate Current as Function of Magnetic Field. Cylindrical Magnetron. Concerning the distribution of potential, the measurements of Engbertl3 are not in agreement with the theories of the various authors. Theoretically, the distribution of potential for the critical magnetic field is given by (see M6'ller) r 2/3 2 10/3 0 = Up 932 +.05 +.014 +..j. (17) We should accordingly have for r/rp = 0.43, Q = 0.71 Up, and Engbert found, in this case, 0 = 0.28 Up. For a little smaller value of the magnetic field corresponding to B/Bcr = 0.96, Engbert finds only 0 = 0.50 Up for r/rp = 0.43. 2. Region Around the Critical Point In this region, the plate current varies with Ia for B = Bcr, even to very small values. The magnetic field is enclosed between Bcr and 1.2

to 1.5 times Bcr. If B > Bcr, some electrons return to the cathode,and equation (15), which gives the density of the space charge, is no longer valid in this form. If one assumes that the electrons leave the cathode with zero velocity and if one holds to equations (1), (2), (3), the plate current must be zero for B > Bcr. If one takes into account the Maxwellian distribution ofo velocities and if the space charge is not large, there can be no minimum of potential between anode and cathode, and we obtain, according to Linder,l4 a plate current of the form Ia = Ia exp - ef rr ( B 2 - B(1r8) where Ia - the plate current for B = Bcr T - the temperature of the electrons k - the Boltzmlanm constant. if one plots In Ia as a unction of B2, the slope of the curve obtained permits the calculation of the -temperature of the electrons. Fig. 3 gives In Ia = f(B2) for various values of Up. This ftuction is not linear but we will re.lace, to a first approximation, the curve by a straight line. It is noterto-hl;iyhr that the slope of the straight line and, therefore, the teimperature of -the electrons depends on the applied plate voltage. To calculate t-he distribution of potential, we cancI make use of equation (15) also in this region, providing we adopt the sumn of the;direct current a(r-d the curne&l; which re-turns instead of -the p1late curr-en-t Ia; we then have IaT Ia wPa (-..( ) 2~~~~~~r ~ ~ ~ ~ (?

10 Ial being the direct current from the cathode toward the anode and Ia2 the current which returns to the cathode. For r, we must take the absolute value, which we can obtain from equation (6). We can then calculate the distribution of potential in the region below the critical point, and we find the same distribution of potential as in equation (17) if we introduce the distance covered to return instead of rp. L J, 3. 50 V 75 V 200V ~~2~~~~~ Fig. 3 In Ia - (B2) 5. Region Above the Critical Point The calculation of the distribution of potential is much more complicated if, at the seane point, electrons are present with various radial velocities. This is the case in the region above the critical point, as we will show on the basis of some measurements. We will see that in this region the phenomena are more complex than in the others. AW'e can analyze the behavior of the magnetron: (a) byr the experimental determnination of the characteristic; (b) by the experimental determination of the potential distribution; (c) by the measurement of thc thenrmal velocity of the electrons.

11 (a) Determination of the Characteristic. Figs. 4 to 6 show the measured characteristics. The magnetic field is the abscissa and. the plate current is the ordinate. In Fig. 6, we have plotted on the ordinate the ratio of the plate current corresponding to a given value of B to the plate current for B = 0. ~O V I, ~. ~~~~~~~~~75 V 1OV 0,2 0,4 0.6 0,8 1 1,2 1,4 Fig. 4 Characteristic, with Up as Parameter In Fig. 4, the parameter is the plate voltage and in Fig. 6, the emission current. We find that: 1. The plate current is never zero, even for a magnetic field B >> Bcr. Moreover, as the plate voltage is raised, the plate current which passes is larger in spite of B > Bcr. mA. ~c 6m Ua=1400 V 7 8 9 10 11 12 13 14. 15 A Fig. 5 Characteristic for a Large Plate Voltage

12 Harveyl5 points out that the plate current is not yet zero for some plate voltages if B > 10 Bcr. 3mA 0,8 87mA 0,6 e 16mA 04],4 \t t40mA 0 150 180 210 240 270 1300 Gauss Fir. 6 Characteristic with the Emission Current as Parameter 2. The slope of the descending part of the characteristic AIa/AB decreases when the plate voltage increases, as we could see already in Fig. 3, as indicated by Linderl4 3. The relative slope -a//a B increases with emission current. 4. As the plate voltage is increased, Ia passes through a maximum for a magnetic field B - 1.1 Bcr. Wie observe, then, an increase of the temperature of the cathode. This well-knownl effect of the heating by returning electrons appears also in a vacuium and without oscillations. This is a particular characteristic of the magnetron. We will return to this in the second part of this paper. (b) Distribution of Potential. This has been measured by Engbert.l1 Curve 1 (Fig. 7) gives the distribution of potential in the region above the critical point. This is independent of the magnetic field if B 3 1.2 to 1.5 times B3cr, is invariarnt with small angles of magnetic field wdith the axis,

13 and. independent of the emission current as lon, as it is larger than a well-determinedd value. 1.0 p 0,8 0,6 02 0,0 0,2 O,, 0,6 0,8 1,0 FPi. 7 Distribution of Potential in the'Region Situated Above the Critical Point The followings empirical formula conveys pretty well the results of the measurements O = U-p) (Fig. 7, Curve 2). (20) Vfith this distribution of potential, we have, according to (2), a constant space charge. Wihen we examine the region below and around this critical point, the distribution of potential (initially ~n r, if the emission current is 2/3 small, and -(r),if it is saturated with space charge) assumes progressively the form of the distribution of potential above the critical point given by equation (20). L. Brillouin has demonstrated that, for the critical magnetic field and zero plate current, equation (20) is compatible with the equations of sections 1 to 3. In the region above the critical point, the electrons return toward the cathode for the values of r less than rp, and one has, between the point of return and the anode, a logaritlhmic distribution of potential.

14 However, in the case of practical interest, the plate current, even above the critical point, is not zero, and one must take account of a density of charge in the entire space. According to the measurements of Engbert, the result is that the constant density of space charge, calculated according to L. Brillouin for the critical point, is valid in the entire region above the critical point. (c) Thermal Velocities of the Electrons. The tC -nperature of the electrons in the magnetron has been studied by Slutzkin,lU igdortshik,7 and, in a very thorough manner, by Linder.14 They have found that the thermal velocity of the electrons does not correspond. to the termperature of the cathode, but to a temperature always much higher. WTe have established a distinction between these two temperatures. 1. For B >> Br the temperature of the electron is proportional to the plate voltage. The author's measurements have sho~vi that the tenoerature of the electrons for a plate voltage of 1000 volts can be of the order of 10-"0K. 2. The temperature is a decreasing function of The emission current; beginning with a certain current, the temperature is constant (saturated space charge). 3. The temperature increases with the magnetic field in the region around the critical point; in the region below the critical point it decreases almost inversely proportional to the magnetic field. The behavior of the characteristic, the distribution of potential, and the Linder effect can be explained in the following manner: If B > Bcr, the point of return of the electrons is found between the electrodes. The electrons do not necessarily return to the cathode, but their trajectories are very long, that is to say, there is a large circulating current around the cathode even when the plate current is small. This current,

15 which circulates around the cathode,is large enough to cancel the electric field at the cathode. Tlis circulating current involves a large space charge, and the electrons which move in the space of the discharge traverse in this way a long path in an intense space charge. There is, therefore, an exihange of energy between the electrons because of the Coulomb forces, and one can no longer consider the space charge a continuum but we must take into account the electric field of the various electrons.* Some electrons gain energy by collision, others lose a corresponding quantity. In this manner, a part of the energy of the trajectory is transformed into disordered thermal energy. This problem is analogous to that of the discharge in a gas.l8 A plasma includes, in effect, a large number of electrons which take energy from the continuous field and transform it to thermal energy in giving it up, through dissipation, to the other electrons. In the discharge in the gas, the density of the electrons is of the order of 109 to 1013 per cubic centimeter,and this is of the same order of magnitude in the magnetron (see equation 55). The problem of calculation of the temperature of the electrons in the plasma is not yet perfectly solved since one arrives at integrals which do not converge. (The Coulomb forces are inversely proportional to the square of the distance between electrons; the probability of collision is proportional to the square of the distance.) These analogous problems are posed in the theory of the widening of the line spectrum by pressure,19 in the theory of the passage of electrons through matter,20 and in the theory of the accumulation of the stars. All the calculations made up to the present lead to results which are too small. If we take account of the interaction between electrons, we realize that the distribution of potential in the magnetron is profoundly modified * We can find a two-dimensional analogue to this phenomenon: that of a heavy point being displaced on a surface of disturbed water.

1; in relation to the consicdertions Cwhichl permit the calculation of equations (6), (7), and (19). For the values B >> Bcr, we can attempt to consider the space charge aUs an electronic gas and apply to it the.aixwell-Boltzmann statistics by using the Hamiltonian function for the electrons. If one neglects the current absorbed by the plate, the density of electrons increases exponentially with the distance (the same as the molecular density in the barometric formula), as pointed out by Lueders in work which has not been published. However, this state is not stable because of the current absorbed by the anode. 9,e carn adm'lit as a first approximation to the distribution of potential in a cylindrical magnetron with small diameter: Q = Up j r 1 (21) T~ie find, therefore, by integration of the equation of motion (30) that the electrons cannot leave the cathode for Ft < 2. For zero absorbei current, iL can become but little different from 2, as showm by the measurements of L'ngbert. A distribution of potential like (r/rp)2 is, as we have said, equivalent to a constant density. The same phenomena are to be fomnd in the plane magnetron: if one takes for the distribution of potential an equation analogous to (21), we must have at _ 2. For the plane magnetron or the cylindrical magnetron with small-diameter cathode, the distribution of potential p' r2 fulfills the conditions at the cathode that p = 0 and 8Q,/a r = O. In the two cases, are have a uniform space-charge density. Thus we are led to assume also a uniform density for cylindrical magnetrons with large cathodes. This is merely a hypothesis that has not been verified up to the present; but we will point

17 out that the latest of the experimental results are in good agreement with the calculations that result. 1We can then calculate the distribution of potential in both these cases. In the region above the critical point, we have p = constant.'We then deduce: for the cylindrical magnetron with small dianieter cathode, 2 q = Un, 0 (22) for the plane magnetron, Q upY ), and for the cylindrical magnetron with a large cathode, r2 - rc2 - 2rc2 In(rir (2 rp - rC2 - 2rc2 ~n(rp/rc) In the region below the critical point, we obtain for e very small plate current (negligible space charge): for the cylindrical magnetron, Up) ~n(rp/rc)' (25) for a very thin cathode, for the plane magnetron,

With a dense space charge, we obtain in the region below the critical point the following approximate values for distribution of potential: for the cylindrical magnetron with a thin cathode, Q =, (28) (a more precise result should be obtained beginning with equation (17)), and for the plane magnetron,' Q = -, (29) (a more precise result is given in (2)). When we pass through the region arounrd the critical point, we pass progressively in the distribution of potential below the critical point (equations 25 to 29) to the one in the region above the critical point (equations 22 to 24). In the region around the critical point, we can represent the distribution of potential by equation (21). For a cylindrical magnetron with small diameter cathode, o <.L < 2 and for a plane magnetron, 1<_AL!2 4. The Electron Trajectories The trajectories of the electrons in a space where we know the distribution of potential can be calculated from the equations of motion.

19 (a) Cylindrical Magnetron with Very Thin Cathode. From (6) and (7) j re find, for the electrons leaving the cathode with zero velocity, r/'Irr =r/rr)N (-_' r/rr)2' 0 rrt being the distance to the point of return; eB 2m EcuaE-i.on (3o) shows that the electrons can leave the cathode only for,x < 2. 7i;a`s is the reason that we cannot have a distribution of potential with In > 2. Equation (30) gives an integration r = rr sin 2- - Lt, (51) and a oLt - - _B t. (32) = "Lt = 2m Fig. 8 represents the trajectories of the electrons for various exponents. For /L = 2/3, we obtain the well-known curve of Hull. For the electronic oscillations it is important to know the time of transit r i.e., the time between the emission of the electron and its return to the cathode.

20 = const. 3 V 2 -t _ (f5) Sin cca(te3f ro (1 ) For = 2/ we obtain175 r = 4.72 r - Fig. 8 Trajectories of the Electrons in the Region Around the Critical Point We calculate it from ( 31) 2 r 2n 2'31 e3 For u = 2/3 we obtain eB We can calculate r with the aid of equations (1), (2), and (3) without use of the hypothesis about the distribution of potential, and one ob'uains, from Pidduck6 r" = 5.o4 eB a value 6 per cent larger than that which we obtain with the distribution of potential by Langmuir ( I = 2/3). For the quadratic distribution of potential, p. = 2, equation (30) no longer has meaning, that is to say, the electrons cannot leave the cathode with zero velocity. This distribution is a limitinrg case which, therefore,

21 cannz'ot occur. Practically, the true distribution deviates slightly, the electrons can leave the cathode, and /z is slightly less than 2. The trajectories of the electrons ca.anot be known exactly and are defirned only statistically because of the extraordinary density of the charge and of the interactions between electrons. What are the possible trajectories for the electrons which have left the cathode with the initial conditions given? In rectangular coordinates, we obtain for IL = 2, y = 27'-.2 y +-71Bx, (35) x = 2x — x -7 By, = (36) which give a differential equation of the fourth order in y. y'+ uc((sc - )nocc2 2 y _ O, (37) eB Equation (37) gives y = Al sin (colt + P1) + A2 sin (ft + o2); (40) the frequencies el and w2 are determined by the equation

22 -2 [ C 1 (41)'2- -" - 2 41 2MC - C which gives, for el and 032 l Wc(1 + -4 ) (42) 2 (C )' ( 4 S) and Ut = 2 DC A1, A2, 1i, 02 are given by the initial conditions. If one assumles %c2<< cc (from T$, 59, tand 12, this is equivalent to B >> Bcr), w;e obtain as a first approximation o1 WC ( (4 - eB (53) rp_ B As a second. approximation, I2 (1 +: ) (44) In the same manner, we obtain, for x, if B>> Bcr, x - A1 cos (colt + 01) + A2 cos (c02t + 02) ~ (46) Equations (41) and (46) show that, in the region above the critical point, the electrons describe epicycloidal trajectories around the catihode,

23 resulting in impulsive motion Mhaose frequency is given by (43, (39), or (45), depniding on the case, and a relative motion whose frequency is given by (42, (38), or (44). The space charge is thus represented as a gas which rotates with a constant angular velocity, co2, arou)nd the cathode. The phase and amplitude of the motion are essentially determined by the scattering of the electrons. We obtain the same angular velocities, el and cu2, if we use, in equation (4), the value of Er corresponding to /L = 2 and if we calculate the angular velocity of the electrons which describe circles around the cathode. The velocities found. are those of the equations (42) and (43). Physically, the existence of a circular motion around the cathode implies an equilibrium between the electrical force en, the centrifugal force mro2, and the Lorentz force. If the intensity of the field is proportil;onil to r, the anlgular velocity is independent of the radius. There are two solutions: (1) the electrons can circulate sufficiently fast so that thle electric force is negligible compared with the centrifugal force; we then obtain equation (38) the cyclotron frequency). Or else (2) the electrons circulate sufficiently slowly ok)u tb- centrifugal force to become negligible compared with the electric force; wrTe then have equation (39) (formula of Posthumus). This equation is important for the oscillations of resonance. Brillouin has made analogous calculations.l (b) Cylindrical Magnetron wth Large Cathode. This case is important since, in particular, all the multicavity magnetrons have large cathodes. We must put into (4) the electric field of (24) and we deduce for r = 0

r rc s4 | B 2[ I+I 1 [ (Ft | (47) The negative sign corresponds to the frequency of Posthumus and the positive sign to the cyclotron frequency. This equation, (47), which shows that the angular velocity depends on the radius, determines the frequency of oscillation in the multicavity magnetron. (c) Plane Magnetron. We obtain analogous relations for the plane magnetron. In the region around the critical point, equations (10) and (11) give Yr Y sin 2 - _ ct (48) f~ = - tcy (49) eB where yr is the abscissa of the point of return of the electron. For,L = 1, we obtain, for example, y Yr (1 - cos ct), (50) x = - t + xo + Yr sin ct, (51) dB 2 that is, the electrons describe cycloids. For the region above the critical point ( Al = 2), we obtain the trajectories of the electrons

25 Y = Po sin ct + Yo, (52) x = p cos t + Yot + x (55) with eD= m i U eB 1 Ucr ) (54) Ucr being the critical voltage. Equation (54) is verified by the measurements of Gutton and Ortusi.'2 These authors use the proper oscillations of the electrons in the magnetron as variable capacity. This capacity is very large if the pulsation of the high-frequency oscillation is equal to w. Their measurements justify indirectly our hypotheses. Note that tile constant velocity in the x direction depends oppositely to the case p. = 1 on the parameter yo. 5. Density of Space Charge and Electronic Current in the Magnetron From equation (2) and the distribution of potential (21), the density of electrons per cubic centimeter is given by: N = 5.6 x 105 e (. (55)r) For example, for Up = 1000 volts, rp = 0.2 cm, and pL = 2, the density of electrons is 6.1 x 1010/cm.3, of the same order of magnitude as in a plasma. In the maGnetron, the electrons produce "I flux> of mag-:netic inrau;tion that one can measure and which M0li].er has calculated.- in t[;!e case w-he.0hre

26 Q " r2/3. The measurement of this flux is a means of verification of the theoretical studies of the magnetron. We will call annular current the current that would induce, while moving in the anode, the same magnetic flux as that of the rotation of the space charge. It is given by: Ir = 5.2 x 10-6 B Up3/2, (56) -1 +AL_ Bcur r(...-r.e Ir is in amperes and Up in volts. i1i3ller23 has measured the annular current around the critical point. These measurements lead to AL = 1 for the distribution of potential. We mean by "magnetron current" the current of the space charge circulating around the cathode above the critical point. It is given by Im = p dr. 7) rc If one puts r = rp in equation (47), we obtain, by first approximation 1 -rc2 Im = 2 Eo Up p (58) 1 - rp [1+ 2 Inrc This current of the magnetron is related to the continuous plate current of a magnetron which oscillates in the region above the critical point. 6. Conclusions We have examined the trajectories of the electrons and. the influence of the space charge to the exhent that is neecse —Jry for studying the mechanism of oscill ation of the magnetron. We have made the hypothesis that

27 the electronic density is constant in the region above the critical point. For the plane magnetron or the cylindrical with small cathode, Brillouin has obtained from the calculations a constant density for the critical field. The hypothesis which we have made leads to the following differences with respect to the calculations of Brillouin: 1. The space charge p is not constant for B = Bcr but only for B 3 1.2 to 1.5 Bcr. 2. p is independent of the magnetic field for B, 1.2 to 1.5 Bcr. 3. The distribution of constant space charge is valid for any rp/rc The two first points are verified experimentally by the measurements of the distribution of potential. In a second article we will show that the third point is in good agreement with the dynamic behavior of the magnetron. We have thus been able to treat all the cases of practical interest. We can distinguish two important cases: (1) the performance in the region around the critical point and (2) in the region above the critical point. In the region above the critical point, the electronic trajectories are normal to the continuous field. We use the energy of tangential motion in introducing an alternating tangential electric field. These are the oscillations of resonance with large amplitude. In the region aroundt the critical point, we use the radial motion of the electrons to produce the oscillations. These are the electronic oscillations of small amplitude. Since in the second case the oscillations are the oscillations of transit time, wre see that the wavelength of the electronic oscillations is

inversely proportional to B' (equation 33), while that of the oscillations of resonance is proportional to B (equation 59).

29 B IBLIOGRAPHY 1. L. Brillouin, Phys. Rev., vol. 60, 1941, p. 385; vol. 63, 1943, p. 127; J. Phlys. Radium, vol. 7, VIII, 19)40, p. 233. 2. M. J. Bethenod, C. R. Acad. Sc., Paris, vol. 209, 1939, P. 832. 3. H. G. MOller, Hochfrequenzteclhn. u. Elektroakustik, vol. 47, 1936, p. 115. 4. B.S.E. Braude, Phys. Zeits. d. Sowj. Jnion, vol. 7, 1935, P. 565. 5. L. Tonks, Phys. Zeits. d. Sowj. Union, vol. 8, 1936, p. 572. 6. E. B. Pidduck, Quart. J. Math., Oxford, vol. 7, 1936, p. 201. 7. G. Grunberg and V. Wvolkenstcin, Tech. Phys. U.S.S.R., vol. 4, 1937, P. 479. 8. E. B. Moulin, Proc. Camb. Phil. S vol. 36, 1940, p. 94. 9. L. Page and N. Aldams, Phys. Rev., vol. 69, 1946, p. 4C2. 10. L. Brillouin, Conference at College de France, June, 1947. 11. A. W. Hull, Phys. Rev.L vol. 18, 1924, p. 31. 12. T. H. Mulert, Hochfrequenztechn. u. Elektroakustik, vol. 41, 1933, p. 194. 13. W. Engbert, Ibid., vol 51, 1938, p. 44. 14. E. G. Linder, J. Applied Physics vol. 9, 1938, p. 371; Proc. I.R.E., vol. 26, 1938, p. 34. 15. A. F. Harvey, igh Frequency Thermionic Tubes London, 1946, Chap. IV. 16. A. A. Slutzkin, S. J. Braude and J. M. Wigdortshik, Phys. Zeits. d. Sowj. Union, vol. 6, 1934, p. 268. 17. J. M. Wigdortshik, Phys. Zeits. d. Sowj. TJnion, vol. 10, 1936, p. 245, 634. 18. Rompe-Steenbeck, Der Plasmazustand der Gase (Ergebn. exakt. Natunriss.), vol. 18, 1938.

30 19. Z. Holtsmark,;.nn. d. Physik, vol. 58, 1919, p. 577; Zeits. f. Pyiyrsik, vol. 51, 192, pT-.-o5. 20. W. B. Bothe, Hanlb. d. Physik, Berlin, vol. 22, 2, 1933. 21. Chandra SekLkar, Astrophys. J.,vol. 94, 1941, p. 511. 22. H. Gutton and J. Ortusi, Conference of Socil't-6 francaise des electslc-iens, Paris, January, 1947. 23. J. I;dller, Hocbfrequenztechn. u. ElektroLakustik, vol. 48, 19c36, p. 141.

31 PART II ON THE PROPERTIES OF TUBES IN A CONSTANAT IMAGNETIC FIELD by O. Doehler Annales de Radiodlectricito, Vol, 3, No. 13, July, 1948, pp 169-183 Sunmmary Starting with the results established in the first part of his article, the author considers the oscillations of resonance with high efficiency, that is to say, excited in the region above the critical point. He examines the differences existing between these oscillations and the electronic oscillations, then establishes a quantitative relation giving the conditions for optimum operation. The efficiency and the input impedances are calculated, and an empirical relation between the anode current and the current of rotation is given. Finally, the dynamic electronic trajectories of a plane magnetron without space charge are calculated to serve as an introduction to the study of the travelling-wave tube in a magnetic field. PART II: THE OSCILLATIONS OF RESONANCE The static behavior of the magnetron has been treated in the first part of this article.* According to the measurements of the distribution of p. 59. P. 39~

potential, of the distribution of the velocity of the electrons, and of the characteristics, we have concluded that the magnetron is the seat of a phenomenon which is negligible in ordinary electronic tubes. There exists a dispersion of the velocity among the electrons in the space of the discharge because of their high density and their long time of existence. In the extreme case (very large magnetic field), we can consider the discharge as an electronic gas. For a stable discharge, the result therefore is that one has, in first approximation, a constant electronic density for the cylindrical magnetron with a very slender cathode and for the plane magnetron.* We assume therefore a constant density for the cylindrical magnetron with a cathode of finite diameter.** This constant space charge exists only in the region above the critical point (small anode current). In the region below the critical point, there exists a distribution of potential which corresponds (without space charge) for a small emission current to the electrostatic distribution and for a large emission current to the distribution of potential of a diode of the same dimensions without magnetic field. In the region around the critical point (Bcr B 1.2-1.5 Bcr), the distribution of potential of the region below the critical point is transformed gradually into that of the region above the critical point. * According to the unpublished measurements of Mr. Gutton, it is probable that the space charge is not constant but has a maximum. This maximum depends on the conditions of operation. For our calculations it is the distribution of potential which is important. This is why the small deviations of the density from a constant value do not have a large influence on the dynamic behavior of the magnetron. I thank Mr. Gutton for this information. *e Ponte3 assumed as early as 1934, because of the large time of existence of the electrons, a distribution of potential which is approximately quadratic (see Fig. 7 of his article3).

33 The resulting trajectories of the electrons show that, in the region above the critical point, the electrons encircle, in general, the cathode. The angular velocity cta is independent of the distance if the cathode is very slender; for a large cathode, wc depends on the distance. In the region around the critical point, the electrons describe cardoidal trajectories. The time of transit of the electrons from the cathode out to the point of return depends on the distribution of potential, i.e., on the magnetic field. As we know, the electronic oscillations of the magnetron with slots and without slots begin in the region around the critical point. Their frequency is independent of the frequency of resonance of the coupled circuit to the extent which the natural frequency of an oscillating system depends on the coupling to a second circuit. The efficiency is small. The wavelength is given by - C (1) C is here a constant which depends on the number of slots, on the type of oscillation, and on the ratio B/Bcr. We shall not occupy ourselves in this article with the mechanism of excitation of the electronic oscillations except to the extent to which we have used them compared to the oscillations of resonance. The present article treats the oscillations of resonance excited in the region above the critical point, that is,the form of oscillations generally employed at present for technical reasons (for example in the multicavity magnetron or in the magnetron with resonance of the segments). The frequency of the oscillations of resonance is given, in general, by the oscillating circuit. Unlike the oscillations excited by the static negative resistance of the magnetron, the conditions for optimum operation of the oscillations of

resonance depend on the wavelength. The efficiency is large (theoretically up to 100 per cent); in practice one has obtained efficiencies up to 70-80 per cent. The oscillations of resonance have been studied experimentally and theoretically in numerous works. 1-15 We shall not occupy ourselves in the following with the electronic mechanism. In this second part, we present a kinetic theory of the magnetron taking account of the influence of the space charge on the distribution of potential. In a third part, we will treat the dynamic behavior for small amplitudes, proceeding from the work of L. Brillouin. In Chapter 1 of the present article, the essential differences between the oscillations of resonance and the electronic oscillations will be examined. In Chapter 2, a relation will be established to give the optimum operation of the oscillations of resonance. This relation will be compared with the measurements published up to the present. The efficiency and the input impedances wrill be calculated in Chapters 3 and 4. In Chapter 5, we shall give an empirical relation between the anode current and the current of a magnetron (1st par., Chapter 5). The dynamic electron trajectories of a plane magnetron without space charge will be treated as introduction to Part III, in Chapter 6. This article does not treat the problems of the oscillating circuit. The latter, in particular for the multicavity magnetron, have been studied, among others, by Goudet,l9 Fisk, Hagstrum, and Hartmann,20 Crawford and Hare.21 1. Differences Between the Electronic Oscillations and the Oscillations of Resonance If the electrons describe a circle around the cathode, the angular velocity w is such that there exists an equilibrium between the electric force eE, the centrifugal force mrto2 and the Lorentz force eraiB. Two limiting cases are then possible:

First Limiting Case (Electronic Oscillations). The centrifugal force is sufficiently large so that, if one neglects the electric force, all the effects of the space charge are equally negligible. This limiting case is present if one uses for the excitation of the oscillations the arrangement of an inverted cyclotron. In the cyclotron, the following process takes place: an electron, wAich is accelerated in the direction of the tangent to its trajectory by an alternating field, attains a larger angular velocity and describes a larger circle, and. therefore a larger centrifugal force. An electron retariled by the tangential alternating field describes, on the contrary, a smaller circle. The electrons retaining their energy will arrive therefore at the anode, and the electrons which lost energy, to the center of the arrangement. This principle is used to obtain oscillations. With an electron gun and a magnetic field there is produced, in the vicinity of the anode, a circulating electronic current in a box split as in a cyclotron. The electrons of unfavorable phase reach the anode and are eliminated. The electrons of favorable phase give up the energy diuring their travel if the angular velocity of the electrons is in resonance with the pulsation of the oscillating circuit present between the slots. Since there does not exist a continuous radial electric field, there is produced, as a result of the equilibrium between the centrifugal force and the Lorentz force, a pulsation ~OH = eB k, (2) m k being the mode of oscillation which is excited.l1920 In the usual magnetron, the radial electric field is not negligible. For a quadratic distribution of potential, it follows therefore, according to

36 equation (1.45) in second approximation, eB 1 eB k ( U ~3: = ~-~l~~) rp This equation (3) is confirmed by the measurements of the proper -Irequencies in the magnetron by Gutton and Ortusi22 which we have already mentioned in the first part. The oscillations of the inverted cyclotron have not been studied experimentally up to the present. The most important difficulty is to be found in the production of the electronic bunches. The modulation of the density of unfavorable phase for the radial alternating field can be avoided by an arrangement analogous to that of the cyclotron.* The oscillations of the inverted cyclotron possess the advantage that the cathode is found outside of the space of the discharge. The backheating of the cathode is therefore avoided. The Second Limiting Case (Oscillations of Resonance). The second limiting case is characterized by the fact that, the velocity of the electrons being very small, the centrifugal force is negligible compared to the electric force. There is therefore uilibrium between the electric force eE and the Lorentz force erSB. For the cylindrical magnetron with a small cathode, the intensity of the electric field is proportional to r in the region above the critical point, from which -we deduce 2 U c. (4) rp2 B * See end of Chapter.

37 If there is a tangential alternating electric field and if an electron is accelerated in the alternatinng field, the Lorentz force islarger and the electron returns to the cathode. If an electron is retarded in the alternating field, it goes toward the anode.* In the region above the critical point, the electrons return toward the cathode to within a small distance from the latter (theoretically zero in the limiting case of a distribution of potential exactly quadratic), and the electrons of unfavorable phase move only in the neighborhood of the cathode, while the electrons of favorable phase reach, in the intense alternating field, to the neighborhood of the anode. The high efficiency of the oscillations of resonance rests on the fact that only the electrons giving up energy can pass through an intense alternating field. The energy which they lose in the alternating field is provicled by the continuous radial field.'There exists still an essential and very great difference between the electronic oscillations and the oscillations of resonance, namely, the action of the radial alternating field. If an electron is moving unnder the action of a r-adial altermating field, this field acts as if the anode voltage [Up in equation (3)] possessed an increased or a diminished value, according to the phase of the field. In the first limiting case, an increase of the voltage produces a decrease of the angular velocity anil, inversely, a depression of the voltage and increase of the angular velocity Equation (3)]. In Fig. 1, the lines of the field are drawm schematically. It follows therefore that, for example, an electron present at point A (radial accelerating field) will be retarded in the tangential direction, and an electron at point B (radial retarding field) will be accelerated. That is, there is produced, * a change of the entergy of the electron has therefore an exactly opposite action on the motion of the electron to that in the first case.

at point C, a focussing of the phase and therefore an increase of the density in the field of acceleration, in a phase unfavorable for the transfer of energy. Fig. 1 Effect of Focussing on the Oscillations of Resonance and the Electronic Oscillations On the other hand, in the case of the oscillations of resonance, the angular velocity, according to equation (4),is proportional to the anode voltage, and therefore, for example, an electron present at point A will be accelerated in the tangential direction. An electron at point B is retarded, that is to say, there is produced a focussing of phase at point D and an increase of density in the retarding field, which is in a favorable phase. 2. The Conditions for the Frequency of Oscillations of Resonance From these qualitative considerations, we can derive a quantitative relation for the optimum conditions. A synchronism must exist between the angular velocity w of the electrons and the pulsation wH of the alternating field. If k is the mode of the oscillations, we obtain XL = ( (5)

39 for a very small cathode, w being independent of the distance. There results, according to equation (1.43): eB Bcr e deduce the well-known formula of Posthus for B r. In the practical E'e deduce the well-known formula of Posthumus5 for B > Bcr- In the practical system of units equation (4) gives x = 946 ~, (7) k Up where B is measured in gauss, r and X in centimeters, and Up in volts. Herriger and H-ilster8 find experimentally 1100 for the constant in equation (7). This disagreement arises from the difference of the distribution of potential in the magnetron as compared with the quadratic distribution of potential of equation (1.25). This distribution of potential is only a first approximation, which is not sufficiently exact. In reality, the intensity of the field is very small and therefore the constant of equation (7) is very large. We can deduce from the measurements of Herriger and Hiilster a distribution of potential $ r1'A75 which, as shown in Fig. 2, is approached better by the measurements of Engbert than the quadratic distribution of potential. Equation (6) (see 1), gives, in second approximation:* * We can arrive experimentally at the second term (term in parentheses) of equation (8) if we excite a form of oscillations which depends less on the oscillating circuit. This is therefore the case for the spiral oscillations (14,23,25). The spiral oscillations appear in the region above the critical point for the magnetron with and without slots. These are excited by the oscillations of the electrons in the direction of the axis of the cathode in an inclined magnetic field. Under the hypothesis of a quadratic distribution of potential, we obtain a condition of the frequency which is given with a constant by equation (6). In Fig. 3, the curves representing equations (7) and (8) are drawn as a function of the magnetic field. The measurements are derived from (24). The constant of equations (7) and (8) have been adjusted according to experimental results.

23 r/crl k = 6[1r]' k U-Y 00, 0 1_?5 0,6 0 V // I 0,2 0,2 0,4 0,6 0,8 1,0 Fig. 2 Distribution of Potential r ~ r2 and. ~ r1'75 Compared with the Measurements. The magnetron with large cathode is particularly important. It was used in the first place by Gutton26 and employed most recently in the multicavity magnetron. It results from equation (1.47) that the angular velocity depends on the radius. In the multicavity magnetron the alternating electric field is found in the neighborhood of the anode, so that it is essential for optimum conditions that there be synchronism between the waves and the angular velocity of the electrons in the vicinity of the anode. If one substitutes for r = rp in equation (1.47) we obtain with (1.47) and with (5) for the optimun anode voltage

41 Up [= i -2 r2B ]F ( f being the frequency of the alternating field. The function F (rcrp) is given by F (r ) = 1 r2 | r), rc L1(+ 2c r 1 (.2... (lo) 140 12 0 100 - 350 Brp2 /o 60./~ o 40 20 100 200 300 400 500 606 Fig. Wavelength of the Spiral Oscillations 20 20 27 Slater, Hartree, and Bloch 7 have obtained analogous relations. Equation (9) corresponds to the equations of Slater. It dciffers in the foom of the function F (rc/rp). The function F (rc/rp) is, according to Slater: e -/ (v e,,*.l (11) We can verify e:,er'imrentally equation (9) for the results concer'ning the rzulticavity magnetron20 and the donutron. 7 The mode k is considered as unlknoi in equation (9) with the result that in the foznmulas given bry Hartree and Slater, the calculation of the optimuml anode volta.ge is not I)ossible with

42 a large number of slots. In order to compare the theoretical and experimental values, Fisk, Hagstrum, and Hartmann20 evidently proceeded in like fashion when they introduced for k the value of the mode which corresponds closest to experiment into the equations of Hartree and Slater. We proceeded in an analogous manner for our relations (9) and (10), and we find, therefore, the values for k are occasionally a little different. The differences are indicated below. Table I, 4th line, gives the measured anode voltage from the American multicavity magnetrons in the region enclosed between X = 50 cm and X = 3 cm. The third line contains the optimum anode voltage calculated according to equations (9) and (10). The first and second lines reproduce, by way of comparison, the corresponding values according to the formulas of Slater and Hartree. In that which concerns the mode, there exist, with respect to (20), the following differences: In the formulas of Slater, Hartree, and in equation (9), we use: for type 4J42 the mode k = 2 instead of k = 1 for type 4J51 the mode k = 3 instead of k = 1 (We believe that there is, in this case, a mistake or typographical error in (20), the value k = 1 leading to considerable deviation.) In equation (9), we use: for type 5J26 the mode k = 2 instead of k = 3 for type 725A the mode k = 4 instead of k = 5 for type 2J48-50 the mode k = 4 instead of k = 5 for type 4J50 the mode k = 5 instead of k = 7 for type 4J52 the mode k = 6 instead of k = 7 We observe that, according to the comparison between the calculations and the experiment, no multicavity magnetron oscillates in the it mode. Also, in the table of Fisk, Hagstrum, and Hartmann,20 the mode calculated according to the equations of Slater and Hartree is not the E mode. In the region of

43 TABLE I ~GNETRON B?.T[~EEN ~ 20 -- 45 cm. Type 700 A-D. 728 A. 5 J 23. 4 J 21-25 4 J 26-30. hJ42 4J~l 5J26. Wavelength 43.0 32.1 28.6 22.8. 24.0. 43.0 32.1 23.4. p (Slater) (kV) 15.8 21.7 24.3 26.5 19.5 21.8 28.8 18,4 25.9 30,9 19.6 27.4 32.6 15. 24, 23,6 " p (Hartree) 15.6 21,2 23.8 26.2 18,9 21.2 27.5 17.8 25.2 30.2 18.9 26.7 31.9 14. 23. 21.9 " p [Eq. (9)] 13,9 17.6 19.6 21.6 15.0 17.2 22.0 14.4 20.4 24.4 15.6 21.0 25.9 13. 18. 22.5 " p (measured) 12,0 19 21.0 24.5 16,5 19.0 24,5 15,5 22.0 26,5 16,5 23,0 27,0 12 23.0 27.0 7 calculated (o/o) 66 82 84 86 72 77 82 70 79 82 72 80 83 66 86 65 ~ measured (o/o) 35 61 65 61 58 58 62 48 53 53 51 54 60 32 65 58 calculated (A) 7.2 13,6 15.0 17,5 18.3 21.0 27.0 17,7 25.0 30,2 18.8 26.1 30.6 8 20.2 62.0 measured (A) 10,0 19 20 28 20 24 33 25 40 48 25 40 46 9 20 46 MAGNETRON k ~ 10 cm. Type 706 A-C. 714 A. 706 A. 714 A. 720 A-E. 4 J 45-47. 718 A-E. Wavelength 9.8. 9.1. 9.8. 9.1. 10.7. 10.7. 10.7. p (Slater) 15,6 16.1 13.8 22.8 25,4 14.8 22.8 24.8 24,7 28.4 32.3 2h,7 32,3 28.4 14,2 21,7 24.9? (Harttee) 14,5 14.9 12.9 21,9 23.3 13,9 21,9 23.9 23.9 27,6 31,5 23,9 31.5 27.6 13,2 20,7 23.9 ~)[Eq. (9)]' 12,4 12,8 11,2 18.2 20,5 11.7 18,1 19,8 19,7 22,7 25,7 19,7 25,7 22.7 11,3 17,3 19.0? (measured) 11.0 11.0 11.4 18,5 20.9 11,6 17.7 19,5 21.0 24.0 27.0 21.0 27.0 23.0 ll.9 17,9 20.0?calculated (o/o) 35 25 44 64 70 52 70 73 77 80 82 77 80 82 50 67 70 i. measured (o/o) 19 17 31 55 55 31 53 47 62 68 68 62 68 68 34 53 54 calculated (A) 11.8 12.7 12,2 19,8 22.4 13,4 20,5 22.5 41 47 53 41 53 47 11,7 17.6 19.6 measured (A) 12,5 12.5 12,5 16 20 12.5 16 20 45 53 65 45 65 45 12,5 16.0 20,0 MAGNETRON k ~ 3 cm. Type 725-730 A. 2 J 48-50 2 J 55-56. 4 J 50-4 J 78. 4 J 52. 2 J 51. 3 J 21. Wavelength 3,2. 3.22. 3,3. 3,4. 3.2-3, 24. 3, 2-3. 3. 3.2. 3,33. 1,25. p(Slater) (kV~ 10.9 13,5 14,3 10 6 13,1 13.8 11.6 24 0 16,7 9.9 11,1 13.7 16 8. ~.. " 0 (Harttee) 10.2 12.8 13,6 10,0 12.5 13.2 10.7 22,7 15,3 9,1 10,3 12,9 15,4 " p[Eq. (9)] 9,5 11,8 12,5 9,3 11,5 12,2 9.9 21.2 12,8 8.5 9,6 12,1 13,3 " o (measured) 10,0 12 13,0 10 12,0 13,0 12,0 22.0 15,0 11,0 12.0 14.3 15,0 7measured (o/o) 44 52 51 44 52 51 50 66 69 42 43 46 37 7calculated (o/o) 60 68 70 61 69 71 52 73 72 50 56 61 51 calculated (A) 9,3 11.5 12.2 9,3 11,2 12.1 11,4 25.4 14.5 10,1 11 13.1 25 measured (A) 10 10 12 10 10 12 12,O 27 15,0 11 12 ]4 15

44 3 cm, Sayers and Sixmith35 demonstrated experimentally that the tube does not oscillate in the Et mode. Analogous observations have been made already by Herriger and Hiilster8 for a tube with six slots with a small cathode. They find, for the it mode, an efficiency n = 18 per cent with a ratio B/Bcr = 1.5, while for the mode k = 2, they find more than 50 per cent for B/Bcr = 1.5. They have not yet given physical explanations. The alternating electric field, for the modes lower than the -t mode, apparently penetrates more deeply into the space of the discharge, with the result that for this mode both the exchange of energy between the electronic current and the oscillating circuit, and therefore the conditions of oscillation, are more favorable. The measurements made on the donutron27 furnish a better check of equations (9) and (10); the possible mode in this case is determined by a preliminary experiment. In Figs. 4 and 5, the straight lines are calculated accor:dling to equations (9) and (0) for the mode and for the indicated wavelength. The experimental points are borrowed from the work of Crawford and Hare.27 Fig. 5 of this article shows that the mode'C" corresponds to the it mode, the mode "B"' to the mode k = 4. Let us compare a value of these curves with the calculated values. For B = 1200 gauss, we have measured Up = 1600 volts, a value equal to that calculated according to our theory in equation (9). According to Slater, we obtain Up = 2700 volts, and according to Hartree, Up = 2400 volts. In another example of Kilgore, ShuLmann, and Kurshan,28 a magnetron modulated in frequency for a wavelength X = 7.5 cm, has the following conditions: rc = 0.97 cm, rp = 1.93 cm, B = 1600 gauss. For the mode k = 4, we should obtain from equation (9), an anode voltage of 825 volts, as cormared to a measured voltage of 800 volts.

45 Equations (9) and (10) shoulld give the optimum a.no.de voltages which are too small to conoar-e rith these measurements. These equations are actually established unlder the hyuothesis of a resonance be-twaeen the EF field. and the angular velocity at the surface of the anodie. But we must, in fact, look for a resonance between the IHF field and the angular velocity of the electron at a certain distance from the anode. The effective voltage is then incleased, as shown by T:able I and Figs. 4 and 5. In neglecting the space charge, Slater assumes a synlchronism between the wT-ave and the velDcity of the electrons at the center of the space of the Mdischarge. hle optimunm anode voltages calculated according to the formula of Slater will thenbx too large. K:8 Up(KV) k- 6.Ocm K 8 1.8 X.m 6,0 cmcm.- 6.1 cm 1,6 X - 64cm / ~ - 6,5cm a0 X z16,6cm.:6,5cm 1,2 X 16,6cm 1.0 v / rp-0.464cm 0, 0,657 0,6 // 16 SLOTS B(Kr) 0,6 0,8 1.0 1,2 1,4 1,6 Fig. 4 Up as a Function of B UpJ (KV) 1,6 oi=8,lcm ~;-8,5cm \ 1,4 - oX,- 8,9cm 1,2 1,0 0,8 0,6 0,8 1.0 1,2 1.L 1,6 B(Kr) Fig. 5 Up as a Function of B

46 3. T'e Efficienucy TTe can evaluate the efficiency from the considerations given in p.arag-raph 1 for the mechallism of the oscillations of resonance. AcCoidilng to the representations given here, the electrons describe, arovud tChe cathodie, epicycloidal trajectories with a frequency given, respectively, by equations (1.43), (1.39), and (1.44) or (1.47).* The frequency of relative circular rnotion whrlich is suoe rposed on the circular motion revolving arotunld the cathode is given, respectively, by (1.42), (1.53), (1.45), and (1.47). If we neglect this relative circular motion, the electrons reach the anode with a kinetic energy (mr/2) rp2 (o2. The electronic efficiency is therefore M 1D1 (1)_ 77 = 1 - U (12) o0 JO r 2- (15) rp and X are measured in centimeters, Up in volts. According to equations (12) and (6), we find, for small cathodes: ( B )02 ( _ 2 B 1 ( ) (14) It follows from equation (14) that the efficiency for the critical magnetic field is zero. The oscillations of resonm.ice canmot be excited for the critical magnetic field, even if the anode current is zero in the static state. e believe therefore tnhat a theory seeking rto etermine the mechanism of * The 1l" refers to Article Neo. 1 of this report.

excitation of the oscillations of resonance beginning with the behavior of the magnetron for the critical magnetic field, is not valid. According to equation (12), the efficiency is determined only by the ratios rc/rp and B/Bcr, if the plate voltage has its optimum value given by (9) and (10). In Fig. 6, the efficiency is drawn as a function of B/Bcr for different values of rc/rp. The measured points represent the electronic efficiencies measured on American multicavity magnetrons with a ratio of radii rc/rp = 0.375 in the region X = 50 cm to X = 10 cm. The measured electronic efficiencies prove to be below the calculated curve, as is to be expected. Lines 5 and 6 (Table I) show the electronic efficiencies of American multicavity magnetrons, on the one hand, calculated according to equation (12), and on the other hand, drawn from Fisk, Hagstrum, and Hartmann. It is noted that the theoretical efficiency is larger than the measured values, particularly for x = 40 cm (types 700-AD and 4J42). n % 100 r - rp80 e 37r 1.0 1.2 1.4 1,6 1 0, Fig. 6 Efficiency As a Function of B/Bcr To a large extent we can explain the difference and obtain a better agreement between the calculations and ihe experiments. It results, in fact, from the measurements on an English magnetron29 th-.t the bock heating for a magnetron in oscillation reaches 3 per cent to 6 per cent of the continuous

48 power. In Fig. 6, the curve of the efficiency is drawn dotted for rc/rp = 0.375 in consideration of the fact that there is a loss of 6 per cent due to the electrons of unfavorable phase. We see that we reach therefore nearly theoretical efficiency. It is important to remark that, according to our theory, the electronic efficiency does not depend on the value of the impedance of the load. The latter changes only the alternating voltage and the power. A variation of the alternating voltage modifies the time of transit of the electrons in the space of the discharge but does not vary the velocity of the electron on its arrival at the anode, largely determining, from (12), the efficiency. These results are Justified by the measurements of Fisk, Hagstrum, and Hartmann (see Fig. 19 of this article [20]). The negligible influence of the alternating voltages on the angular velocity explains the fact that the shunt impedance of the cavities and of the load does not have an appreciable influence on the continuous voltage and the magnetic field corresponding to optimum operation. This explains that, for very different relative values of the loss, the optimum conditions correspond well to the indicated experimental results. The important differences mentioned between the calculations and the experiments for the magnetrons 700 AD and 4J42 are, in our opinion, bound to a construction that does not correspond to optimum. We shall return to this in Chapter 4. Posthumus,5 Herriger and HMilster, Fischer and Liidi,l2 and the Hartree group29 arrive at analogous conclusions for the electronic efficiency. 4. The Impedance of the Magnetron From the measure of theimpedance of the magnetron, we can obtain conclusions on the behavior of the magnetron in oscillation, this impedance

49 being measured between the slots of a magnetron which does not oscillate. Jhnke30 and Harvey31 have carried out the measurements of the impedance of a cylindrical magnetron with slots and a small cathode by exciting the magnetron from an external generator. We can establish quantitative results giving the order of magnitude of the impedance and its variation with the data of the operation for a cylindrical magnetron with a small cathode. In this case, the angular velocity is independent of the radius. The current of the magnetron calculated in Chapter 5, 1st par., undergoes a phase-focussing action produced by the resonance between the angular velocity of the electrons and the phase velocity of the wave which is propagated around the cathode in such a manner that the electronic current gives the energy to the HF field. Supposing the condition of resonance to be realized, we make two hypotheses in order to calculate the impedance: (1) We assume that, in the radial plane which passes through a slot, the potential undergoes a discontinuity equal to 2 AUp(r/rp)2 and that between these radial planes, it depends only on the radius. We have, therefore, the distributions of potential given alternatively by 0 = Up (l + A) (r) or 0 = Up (l A) p o r -r This condition is not fulfilled for the multicavity magnetrons because the breadth of the slots is of the same order of magnitude as the breadth of the segments. In the magnetrons with small cathode and a small number of slots, this condition is only fulfilled in the vicinity of the anode. But the error is not very large because the exchange of energy is accomplished principally in the neighborhood of the anode.

50 (2) All the space charge present in the space of the discharge is focussed in such a way that it is found only in the negative phase of the alternating field propagating around the cathode. The current density at point r isp rw. According to our hypothesis we find in the negative phase a current density twice this value. In the positive phase there is no current. The current; of the magnetron as a function of the time of transit is given therefore by Fig. 7. The amplitude is 2 Jm; Jm is given by equation (1.57). J4) Fig. 7 Current as a Function of Time The Fourier analysis of the current (Fig. 7) gives (4/it) Ji for the fundamental. The balance of the power is therefore r C AUp Gm = - pr AUp(dr, (15) where p is the magnetron space-charge density, Gm the negative admittance, and I the length of the anode. p is,from equation (1.22) p4E U- (16) p I-'. follows from equations (15) and (16) that the resistance is negative. Rm = Gm 2 -HUpEol Ue (17) P

51 From this it follows that: (1) the admittance Gm is inversely proportional to the amplitude of the alternating voltage; (2) from equation (7), for B >> Bcr, X and Rm are proportional to the magnetic field for a constant amplitude of the alternating voltage. These conclusions are valid only in first approximation. With the aid of the method of Brillouin, we will calculate exactly, in the third part of this article, the impedance for small amplitudes. Equations (15) and (17) are confirmed by the measurements of Harvey and of Jhnke. In Fig. 8, the admittance of a magnetron with an external IF source is drawn as a function of Up from the measurements of Janke. 12 lx 10-6 mho 10 8..4 *2 1 1 2 3 4 5 6 7 8 9 10 Admittance of a Magnetron as a Function of Amplitude According to Janke We obtain the desired straight line. Fig. 9 represents the negative resistance of resonance as a function of the magnetic field for a constant amplitude of the alternating voltage, from the measurements of Harvey. According to equations (7) and (17), Rm must yield a straight line as a function of the magnetic field. This straight line passes through the origin. In Table II, we have compared the values measured by Harvey with equation (17) for a magnetron with rp = 0.5 cm, R = 2 cm. A coincidence, if exact, can be only fortuitous. We cannot interpret quantitatively the measurements of Janke,

52 the length of the anode not being known. From equation (17), the length of the anode should be 1.2 cm, a length certainly too small. K f CALCULATED FROM EQUATION 17 60 50 40 30 600 1200 1800 2400 3000 Fig. 9 Impedance of the Magnetron as Function of B, Measured by Harvey TABLE II Up AUp X Rm (V). (V). (m. measured calculated (kS2). (kS2). 100 29 9,27 7,0 9 100 29 14, 7 12 14,2 100 29 23,2 18 22,8 100 29 33,1 25 32,5 100 29 43,3 35 42, 5 100 29 61,8 45 61,0 100 29 80,3 60 79 100 13,6 10,0 3,5 4,6 100 13,6 19,0 5,5 8,3 100 13,6 60,0 19 25 120 78 61,8 115 127 97 19 61,8 24,8 27,4

53 The behavior of the cylindrical magnetron with small cathode for an HI which is not in resonance with the angular velocity has been studied experimentally by Janke. Fig. 10 represents, according to Janke, the admittance of a magnetron with four slots, in the complex plane, for different amplitudes. We find first that the largest real admittance is associated with an inductive component. As we will show in Chapter 6, this inductive component can be explained by a focussing of the electrons in the radial direction. This focussing of the electrons has the same phase as the electrons which induce an inductive component. Unlike other well-known transit-time generators, the oscillations of resonance have a negative resistance for all frequencies. This fact is one of the most remarkable qualities of the oscillations of resonance. For the electronic oscillations, for example, this impedance is negative only for certain ranges of frequency. In first approximation, J0Hnke substitutes for the experimental curve a circle, and thus finds a circuit equivalent to a resonant circuit with an admittance given approximately by equation (17). The difference between the circle and the experimental curve is considerable. Malter and Moll32 assume the same equivalent circuit for a ma;netron with large cathode. Gni ind 6xO 160 ho ",.." Cap Fig. 10 Admittance of the Magnetron According to Jgnke

The difference between the experimental curve and a circle is qualitatively comprehensible. A circle would signify that the magnetron current would have the same intensity as if there were no resonance between the angular velocity and the HFi alternation. There should exist only a change of phase between the alternating field and the current of the magnetron if the curve is a circle. It is not easy to calculate the impedance of a magnetron with large cathode, the angular velocity depending on the distance. The admittance is probably proportional to the magnetron current, calculated in the first part. The admittance depends on the amplitude in a very complicated manner because the electrons travel a shorter time in the space charge when the amplitude increases,and, therefore, the change-of-phase of the electrons, with the alternating field propagating around the cathode, diminishes. Qualitatively, we find that the efficiency and the negative resistance of the magnetron increase with B/Bcr, but thus, the efficiency of the circuit decreases with increase of magnetic field. It follows that there is, for a given circuit resistance, an optimum magnetic field for the maximum total efficiency. On the other hand, the negative resistance of the magnetron must increase and the efficiency decrease if one increases rc/rp. If the distance between the cathode and the anode becomes very small, larger alternating fields are necessary in order that the space charge not be out of phase during its travel in the space of the discharge. The measurements published in (20) confirm this result. The coupling of the load for a maximum efficiency must be so much less when rc/rp is larger. In (20) the Q of the loaded circuit has the following values:

55 rc/rp Q 0.27 280 0.37 100-150 o.50 280 0.67 350 Except when the value of rc/rp = 0.27, Q is proportional to rc/rp. The tubes 700 AD and 4J42 correspond to the value rc/rp = 0.27, X = 40 cm. As we have already mentioned, these tubes show an efficiency deviating very considerably from the theoretical possibilities. Apparently, in these tubes the different modes are very near to each other. The optimum coupling of the exterior load cannot be obtained without the danger of "demoding" (variation of the mode during normal operation). The impedance of a magnetron with large cathode calculated according to the method for the magnetron with sma1ll cathode, devriates from the measurements more as rc/rp is larger. According to the unpublished measurements of Bruck and Hulster. the alternating voltage in a m:gnetron with 8 cavities is 10 kv for a continuous voltage of 14 kv, rc/rpo = 0.375 and a power of 10 kw. It follows that the impedance is of the order of magnitude of 1000l. From the calculation with the method for the magnetron with small cathode, we obtain the impedance of the order of magnitude of 100 - 500 Q. 5. The Anode Current Kilgore, Shulmann, and Kurshan33 calculate from the measurements the anode current in an oscillating magnetron. They find in this way that the anode current is of the order of magnitude of 1/10 to 1/50 of the anode current of a space-charge-limited diode, without magnetic field and with the same dimensions. The anode current depends on a large number of parameters. We can, nevertheless, expect that it will be proportional to the mnagnetron current (1.51). The anode current should therefore be in practical tunits:

2 56 () Ja = Xk r2v (18) 1-(-) (1 I +Inr~) J, being in amperes, U in kilovolts, and the wavelength, X, and the anode length, R, in centimeters. The numerical constant of equation (183) has been chosen in such as way as to represent the experimental results. WIe find therefore that the anode current is of the same order of magnitude as the magnetron current, as is in agreement with the measurements made with an American multicavity imagnetron. The anode current measured and calculated from equation (18) is shown in the 7th and 8th lines of Table I. Donal, Bush, Cuccia, and Hegba7i34 bive an anode current of 0.73 A for a magnetron of 1 k7i with frequency modulation. We obtain 0.81 A from equation (18). The magnetron of Kilgore, Shulmann, and Kurshan2s has an anode current of 62 mA. Equation (18) gives us a value of 310 m\. The error is considerable. This shows that equation (18) is only a first approximation. We can give an approximation to the trajectory of an electron in an oscillating magnetron. The amplitude of the current is in the neighborhood of 2 Jm (FIg. 7). The anode current is approximately Jm. It follows that the electron travels in a tanvential direction, double the distance between the anode and the cathode. 6. Calculation of the Trajectories of the Electrons in a Plane Magnetron Without Space Charge for Small Amplitude of the Alternating Voltage The relative circular motion of the electrons of angular'selocity Wc has been neglected in our preceding considerations. For B >> Bcr this amplitude is small. In practical cases, B' 1.5 Tcr, with the result thrtthe motion again has a positive influence on the behavior of the tube. Therefore, confining, ourselves to the case without space charge and with a small alternatinrg electric field, we shall calculate the electronic trajectories in the

57 plane nagnetron, a case where the relative motion plays an essential role and where the calculation is still rather simple. But it is not possible to reach the conclusions for the case with space charge since in this case the velocity of the electrons depends on the distance from the cathodle s. In spite of all, the results give a first general idea. In the stationary case, the motion of the electrons is -iven, from equations (1.50) and (1.51), by y = p(l - cos Wct) (19) x = p sin wt - p wt ( (20) wh~re UDp7? =W eB P d W' c m An electronagnetic wave with phase velocity ph = is moving in the x direction (perpendicular to the continuous electric field and the maqnetic field). The alternating voltage at the anode y = d would be i (t -'x) The alternating field ~ is therefore given in the space of the discharge by U shfy ei(wt - x) (21) For r d<<l, we obtain Ey 0 (22) Ex = -_ ei(WHt - Fx) (23) Exd d () t;e can nor obtain the equations of motion

58 = - WC - i7 —d ry i(wHt - rx) (25) If one introduces in the alternating term of equations (24) and (25) for x;and y, the static trajectory (equations (19) and (20), we obtain a factor ei'(p Sin wct), which, developed into a Fourier series, gives -irp sin cot LC1.j S X En J2n (rp) cos 2n w t n-o (26) - iE2n+l J2n+l (fp) sin (2n+1) Wct, where En = 2 for n > 1, E0 = 1, Jn = Bessel function of order n. If we assume a synchronism of the velocity of the electrons in the x direction with the phase velocity of the progressive wave, we then obtain H rp w,. s(27) Equations (19), (20), (26), and (27) will be introduced into equations (24) and (25). The initial conditions are x = 0 x = 0 y = 0 y = 0 for t = 0 fquations (24) and (25) are then integrable. If we take into consideration only the first two terms of equation (26), we obtain the solution - = (1 + CEJo(Pp) + sin Wct - (1 +EJo(rP)) Wct p 2 c iE [Jl(yp)p r JoP(P C) t sin Wc - rpjl(rp) Cct cos wct (28) - 2iE [-(r) T) " (rp )1 1l-cos 0t) +Erp.irJ) s in w t 3

5) LEo(Y ~ JE(I'P) J1() ~E i J1() EiJO (P )] (1 - co CUt); + -iE(P )Jo(rp)wct +Jr c) rp T'Lp)w9t sin Jt (29) +i~ ~-J(F) + p J sinUt + iE [J1(r) F r jo(lp c t COs Wt u iT where E = P e,Obeing the phase at time t = O and for x = 0. Up Equations (28) and (29) are represented in Figs. 11 to 14 for E= 0.1, and rp = 0.5 and 1. These valuesfrp = 0.5 to 1 are not comrapatible with the hpothesisr p << 1. The fields being proportional to sh( p) or to ch(ro) aLnd the value of these functions not departing appreciably from r p 4and from 1 for rp < 1, we have chosen these large values to lmake evident the influence of F o on the motinr. o? the electrons. Fig. 11 Trajectories of the Electrons in the Direction y PF 0. 5 E ~= o.1 In -the nulticavity inar-netron, we have for the Tr mode: = t/b, w-here b is the distance between tGwo slots -. Por the k mode: r = (2n/b)(k/!) (T= nuz.ber of slots). The distance being the difference of the radius between the anode ani the c-athode, we find therefore empirically that, for the America<n multicavity mnagnetrons rd N 1.9 ~ 0.5,

60o that is to say, k 1 ) 1.9 (+ 0.5) (30) x Fig. 12 Trajectories of the Electrons in the Direction X r = 0.5 E= 0.1 Figs. 12 and 14 show the motion of the electrons in the direction of the travelling wave for different phases ~. There is a focussing of the electrons in the negative phase of the travelling wave, as we have already explained qualitatively in Chapter 1.,v\,/ Fig. 13 Trajectories of the Electrons in the Direction Y r = 1.o E= 0.1 Figs. 11 and 13 show the motion of the electrons in the direction of the anode. The electrons of favorable phase go to the anode and the electrons of unfavorable phase to the cathode. For large r p the amplitude of oscillations of the electrons of uvnfavorable phase (Fig. 13) is snmall; the electrons reach the cathode with a small velocity and cause therefore little back-heating, The electrons of favorable phase, on the other hland, reach t'hn

61 anode with a large kinetic energy and therefore decrease the efficiency. Moreover, for small values off p, the back-heating is larger, and the kinetic energy of the electrons which reach the anode smaller.'We can therefore find an optimum value of r p. There are other reasons leading likewise to an optimun value of p, as Goudet has already indicated. If, for a constant number of slots, the cathode-anode distance is too great, the IDt field cannot penetrate very far into the cathode space. If the cathode-anode space is too smaill, the capacitance of the circuit is too large and the strength of resonance too small; the tangential alternating field disappears. Goudet indicated that the optimum distance is of an order of magnitude equal to the distance of the slots. JWe obtain, therefore, the ratio rc 2x -- ~1: - - (31) rp N (N = number of slots). Slater33 gives an empirical forrula for the optimum ratio: rc N - 4 =r + N 4~ (32) Equations (31) and (32) do not differ much from each other. Fig. 14 Trajectories of the Electrons in the Direction X rp = 1.o ~= 0.1

62 There results from the motion of the electrons, in Figs. 11 and 13, that there is a weak focussing in the y direction. This focussing has an inductive phase which can explain the inductive component of the magnetron impedance for the case of resonance (Fig. 10). 7. Conclusions In this report, we have obtained quantitative relations for the efficiency (13), the optimum conditions for operation (9), and the magnetron impedance with small cathode (17). We have developed semiempirical considerations for the anode current (18) and for the optimum ratio rc/rp (30). We are now going to indicate briefly the method of calculation of a magnetron, the wavelength of the anode, the plate voltage, and the mIl power being given. We choose for examples the American multicavity magnetrons, indicated in (20) and (34), i.e., magnetrons with wavelength and construction very different. The ratio rc/rP may be calculated from equation (30) for the different modes, that is to say, for the different number of slots. The continuous power and therefore the anode current are derived from Fig. 6, for a given ratio B/B3c with a circuit efficiency of 90 to 95 per cent. The anode length is calculated from equation (18) and the radius of the anode from equation (9), for a given mode and a given ratio 3/3cr. According to this method, some tubes have been calculated in Table III for B/3cr = 1.4 to 1.5. The dimensions and the information of the operation of the American multicavity magnetrons are shown in the last column; the information on the operation and the dimensions for the different modes, in the other columns. The agreement is satisfactory. Apart from this information, other parameters influence also the design of the magnetron: the strength of the magnetic field, the specified cathode density, the back-heating of the cathode, the maximum anode load, and, for large ratios of rc/rp, the efficiency of the circuit.

TABLE III TYPE 720 A-E. B X = 10 cm, = 1,4. Bcr Mode 3. 4. 720 A-E. Up (kV) 21 21 21 P (kT) 550 550 550 Ja (A) 47,5 47,5 42 a (cm) 4,1 4,1 4,0 r " 0,64 0,97 0,70 r 0,24 0,36 0,27 B (F) 2500 1900 2300 TYPE 4 J 21-25. = 22,8, - = 15. Bcr Mode 2. 4. 4 J 21-25 Up (kV) 2' 22 2-.- 22 P_ (kwi) 50 50 50 J, (A) 4 40 40o 40,t (cm) 7,5 7,5 7,5 4,9 r " 1,05 2,0 3,2 1,5 rc 0,05 0,75 1,66 0,55 B (r) 1620 880 620 1200 TYPE 2 J 53-56 7. = 3 cm, = 1,5. Ber Mode 4. 5 6. 2 J 55-56. up (kV) 12 212 12 P- (kW) 50 i) 50 50 J, (A) 9, 3 9, 9, 31 (cm) 0, 42 0,42 0,38 0,62 rp 0,23 0,26 0,29 0,31 rC 0,12 0,18 0,18 0,16 B () 6400 4350 450 4750 3350

64 TABLE III (cont'd) 1 kW MAGNETRON ACCORDING TO [34J. X = 34,5 cm, B = 1,5. Mode 3. 4. 5. Measured U (kV) 2,5 2,5 2,5 2,5 P (kW) 1 1 1 1 Ja (A) 0,80 0,8o 0 o,80 0,73 t (cm) 2,0 2,0 1,8,3 rp 1,15 1,62 2,25 1,63 rc 0,42 0,82 1,40 0,83 B () 580 410 36o 4o00

65 B II3LIOCTR0GPHY 1. K. Okabe, Proc. Inst. Radio Engrs., vol. 17, 1929, p. 652; vol. 18, 1930, p. 1748. 2. A.,A. Slutzkin and D. Steinberg, Ann. der Phys._ vol. 1, 1929, p. 658. 3. M. Ponte, L'onde Electrue, vol. 13, 1934, p. 993. 4. G. R. Kilgore, Proc. Inst. Radio Engrs., vol. 20, 1932, p. 1741; vol. 24, 1936, p. 1140. 5. K. Posthumus, Wireless Engr., vol. 12, 1935, p. 126. 6. 0. Pfetscher and W. Puhlmann, Hochfrequenztechn. u. Elektroakustik, vol. 47, 1936, p. 107. 7. E. Ahrens, Hochfrequenztechn. u. Elu. ektroakustik, vol. 50, 1937, p. 181. 8. Herriger and Hulster, Hochfrequenztechn. u. Elektroakustik, vol. 49, 1937, p. 123. 9. Lammchen and A. Lerbs, Hochfrequenztechn. u. Elektroakustik, vol. 51, 1938, P. 87. 10. F. W. Gundlack, Hochfrequenztechn. u. Elektroakustik, vol. 149, 1937, p. 201. 11. E. W. B. Gill and K. G. Britton, J. Instnr Elect, Enrs. vol. 78, 1936, r. 461. 12. F. Fischer and F. Luedi, Schweiz. Elekt. Verein Bull., vol. 18, 1937, p. 277. 13. J. S. McPetrie, J. Inst. Elect. Engrs., vol. 80, 1937, p. 814. 14. E. Zieler, HIochfrecuenztechn. u. E]ekrtroakustik, vol. 60, 1942, p. 81. 1i. J. Voge, L'onde Electrique vTol. 26, 1946, p. 345 and 374. 16. L. Brilloui n, P'hys. Rev. vol. 60, 1940, p. 385; J. Phys. Radium, vol. 8, 1940, pI. 25D3. 17. J. P. Blexvett and S. Ramo, Phys. ev.L vol. 67, 1940, p. 635. 18. W. E. Lamp and M. Philipps, J. Appl. Pys., vol. 18, 1947, p. 230. 19. G. Goudet, rique vol. 26, 1946, p. 49. 20. J. B. Fish, H. D. Hagstrum and P. L. Hartrnann, Bell Syst. Tech. J., vol. 25, g1946, rp. 167.

66 21. F. Crawford and M. Hare, Proc. Inst. Radio Engrs., vol. 35, 1947, p. 361. 22. H. Gutton and J. Ortusi, Conference. la Societg des Electriciens francais, janvier 1947.. 23. A. Slutzkin and D. Leljakov, Phys. Z. d. U.S.S.R., vol. 5, 1935, p. 314. 24. 0. Doehler and G. Lueders, Hochfrequenztechn. u. Elektroakustik, vol. 58, 1941, p. 29. 25. E. Megaw, J. Inst. Elect. Engrs., vol. 72, 1933, p. 326. 26. E. C. S. Megaw, J. Inst. Elect. Engrs., vol. 93, Part III, vol. 17, 1946, P. 977. 27. F. H. Crawford and M. D. Hare, Rep. of Rad. Res. Lab., Harvard Univ. 28. G. R. Kilgore, C. I. Shulman and J. Kurshan, Proc. Inst. Radio Engrs. vol. 35, 1947, p. 657. 29. W. E. Willshaw, L. Rushforth, A. h Stainsby, R. Tatham, A. W. Balls and A. H. King, J. Inst. Elect. Engrs., vol. 93, Part III, A, 1946, p. 985. 30. M. Janke, Hochfrequenztechn. u. Elektroakustik, vol. 54, 1939, p. 73. 31. A. F. Harvey, J. Inst. Elect. Engrs., vol. 86, 1940, p. 297. 32. L. Malter and J. L. Moll, R. C. A. Rev., vol. 7, 1946, p. 414. 33. G. R. Kilgore, C. I. Shulman and J. Kurshan, Rep. R. C. A. Lab. Div. Princeton Univ. 34. J. S. Donal, R. R. Bush, C. L. Cuccia and H. R. Hegbar, Proc. Inst. Radio Engrs., vol. 35, 1947, p. 664. 35. H. A. AH. Boot and J. T. Randall, J. Inst. Elect. Engrs. vol. 93, Part III, A, 1946, p. 928.

67 PART II I THEi TRJVELLIftG-'WAVE TUBE IN A MkGINETIC ihIELD by J. Brossart and 0. Doehler Annales de Radioeldctricit6 Vol. 3, No. 14, Oct., 1948, pp. 328-338 Summary In this article the authors study the behavior of the magnetron as a travelling-wave tube; they describe a new type of tube, the magnetron travellingwave tube, and, neglecting the influence of space charge, they calculate the g.in of this tube used as an amplifier. Finally, they point out the essential differences between the traveling-wave tube of the Kompfner-Pierce type and the magnetron travelling-wave tu'oe. The first part of this investigation was devoted to the study of the static characteristics of the magnetron; we have shown, in particular, that for the plane magnetron, in the absence of space charge, the electrons move with a velocity vs perpendicular to the electric and magnetic fields, and it has been found that Vs ='dB (1) where Up is the anode voltage B is the magnetic field d is the anode-cathode distance

68 On this motion is superposed also an oscillatory motion of pulsatin xc= eB. It is interesting to note that vs is independent of y, the distance m of the electron from the cathode. When the space charge is not negligible - always for the plane magnetron - we have seen that this space charge has a constant density in first approximation; the electronic trajectories remain similar to the preceding case where we neglect the space charge, but the velocity of the electrons perpendicular to the electric and magnetic fields is written as 2Up y (2) and depends on the distance y of the electron from the cathode; in addition, the oscillatory motion superposed on the continuous motion depends slightly on U The oscillations of resonance are excited by a high-frequency electric field that follows each of the electrons and constantly takes energy from the aggregate of the space charge which is of the same rotation. The principle 4 of the travelling-wave tube is analagous; we obtain amplification of a highfrequency wave owing to the interaction between the bunches of electrons and the electric field of the wave which is propagated in the same direction as the electrons and with a velocity similar to the latter. We are therefore led to believe that it can be possible to use the magnetron combined with the principle of the travelling-wave tube and obtain in this way an amplifier tube for U.H.F. In the remainder of this study we shall call the new tube Magnetron Travelling-Wave Tube and we shall describe two modes of possible realization, for which we will calculate the gain, limiting ourselves to small signals. 6 Fig. 1 gives a first realized form. C is the cathode, A is the anode; which is made up of a delay line, for example, a helical plate; E is the entrance for the electromagnetic waves; S is the exit. In the direction of the axis is found a constant homogenous magnetic field; 0 is an arrangement for focussing. When the velocity of the electrons leaving the gun is so great that

69 the force duxe to the electric field between A and C, augmented by the centrifugal force, balances exactly the Lorentz force, we have ( e E - 2 = wr77rB rw= m' and the electrons rotate around the cathode. If, in addition, we can neglect the centrifugal force, that is to say if rw~-E, or if we are considering a plane magnetron, we obtain for the velocity of the electrons the relation (1); the electronic trajectories are those of circles concentric to the cathode, and eB the motion relative to the frequency wc= -B is negligible. The influence of C m the centrifugal force and the deviations of the trajectory beginning with the trajectory of equilibrium have been examined in the first part of this investigation. The electronic bunch is finally captured by the collector, K, which is serving likewise as an electromagnetic screen between the entrance and the exit, so as to prevent parasitic coupling. with Circular Trajectories with Epicycloidal Trajectories ]E Fig. 1 Fig. 2 Magnetron Travelling-W-1,ave Tube Magnetron Travelling-Wave Tube with Circular Trajectories with Epicycloidal Trajectories

70 Fig. 2 gives a second realized form. The cathode C is emissive in many places. The anode is made up of a delay line with entrance, E, and exit, S, and a collector, K, to absorb all the bunches and form a screen between the entrance and the exit. Experience gained on such a tube with emissive cathode at the entrance has shown the influence of the space charge to be negligible when the current does not exceed approximately 10 Tm\. On the other hand, for currents much larger, we can imagine that the space charge plays a considerable role, and we will take account of this in assuming that the space charge is constant (except perhaps in the neighborhood of the collector because of the absorption of the electrons), as has been established in reference (1). For this arrangement (Fig. 2), we will have epicyloidal electronic trajectories, and unlike the preceding type of tube, it will be necessary to take account eB of the motion relative to the frequency c = when the space charge is negligible. The magnetron travelling-wave tube presents important advantages in relation to the travelling-wave tube, and it seems justifiable to study the practical realizations of this tube in spite of the many technical difficulties which one will have to overcome. In effect, we have shown5 that the travellingwave tube is not usable for an amplifier of high power, for its efficiency cannot be greater than a few per cent. This is due to the fact that, if one wishes to obtain a gain, one cannot give to the electrons a velocity much larger than the phase velocity of the forced wave which they accompany, and only the surplus of kinetic energy corresponding to this difference of velocity can be transformed into electromagnetic energy. In the magnetron travelling-wave tube, on the other hand, the electrons that would have to give up energy to the high-frequency field constantly absorb energy from the constant field, as we have shown

71 in the second part of this investigation. We would thus have an output very much larger. On the other hand, we could use, in the magnetron travelling-wave tube, an electronic current very much larger (of the order of one ampere) than in the travelling-wave tube (10 to 20 ma.). We are going to study, in this article, the gain of the magnetron travelling-wave tube for small signals. The method used will make possible, in addition, the solution of a problem which has been posed, but not solved, in the second part of this study2, namely: the determination of the starting resistance for a magnetron with large cathode. The calculation of the principal characteristics of operation of a magnetron travelling-wave tube will be done following the same method as that given at the end of the second part. 1. Hypotheses Compared with the travelling-wave tube, the study of the magnetron travelling-wave tube is more complicated because, in the first place, it concerns a cylindrical problem in two dimensions and in the second place it is necessary to take into account the essential role that the space charge plays (see the first and second parts of this study), particularly for the second form of construction (Fig. 2). In consequence, we will not be able to treat the question without a number of important simplifications and approximations: 1. We shall restrict the study to small signals. 2. We shall limit ourselves to the case of the plane magnetron, and to the case of the cylindrical magnetron when one has rp/rc = 1 (rp radius of anode, rc radius of cathode). It is necessary,nevertheless,to note for the magnetron travellingwave tube, that one must already realize this to be considerably different from the limiting case of the plane magnetron.

72 3. For the arrangement represented in Fig. 1 and for the arrangement represented in Fig. 2, when the electronic current is not very large, we shall assume that the influence of the space charge is negligible. In the arrangement represented in Fig. 2 with large current, we shall take account of the space charge by assuming that the density is constant. 4. In the arrangement represented in Fig. 1 we shall neglect motion relative to the frequencywc =iqB, andassume that the electrons describe circles concentric with the cathode. 5. For the arrangement represented in Fig. 2: a) We shall neglect the influence of the space charge but take into account the relative motion of the electrons. b) We shall take account of the space charge by assuming that the density is constant, but we shall neglect the relative motion of the electrons. 6. We assume 77B >> Ic rVc, vc, being given by equations (1) and (2), is the velocity of the electrons. Fis the phase constant of the wave which is propagated in the anode-cathode space. w is the angular frequency of the high-frequency wave. In the case where the space charge is negligible, the preceding inequality is always valid, because we have a gain only when the velocity of the electrons is approximately equal to the phase velocity of the wave. On the other hand, when the space charge is important, the velocity of the electrons depends on their distance from the cathode, so that in the vicinity of the cathode, the preceding inequality is valid only for a very intense magnetic field, BB>.cr. On the other hand, in the vicinity of the anode, the electric fields are important and the inequality is always valid.

73 7. We shall neglect the effect of the boundaries and assume the system to be indefinitely extended along the axis. 8. For the case with space charge, we shall assume rd<<l, which signifies that the wave length of propagation of the wave is large compared with the product of 2nt times the anode-cathode distance. We note that the preceding hypothesis is not valid in the multicavity magnetron. The influence of'd on the electronic trajectories has been discussed in the second part (Section 6). 2. Study of the Magnetron Travelling Wave Tube with Space Charge Neglected A. Study of the Arrangement of Fig. 1: We Neglect the Motion Relative to Frequency wc =77B. The calculation of the gain is done according to the method used in (4). 1. Determination of the Electronic Trajectories. - The system is represented schematically in Fig. 3, with the three axes of reference. A constant electric field, Up/d, is directed along - y and a constant magnetic field, B, along + Z. We assume that the electrons enter in the system parallel to the direction + x with a velocity vo given by (1): Up o = dB PLANE OF ANODE -Z ~~~DIRECTION OF TRAVELLING WAVE --- PLANE OF CATHODE wig. 3 Coordinates of the System

In the absence of the high-frequency field their trajectories are then defined by = VO (3) Y = yo, with r = t - to, to being the time of passage of the electron considered in the plane x = o, and t being the instant when this electron is found at X. If now we assume that a high-frequency wave be propagated in the direction x with a phase constant r= Y - ik, we can obtain the high-frequency electric fields with the aid of a simple theory which neglects the influence of the potential vectors* and we arrive at the expressions: Ex - rAup shi) eiwto eiwt+Px, X -UP sh ify) (4) Ey -= iAUp ch(iry) eiwto eiwrT+rx, =Y ~ -ih(i+rd) with r -Y - ik, k V V being the phase velocity of the wave. AUp, having the dimensions of a voltage, serves to define the amplitude of the field; when this amplitude is small compared to that of the electric field Up/d, the trajectories of the electrons in the presence of the high-frequency wave can be considered to be the trajectories (3), perturbed as Y = Yo + S Y, (5) X = VOT + 8 X. Supposing the phase velocity of the wave approximately equal to vO, S x andsy * A more complete theory beginning with the equations of Maxwell is developed in the appendix; for Fr' 2 ~ 2/c2, the case which interests us here, this gives practically the same expressions for Ex and Ey as the simple theory.

75 satisfy to first approximation the equations +w 8x = iWrUn chhiryo) ieto ~ r Y + C = irAu' P sh(irif e e (6) x- A = U sh (iryo) eito ei, where we have used e i Vo 77= m c = W7 B, = =wp - ivoY 1 _- VO If we have ll << w, it follows that Sy I= Iaupc -. i..rY it ei.T WCC sh(ird) (7) 8x = ri sh(irfyo eiwto eir, =~~U-kn sh(irad) 8vx = i8X, 8vy = (8) We must remark here that Sx and By are proportional to 1/c, whereas, in the travelling-wave tube, the perturbations in the trajectories are proportional to 1/~2 [see (3), (4)]. On the other hand, Sy is proportional to the field Ex and Sx to field Ey. 2. Calculation of the High-frequency Current. - The unmodulated electronic beam is supposed to occupy the interval yo, yo + Ayo and carry a constant current I. The constant density of charge is given by 0, I — 00 = vIy Let us now apply to the modulated beam the equation of conservation of electricity: V (pv) = t

76 Restricted to small motions, this equation is reduced to vo -(P - Po) + iw(p - pO) = O and shows that one has everywhere p = Po. In other words, in the modulated beam the density of charge remains constant and equal to the density of the unmodulated beam. The alternating current ix is therefore due only to the variation of the cross section of the bear., and one can write Yo + AYo + 8(Yo + AYo) I + i =x = Po [ v0 + 8Vx(y)] dy Yo + 8(Yo) so, since Ikl< w and Ayo is very small ix = iIdF u- ch s(irYo) eiyo eiEr. (9) TJ., g sh(ipd Remarks: a) We can find directly the expression for the current by applying the conservation of electricity in the direction x, that is to say, by writing I + ix = I dt_ dt' but this method does not show directly the interesting physical fact that p = Po throughout the beam. b) The alternating current iy can be neglected in first approximation; in effect, we do not have the continuous component of velocity along y, ancl the current crossing an element of height, q, situated in the ordinate y is written s imply my = X

77 a term which is negligible compared to ix, since ~ is very small compared to w. 3. The Energy Balance. - Let: - dP be the apparent power given up by the electronic current along a short distance d x; + dP1, the apparent power consumed by the line along the distance d x; + d P2, the increase, along d x, of the apparent power which is propagated in the direction of the wave. From the law of conservation of energy, we can write dP + dP1 + dP2 = 0, (10) the equation which is going to permit the calculation of the propagation constant of the forced wave,beginnign with the characteristics of the tube (current, voltage, magnetic field, etc.) and the propagation constant Y -ik of the free wave capable of being propagated along the delay line in the absence of electronic current. - d P is given by Exix*dx and is obviously written, by neglecting Y in comparison with ik in the hyperbolic funct ions 02Yy -2 hkcvwiw2 -dP: I)c2 d2 ) + (12) For the calculation of d P1 and d P2, it is necessary to recall that the apparent energy carried by the wave can be written p ExEx* (13) 2X (13) Rx being a constant characteristic of the delay line (the coupling resistance). (In the Appendix, we shall calculate the electromagnetic fields and the coupling * indicates complex conjugate.

78 resistance for a helical plate.) Under these conditions, we have EE= x (X+ ik) dx, (14) dP2 R= (y+ ik) dx. (15) Cancelling separately the real and imaginary parts of equation (10), we finally obtain the two relations (y) (Y2 +2 22) dRx coth (kyA) (16) V2 2Zo0 vo dRx coth (kyo) 2 (k - k)(y2 + 2 p) = d oth (17) vo ~ 2Z0 where Zo = _ represents the impedance of the beam. Relations (16) and (17) now permit the calculation of the gain of the tube and the phase velocity of the forced wave. Relation (17) shows, in particular, that if k = k, we have p = o; in other words: if the velocity vo of the electrons is equal to the phase velocity v of the free wave, the phase velocity v of the forced wave will be the same as that of the free wave. This is the profound difference in the mechanism of the operation of the magnetron travelling-wave tube and that of the travelling-wave tube where the phase velocity of the forced wave is always different from that of the free wave. Relations (16) and (17) are discussed at the end of paragraph B. B, Study of the Arrangement in Figure 2: We Will Take into Account the Relative Motion of the Electrons. -- In the preceding paragraph we have neglected the relative motion of the electrons, because we assume that an appropriate arrangement of focussing injected the electron beam into the cathodeanode space of the magnetron with a velocity v0 parallel to the cathode and * indicates complex conjugate.

79 equal to Up/dB. Now, it is evident that this arrangement will be difficult to realize,and, as a practical matter, we will rather put the source of electrons in the plane of electrode C (Fig. 1). In this case and in the absence of space charge, the electrons describe the static cycloidal trajectories calculated in the first part of this study [equations (50) and (51)]: y = x (1 - cos eCT), (18) x = vor + 2r sin NcT, (19) where vo is given by (1), Yr=d (B —) being the amplitude of the cycloid and r = t - to is the time of transit of the electron entering at time to. In the Second Part, paragraph 6, we have calculated the dynamic trajectories of the electrons when the anode-cathode distance is small, for example r d = 1 ( is the wavelength of the RF in the cathode-anode space); we have found in particular that for small values of r d, the amplitude of the relative motion of the electrons in favorable phase is small, also that the electrons in unfavorable phase have a relative motion of large amplitude (see Part II, Fig. 11). For the values of rd of the order of 1, we have likewise found that the amplitude of the relative motion of the electrons of unfavorable phase is large, which entail a small efficiencyan important part of the electronic current being absorbed by the cathode (see Part II, Fig. 13). As a practical matter, the American magnetrons have the values of rd of the order of 1.9 + 0.5. For such values of rd the calculation of the dynamic trajectories of the electrons is complicated; these trajectories have components of pulsation w(pulsation of the HF wave), Wc = 77 B, 2Wc etc., also Pc+ + qw (linear combinations of the coefficients of integral number of pulsationsw c and w). In that which follows, we will take account only of the components of pulsation W, since only these components will give rise to a current frequency F = 2-. Once the HF current

is calculated, we will be able to determine the gain of the tube, providing we do not take account of the absorption of the electrons by the interior and exterior cylinders. Also, we have seen (Part II, paragraph 6) that the components of frequency Fc = 2-c should have a large influence on the electron trajectories, for these are the ones which cause an absorption by the electrodes. We can neglect this absorption when Yr is smaller than d and when the interior cylinder is at a negative potential with respect to the cathode. Suppose that these conditions are fulfilled and assume that an HF wave is propagated in the direction + x with a phase constant = Y -ik, then the electric fields By and Ex are given by the relations (4) and the trajectories of the electrons will be the trajectories (18) and (19), perturbed, and the perturbations Sy and ax must then satisfy the relations: 8Y + c8x = iThAUp chiya)- e to eiwT+Px ~~P sh ~(f~rdT ~(20) 8 5- c8Y = nrU sh(Tp I ) eito eiwr+rx sh(iJrx) Substituting in equation (20) x and y from the non-perturbed values (18) and (19) and neglecting in the development of ch(iFy)e r and sh(iFy)e PX all the terms which contain wc, we can write equation (20) in the form 8Y + c8x = Ae ito ei (21 (21) $x - wcSy = B e ito eiT, where A = i7 AUp sh( (22) B =77 rAUp T (23) JQ = o( 2 1 + 1-6(iP r) + 24.128 (irYr) + 720.2ix(i Yr)6 J+ 2(4 2r[( 21) +-FZ7 (iPYr)4 + (i49r ) 611 (24)

81 T( - ( )-r + i-.Yr (ifYr)3 + 206 (ir)] +J2 (4__2-) [ir + 120.6i ( (25) where w + W -irvo (same as the preceding value). Under these conditions the forced solution of (21) gives: (A C -A - i —-B ir y = eto e &,.C. -B+ iA = $ -2 8_-2 eiuto ei&T, or, practically, whenlliw<<c, By = iB eiwto ei$T, (26) CWc 8 = -iA eiwto ei$r (27) Applying the conservation of electricity in the direction x, we can then calculate the ES current ix in the same manner as in paragraph 2-A. But at first it is necessary to clarify the following point: as a result of the cycloidal motion of the electrons, we no longer have a density of continuous current constant in the x direction, but the density depends on y, the distance to the cathode. In particular, for y = o, the continuous density in the x direction is zero, so that, in the vicinity of y = o, the approximations made for the calculation of ix are no longer Justified.

82 In that which follows, we will admit, in any case, that the density of continuous current is constant in each section and equal to the mean value This permits us to simplify the calculation of ix without involving too large an error being made in the remainder under the hypothesis of small signals. We find then Y Yr ix =J dix, (28) y= o with dix = iI A rQ eiwto ei(T dy (29) ~X Yr Up sh(ifd) The energy balance is written as before: dP + dP1 + dP2 = o, (30) with Y = Yr -dP = x d (31) = P (y+ ik) dx, (32) dP 2ExEx ( + ik)x (33) x Let us recall that we have Ex = -ru sh(id eiWto ei&T see equation (4) k sh -kd(ia Rx = h c t / e-k sh() see Appendix, (15') inih tcot sh(kd * indicates complex conjugate.

83 Cancelling separately the real and imaginary parts of equation (30), we obtain finally 2 w2 2 -(4 (r _ )(72 + 2 p2) _ DY _2 (k - k)(y2 + v p2) D p, (35) with -kd D Q ch(kYr)-l sin' jgjf de X 2YrZo sh( - cot' V o h vo Vo vo Ak p = 1- = u(36) v vk (6) V u = 1 - Ak = k - k The relations (34) and (35) are identical to the relations (16) and (17) obtained for the case where we neglect the relative motion of the electrons. Only the value of D is different in the two cases. Discussion of the Results: Let us assume from the first that p = o. We have then propagation of an amplified wave of amplification constant: r- k de-kd 2 ~= + -B + (37) 2 2YrZosh(kd cot/ Eo h Vo with f(kYr) = rhk- 1 () + (kYr) +- 14 {. 2)1 (kY) + 10 (kYr)+] +J2L ) + 2.52 (kYI ) + 2 (kY )6 1

84 For variation in the ratio B/Bcr from 1 to 1.3, we find that Y is maximum for values of kd near 1. Let us assume that Y = o (the delay line has no attenuation). There are three possible values for Y, and to each of these corresponds a value of Ak:?= o, with Ak = D, vo 2 ) = - D - ( u U\ w u -v 2) with Ak = vo 2 (38) 2 vo 2 T= (vo with Ak = 2 If we are in the region where we have amplification, we will necessarily have D>( u); consequently, the wave corresponding to Y = o cannot exist, for this would correspond to Ak imaginary, and this is contrary to the hypotheses of the calculation, and there is no possibility of propagation except for two waves: the one strongly attenuated, 2 the other strongly amplified, - / Wu\ 2 ='D+ v 2)' each one of these having a phase velocity v equal to the arithmetic mean vo+V 2 between the velocity vo of the electrons and the velocity v of the free wave. This constitutes, therefore, one more essential difference between the operation of the magnetron travelling-wave tube and that of the travellingwave tube where one always has three waves. The existence of only two waves coincides also with the fact that the gain of the magnetron travelling-wave

85 tube is proportional to I1/2 and not to I1/3, as in the travelling-wave tube, and that the gain is reduced to the value and no longer. Study of the Initial Conditions: In the travelling-wave tube, the three waves fulfill, e.t the entrance, the following conditions: Z fields = field of the injected wave, E F currents = 0 IF velocities = O. In the magnetron travelling-wave tube, the two waves will fulfill, at the entrance, the conditions: X fields = field of the injected wave, X HF currents = 0, E HE velocities = 0. The two first conditions give E1 + E2 = Eo (39) El E2 E1 2 = 0, so that 1- = Eo E1 = - For p = o, we obtain - ~ -~~ EO E1 = E2 2= and the gain in decibels is written Gdb = 8.7Y2 - 6 (40) r being given by (37) and f being the length of the delay line.

86 Remarks: The condition 2 velocities at entrance = 0 is fulfilled automatically; in effect, in a magnetron travelling-wave tube, there is always, in addition to motion parallel to the cathode, a relative motion a function of w c 2 wc, etc., and the amplitude of this relative motion depends on the phase of the EF wave and is determined by the condition I velocities at entrance = 0. Conclusions In closing, we shall recapitulate the principal differences between the magnetron travelling-wave tube and the travelling-wave tube. 1. In the magnetron travelling-wave tube, the current is proportional to 1/t, while it is proportional to 1/42 in the travelling-wave tube. This involves a smaller gain for the magnetron travelling-wave tube than for the travelling-wave tube. 2. In the magnetron travelling-wave tube, the gain is maximum when the electrons have the same velocity as the forced wave,and, in these conditions, this velocity is likewise that of the free wave. In the travellingwave tube, on the contrary, we have a maximum gain when the velocity of the electrons is equal to the velocity of the free wave. 3. There is no modulation of density in the magnetron travellingwave tube, while this modulation exists in the travelling-wave tube and has important consequences (9). 4. The efficiency of the magnetron travelling-wave tube is large, while that of the travelling-wave tube is small. 5. In the magnetron travelling-wave tube, there exist two forced waves; in the travelling-wave tube there are three. 6. The existence of two waves instead of three involves: a) that the gain of the magnetron travelling-wave tube is

87 proportional to I1/2 instead of I1/3 for the travelling-wave tube; b) that the attenuation of the delay line comes into play appreciably by the factor?r/2 (instead of Y/3 for the travelling-wave tube) to reduce the gain; c) that the amplitude of the injected wave is divided into half for each of the forced waves (in the case of maximum gain), while this is divided by three for each of the three forced waves of the travellingwave tube. APPENDIC In the following pages we shall calculate the resistance of coupling (see equations (13), (32) with the aid of an approximate method used by Rudenberg8. Fig. 4 shows a delay line such as is practically realizable for a magnetron travelling-wave tube: it acts simply as a helical plate wound on a cylinder of rectangular cross section; the wires of the surface of the helix turned toward the cathode make a constant angle 4 close to i/2 with the direction x. In order to simplify the theory we suppose that the HF fields in the cathode-anode region are due only to the Fi currents circulating on the surface closest to the cathode, the influence of the upper surface and the lateral surfaces being considered as negligible. We can, therefore, to a first approximation, replace the helix by a plane located at y = d indefinitely extended in the directions x and z and infinitely conducting in the direction making the angle 4 with x; moreover, it is necessary to impose on the fields the following conditions in the phase to take account of the geometrical structure of the helix. p being the pitch of the helix., s being the length of a turn of the helix,

88 we should obtain E (x + s cos4, z + s sin ) = E (x + p, z) H (x + s cosTS, z + s sink ) = H (x + p, z) Fig. 4a Helical Plate Finally, we have to find the HF fields produced by the surface of current described above which separates the space into two regions: Region I is the interior region defined by 0 6 y ~ d Region II is the exterior region defined by y > d The boundary conditions, which it is necessary to take into account, are: a) On the cathode (for y = 0) Ex = 0, Ez = 0, (1') conditions which express that the electrical field is normal at the cathode. b) At the anode (y = d) Ex cos + EZ sin I = 0, (2') a condition which expresses that the electric field is to be normal to the current in the crosspiece surface. FxI = ExII, EzI = ExiI E (3') conditions which express that the tangential electric field is to be continuous at the crosspiece of the surface of the current. HzI sin + HxI cos = HzIT sin + HxII cos, (4') a condition which expresses that the current of the surface is parallel to the direction making an angle k with x.

89 E (x + s cos, z + s sin4' ) = E (x + p,z), (5' a condition already expressed above. Z /+s cs I /' / Z +S SIN AP / / / / /' / / / / / / Fig. 4b Replacing the Helix by a Plane System c) At infinity (y = cD). The fields should be zero E (y = Oa) = O (6') H (y =oD) = O It remains, therefore, to integrate the Maxwell equations: V x E = _ H, V x = o V. E = O V. = 0o in the case where the fields depend only on x and t by the factor ei(wt - kx), Letting,%O EoC -2 = K2 = we obtain in particular for Ex and Hxj -2Ex +. + (K2- k2) Ex = 0, Ay Az2

90 equations which show that the fields depend only on z by the factor e-igz. Finally, taking account of conditions (1'), the fields in the interior are written EYI = i B1 Al ch ay [k2 _ K2 l 2 - K2 ] ch ExI = B1 sh cy Ezi = k2 _ B1 + i k2 A1] sh ay(t-k- z). HY = Lk2ia A1 + coEo B1] sh czy k2 - K2 k2 - K2 Hx1 = A1 sh cy'ZI =Al h ay k2 2 1 k2 2 1] chcy Taking account of conditions (6'), the fields in the outside are written EyII [ ikc- BK 2 Ak]2 K 5xII B2 EzII = [ - B2 i z ~, A2 EkIk - K[ 72 - K2e-y i(Wt - kx - z), ikc c rII = 2 A2 + k -K B HxII - A2 HzII = A2+ i 2 B2] k HL ks2 - K'a, p, k being related by the equation 2 = a2 + K2 - k(7') k2.~~~~~('

91 The conditions (3'), (5'), (1') and (4') give,respectively,.2 = - A1 e d sh (ad) and B2 = B1 ead sh (ad), (8') = = - si-n + cost) (9') iE2 - K2 in A1 + [k2 sin + cosj 31 = 0, (10t) _K2 - K2 k2 _ K2 Il[k K Cos- A Al - wEc2 sin4 B1 = 0. (11') k2 - K2sin + cosk k 2-K2 The fields are now known without ambiguity and in combining their expressions with relations (7') and (11'), we can express them as a function of a single constant which, in addition,we will be able to relate to the current density of the surface with the aid of Ampere's law. We have therefore: a) In the interior EyI = si - A1 ch cay F2cI =I_ i A/ 1 sh ay EzI = i /-~- cot4' A'^!1 sh cry j ei(wt - kx - hz). (12') Hy, - iwrl oh coy s in I HyI = A1 ch ch y HzI = - cot- A1 ch y7

92 b) In the exterior - ead sh acxd yII \o A ExI = i e sh add A EzII = i ead sh ad cot A e-a ei(t - k - kx z) (13') ied sh d A Hx II = - ead sh d A1 HzII = ead sh ad cot~ A1 with ( p si~ - cot )nK' p ers: We s coth K (14o ) Remarks: We see that for near to th near t o e expression for the field on the interior are approxirnately the same as those given by relations (4), which would be obtained beginning from a simple theory which neglects the influence of the vector potentials. Calculating the flux of the Poynting vector P = 1/2(E x across the right-hand cross section, we obtain the power transported,'., for a helix length, h, W~ = AJ / s_ n cot ~ [sh2 cxd + ch2czd - 1

93 and the resistance of coupling Rx can be deduced from the relation.T Exx-* -~J 2Rx which allows us to write a c o sin ec-adh2 (15') Rx: h Co cot~i shad a

94 B3IBLIOGtRPHY 1. O. Doehler, Ann. Radiodlectricite, vol. 3, 1948, pp. 29-39. 2. 0. Doehler, ibid., vol. 3, 1948, pp. 169-183. 3. J. Bernier, ibid., vol. 2, 1947, pp. 87-101. 4. 0. Doehler and W. Kleen, ibid., vol. 2, 1947, pp. 232-242. 5. 0. Doehler and W. Kleen, ibid., vol. 3, 1948, pp. 124-143. 6. Patent application C. S. F., deposited 9 January 1947, under number 528014. 7. Patent application C. S. F., deposited 13 June 1947, under number 536160. 8. R. Rudenberg, Applied Physics, vol. 10, 1940, pp. 663-681; vol. 12, 1941, pp. 219-229. 9. 0. Doehler and W. Kleen, Ann. Radiodlectricitd, vol. 3, 1948, pp. 184-188.

95 PART IV TIE TRAVELLING-WVAVE TUBEW WITH A MAGNETIC FIELD Extension of the Linear Theory, The Effects of Nonlinearities and the Efficiency by 0. Doehler, J. Brossart, and G. Mourier Annales de Radiodlectricit6 Vol. 5, No. 22, Oct., 1950, pp. 293-307 Summary The authors study again the linear theory of travelling-wave tubes with constant magnetic field without taking into consideration one of the assumptions made in the preceding issue. Two additional waves are found in that case, which are neither amplified nor attenuated. By computing nonlinear effects, they point out that the essential fact is the absorption by the anode. The efficiency is then evaluated. When the electron beam is sent with a velocity small with respect to the anode voltage, its value is larger, and a simple expression can be found for it. (U.D.C. 621.385.1.029.) Introduction In a previous publication, Warnecke and Gudnard have pointed out, along general lines, the properties of a new type of amplifier for UEF, the travelling-wave tube in a magnetic field. The first experimental results obtained on these tubes have been mentioned in reference 2.

96 One of the forms arising from this, the plane magnetron travellingwave tube, is represented very schematically in Fig. 1, in a plane perceni.cular to the direction of the magnetic field, B. An electron gun produces a beam of electrons, F, which is caused to be displaced between a delay line for electromagnetic propagation, Ln, and a plate, P2. Ln is maintained at the high voltage of the tube, the plate, P2, being at a potential near to that of the cathode. The space in which the beam is displaced is also the seat of an electric field and of a crossed magnetic field, uniform and constant; they form with the direction of propagation a "trihedral trirectangle" (three mutually perpendicular vectors). We analyze generally the motion of the electrons in this case in two parts: a) A uniform rectilinear drift motion of which the velocity is perpendicular to the electric field, E, and magnetic field, B; the velocity, vo, is given by the ratio E/B. It is characterized by equilibrium between the force exerted by the electric field and the force due to the magnetic field. The equilibrium velocity vo is independent of the potential of the plane in which the electron is moved. b) With respect to a system of reference endowed with a drift motion, a relative motion of free oscillation in two dimensions whose pulsation, ock is defined uniquely by the magnetic field and which can be avoided by proper initial conditions for the electrons. We will call "optically ideal" an electron gun which produces a rectilinear and laminar beam, that is to say, where the relative motion is eliminated, and we will limit ourselves to this case. In Fig. 1, the beam is extracted from the cathode, C, by a positive plate, P1, held at a potential lower than that of Ln. The electrons move with a velocity very near to that of a wave guided by the delay line, in such a manner that they remain practically in the same phase during all of their

97 transit in the tube; the HF field produces a progressive bunch in the midst of the beam and borrows from the energy of the electrons. These latter should be retarded longitudinally as in the travelling-wave tube if the presence of the magnetic field were not converting this longitudinal slowing-up into a transverse motion which g;rows continually toward the line where the potential is higher, allowing all to have practically the same velocity. Thus, the electrons take the energy from the continuous electric field, which they do not keep but which they give immediately to the HF field. EL ~~n 5 Ur -. l3999 -Ln Fig. 1 Essential Elements of a Plane Magnetron Travelling-Wave Tube Fig. 1 Essential Elements of a Plane Magnetron Travelling-Wave Tube Ln, "slow-wave" line (anode); En, input of HF circuit; S, output of HF circuit E, continued electric field; B, continuous magnetic field; F; static trajectory of electronic beam. One deduces immediately an estimate of the maximum theoretical efficiency of the tube if the beam is very small. If it enters into the condenser made up of the plate P2 and the line Ln with a potential Uo and if Ln is at potential Up with respect to the cathode, the energy given to the electrons in the space of the discharge is e(Up - Uo). Let us assume that all the electrons reach the line with the initial velocity %v (if the signal is very small, the velocity imported to the HF field is negligible); the field, therefore, receives all the energy e(Up - Uo), and the electronic efficiency is e(U Uo ) U = 1 U eUp Up

98 In the plane condenser, the potential grows linearly with the distance to the electrodes, and one has accordingly 7el = 1 d yo being the initial distance of the beam to the plate P21 and d the distance between Ln and PI2, provided that P2 is at potential of the cathode. The efficiency is therefore bound in a very simple fashion to the geometrical characteristics of the tube. Contrary to the situation prevailing in the tube with linear progressive waves, one can thus transform into kinetic energy almost all of the energy derived from the source of voltage, provided that the beam passes sufficiently close to the plate P2 in the static condition. In addition, experiment has shown that one could easily cause sufficiently large currents to pass in the space of the discharge. One has therefore in the magnetron travelling-wave tube an amplifier with large power output and high efficiency. The nature of the known quantities concerning the operation are such that if one bends a plane tube to resemble the one we have described, giving it a diameter of the order of one-half wavelength, the centrifugal force to which each of the electrons is subjected is very small compared to the two other forces which would be exerted on it primarily in the static operation. One can therefore construct from plane tubes circular magnetron travellingwave tubes which will show only minimum differences. A first theoretical study on the magnetron travelling-wave tube has already appeared in this journal (3). We plan to complete this now: a) Up to now we have limited ourselves to the case where -I = Wc >> ic+vo (1)

99 e and m being, respectively, the charge and mass of the electron, B the magnetic field,w the angular frequency of the amplified signal, vo the electron velocity, and r = r - ik the propagation constant of the wave in the direction of the beam. This relation indicates that the angular velocity of relative motion of the electrons is large compared to the modulus of the angular frequency of the wave seen by an electron according to a linear theory; let us remember that this apparent angular frequency is a complex quantity if the applied signal varies exponentially along the electron trajectory. This hypothesis is certainly always very well verified. In fact, wc is of the same order as w or even larger; Xis small compared with k (if one has Y = k, the gain would be 55 decibels per wavelength in the line, much smaller than the wavelength in vacuo'), and jw + r v is little different fromw- kvo; w - kvo is zero if the electrons are in absolute synchronism with the wave (vo = w /k, velocity of the wave); even if the synchronism is not absolute, this quantity is smaller than w and therefore smaller than Wc by at least a factor of ten. Nevertheless, we have been led to abandon this hypothesis for the following purely theoretical reason: in reference (3), we found two waves; that is, there are four initial conditions: one for each of the two components of HF velocity, and one for the current and one for the field: X fields = injected field (2) C currents = injected current; (3) as in (3); but, on the other hand, 8 Yo = O, = o. We will find here two supplementary waves and we will have four unknown quantities to determine in the four equations. This will permit us to explain a remark in reference (3) relative to this matter. We will see that the supplementary waves involve a motion with an angular velocity very near to

100 c}, which characterizes the relative motion of the electrons in the purely static case. The small difference from the static case is due to the mechanism by which these waves are coupled by the fields of the amplified end attenuated waves. We car compare these last to the forced oscillations of an oscillating system and the two supplementary waves to the free oscillations.. b) In (3), we wrote (see Notations) that the power given by the beam along the element dx is -dP Exy o)ix - dP 2 dx. The field Ex varies with y, and the electrons do not keep the coordinate yo but withdraw by y; one must therefore adopt Ex(yo + y) instead of Ex(yo). This effect, that is studied for the travelling-wave tube in (4j) and (5) leads to an electronic gain in decibels J2 times larger. c) In the Appendix of (3), we have given a calculation of the coefficient of coupling of the beam and the wave, defined by Ex(yo)E*(Yp) Rx 2P We will give here a calculation corresponding to the conditions nearer to those which are effectively realized, in using a method given by Pierce6. d) But the most important part of this article treats the nonlinear effects in the tube and leads to an evaluation of the efficiency which shows that the method given in (2) is Justified. It is remembered that, in the first theoretical study (5) we limited ourselves to small signals, assuming that the HF motions of the electrons would be sufficiently small so that they remain almost in the same phase of the field. This is only for comparison with this hypothesis which leads us to the linear equation. As has been done for the linear travelling-wave tube7, we shall use here the method of successive approximations and we shall limit ourselves to the third. Actual

101 practice differs very much from this approximation; however, we can imagine that some other effects, of which these initial hypotheses do not take account, influence appreciably the power and the efficiency (for example, the absorption of electrons by the electrodes); the gain calculated up to the third approximation must be sufficient to isolate a certain number of essential parameters and their influence. This method has the advantage of being more general and valid for all types of magnetron travelling-wave tubes than that of Nordsieck, which is more precise. Hypotheses The initial hypotheses are the following: 1. We restrict ourselves to the plane magnetron, the case which is approached sufficiently by the practical realizations. 2. We neglect the effects of space charge, continuous as well as alternating. 3. We neglect the boundary effects, that is to say,we treat the magnetron travelling-wave tube having two dimensions. 4. We have an ideal optical system: the static trajectories are rectilinear. 5. The beam is thin, that is, Ir Ayo<c 1l (Ayo, being the thinness of the beam.) 6. The propagation constant r = Y -ik is such that r is clearly smaller than k. 7. For the calculations of large signals, we will assume eB = >> i + rvo 8. The velocity of the electrons, vo, is small compared with the velocity of light.

102 9. The resistance of coupling is zero for the harmonics of the fundamental wave, 2w, 3w, etc. 10. We do not take account of the perturbations applied to the HF fields by the absorption of the beam along the line. 11. We assume that the HF line possesses a continuous structure; actually it may be either a helix or a line with waves; in this case, the free wrave allows a large number of components tightly bound together; our hypothesis considers only the largest and most rapid among these, although the others undoubtedly are effective if the behavior of the tube is nonlinear. This approximation is less valid when the "opening" of the circuit is larger (case of large accelerating voltage). Notat ions vo - velocity of electrons v - velocity of free waves v - velocity of forced wave P=1 -4 v ro = Y -ik - propagation constant of free wave r = Y -ik - propagation constant of forced wave B = magnetic field Up - voltage between anode and the interior cylinder Uc - voltage between anode and cathode eB c m W - high frequency angular frequency - efficiency Rx - resistance of coupling ~ /cm2 Ex - electric field in the direction of the beam

103 - electric field in the direction of the continuous electric field pO - density of the continuous space charge P1,2 - density of the alternating space charge in first and second 21' 2 approximation VXl' Vx2 }..., alternating electron velocities in first and second zVy1,;Y2 approximation d - distance between anode and interior cylinder 4= Wto -I- u pr ~ = wt - x T = t - to = transit time a -=, —-..... Y, b = k,. 3rb p2 + 2 -2'" k2 p i- /ko C P-1 —,- _ __Y/ko c 2 - 2 g= 2 -. ~p + /k p + 9yd/k 4. Linear Theory (a) Determination of the Electron Trajectories. The system is represented in Fig. 1 with the three axes of reference. A constant electric field Up/d is directed along Oy and a constant magnetic field along + z. The electrons enter into the system parallel to the H x direction. In the absence of an H' field their trajectories are therefore defined by Y = Yo x = vo T, (4) with r = t - to, to being the instant of passage of the electron in the plane x = o, and t the instant when this electron is found at x. The velocity vo is given by vo = (5) 0u

A lTi wave is propagated in the x direction with a propagation constant r = r - ik, withc<< k. If the velocity of this wave is small compared with the velocity of light we can derive the fields from a scalar potential and we have E~y - - ~~U oo h (ky) iet + (6) E k Ap sinh (kd) s inh ky) ict + lx + ik AUp sinh (kd e Let us consider the trajectories in the presence of the HF wave as the perturbed static trajectories y = Yo + 81Y + 82Y+ *.**. (7) x = VoT + lx +- 82X +.... and we obtain for the first approximation inSx andsy the equations: + c S = k AU cosh_ ky_) eiwto + ier c UP sinh (kd(8) _~.. Wsim (kyo) iwto + icT x - (cS = -il7k AUp ( - c- i7k Up sinh (kd) e with - wp - ivor (9) There follows from this that: _1 coth (iryo) iWto + itr, - i ~ rAv-p sinh i ) 1 Wtc th e (10) 8x = P cosh kilyo) 1 += tanh (i yo ) eit~ + ir, COc sinh (i1r and Svy = iCsy, 8v - ix, (11) - i~~~'Sx ~~(11)

(b) Calculation of the Current. The unmodulated beam is assumed to occupy the interval yo, yo + AYo and to carry constant current Io. The alternating current is then given by a. i = p(v0 +Si) (1 + ay 8 ) Ay\ -II. (12) The term in ~ (By) represents the variation of the cross section of the beam due to the action of the HF field. The density Pi is given by the equation of conservation of charge V Pv = - P (13) at' which gives as first approximation vax + i -- = - Poa 8y + 8 Vx (14) O ax at 0o ay y aj;x The introduction of equations (11) into (14) gives P1 = o, (15) that is to say, the alternating charge density is zero in first approximation and we have, for the current, ix m_ __ cosh (iyo) 1 + j tanh (iFo ) iw tceikr ix= iIfd-iB cgjy —_e (16) UP s inh (ird( 2 (c).nergy Balance. In reference (3), we determined the power given up by the beam along the element dx by -Ex(yo) ix LJy) dx. (17) In reality, in writing - dP in this form, we have ignored the fact that the beam is displaced along y in a field Ex as a function of y, and it is necessary to write

-o6P yx L 00 1 + a [lF Sy(yo)] p vo dyo0 (18) 2 x Ll + o +Y ~o Yo We obtain therefore, in studying the balance of energy as given in (1), r-r0 i= idRa, coth y c tanh j P ) i+ coth Zo X C 1 zo..... 2 - 1 + coth.2 )] (19) where fo = i - ik, i.e., propagation constant of the free wave. In replacing 6 by its value (9), we obtain an equation of fourth degree in r, which involves the existence of four waves. Wc determine the solution of (19) by approximation in assuming that we have a small current to deal with, i.e., Zo =- is large. We have I therefore two groups of solutions: 1. I 1 small; we neglect ( i) compared with unity and (19) is written (y( 2 + 2 2 Rx cot ) - 2 (20) (-Lk)(/ 2 2) Rx coth ky p (21) P Z, =;- P. (21) o Vo Equations (20) and (21) are identical with a factor of nearly two to relations (16) and (17) of reference (3). The coefficient of amplification ) is therefore 2 times the value given in reference (3). 2. 1 - 2 is small. We will obtain F by writing first l - = 0, and in carrying the valueqwe find rF =. -i... — (22) C WC~~~~~V

1l7 in the second member, which finally gives:. = - WC/- + i d Rx oth - w[1 - tanh (ky.) + coth (kyo) ~3,4 = - - +U i 31 42o2 zo J2 1 C.coth_ 11]j 1 (23) -2 + 2 -. We see therefore that the two supplementary waves are neither attenuated nor amplified but that they have a large dispersion of velocity. These resemble two waves found in the linear travelling-wave tube when we take account of the radial field,5 5. Study of Large Signals In order to define the efficiency of an electronic tube, we can assume different points of view. In the arrangement where the HF energy of the current is collected by a single circuit (klystrons, amplifiers with grids), this energy is written U2, U being the HF voltage and i the fundamental of the electronic current. UoIo being the applied power, the efficiency, in the case where U = UO and where i is in phase with U, is written lil =7 2Io (24) The maximum of i is therefore 2Io, which would correspond to the bunches of electrons infinitely small and infinitely dense; and in this case the efficiency is equal to 100%. Now, in the travelling-wave tube, we have seen that i is of the order 1.2 Io; if we therefore take (24) to define the efficiency, this can attain 60 per cent. But we know that in the travelling-wave tube, the electrons lose only that amount of kinetic energy which corresponds to the difference between this velocity and that of the forced wave; as the result, finally, the HF power given to the forced wave is very small compared with the applied power,

108 We will consider, on the other hand, a current which is of the form represented in Fig. 2. \' /' lines in Fi. 2) The field and the bea hving always the same velocity, the electrons will finish by giving up all o their kinetic energy, and an effiplace in the omanetroh n travelling-e tube for the electron, s of favorable phase: These electrons will form bunches, each located in a retarding field and displaced with the same velocity as that of the field; they will, therefore, have a tendency to be slowed don, which will bring them closer to the anode. In this process, they will, on the one hand, have given up high-frequency energy and, on the other, have taken energy from the continuous field in such a way as to overtake their retarded veis ocity; this mechanism will be which will be in an intense retarding fieldn will arrive quickly at the anode and will have a transit time shorter than those which will be in a weaker retarding field. As a consequence, in order to determine the efficiency of the magnetron travelling-wave tube, it is necessary to determine the percentage

109 of electrons captured by the anode as well as their velocity of arrival at the anode; this allows~ thenthe following course: 1. Determine the pertwubed trajectories of the electrons. This will be done up to the third approximation. 2. Determine the velocity of the electrons as a function of y. 3. Calculate the density with the aid of equation (13). 4. Determine the influence of the nonlinearity on the gain. 5. Determine the velocities of the electrons which are absorbed by the anode and take account of the nonlinearities on the gain. 6. Determine the percentage of electrons captured by the anode. 7. Finally, calculate the amplitude of the alternating current. As we have said, this quantity does not play an essential role in the determination of the efficiency; it gives only an estimate of the amplitudes. 6. The Perturbed Trajectories We have the equations of motion Sy + ioS~ = - i7raU cish (ir) eit ei, (25) y c~~ 177A sinh (ir d) e KI - 8 AU sinjiIFy) eiwtoiiw r+PIx (26) W - 8 = - p sinh (ird) e (26) Except in the exponentials erx we shall replace everywhere in (25) and (26) r by ik. to = time of passage of the electrons into the plane x = o; T = transit time. Equations (25) and (26) are solved only by successive approximations. We develop the solution in growing powers of AUp and we put, according to (3), Y = yo 1 + 2yy + &2y, (27)

110 x = Xo + vot + 81x + 82x + 83X, (28) which we introduce into the terms on the right. Equations (25) and (26) are, in this way, made linear and easily integrated, a condition on the basis of which one calculates real quantities, the complex notation being valid only in the linear equations. In this calculation, we are concerned only with the amplified wave, and we neglect all of the terms which are small compared to the terms in 1. We obtain: 1~Y = dAU sinh (kyQ) voTr (a cos ~ - b sin), (29) Psinh (kd7 2 kd /ALj Wb sinh 2Y d 4 sidnl-' (kd) U V sinh (2kyo)e7 0 (30) 3= - d (kd)2 Xisinh( kY~) [ + b 2 (b+ia)- 2iab] + sinh (3kyo) [2 2 Wb2 (b-ia iwtc v i w (31) and i1x = d A__ cosh (kyd) eVoT (a sin + b cos 4 ) 81X U U p sinh (kd 6 (32) 2 82x = d 2 sinh (kd) (2ky)e 2 -d 4 sinh kd Ut eUp [ (a-b) sin 2 + 2ab cos 2~,

111 3 2 AU (k2 -okd) e- StO+iwpT 3Xx -d A \ Up 4 sinh3 (kd) V0Y +ip Xcosh2 (ky) [ 3a2+b2+i2ab+- W- (b+ia)] (34) cosh (3kY)v (b ia) + (a-ib)2] where SA Y o a = b 2 2 c = _ 2 p2,+ g 2 212X 2 j O_ In equations (29) and (32), we have neglected the terms in ei3at. Equations (29) and (34) determine sy and8x as functions of the transit time T and of y, which is the ordinate of the entrance of the current into the space of the discharge. As a result of the nonlinearity of equations (25) and (26), 83y and8 3x are the terms of frequencyw; but they contain the exponential factor e3Vo T, which entails also the existence of a term in (AUp)3 e3Vo tfor the current. As a result of the coupling, we should therefore, necessarily have in addition to the field given by (6), a field in (AUp)3 e3 x, given by Ey -kA. AUP-)- cosh (k) e3'X ei(Wt-x) y Up2 sinh3 (kd) E = ) sinh (kx) e3Yx i(wt-kx) Ex ikA U2 sinh (kd)

112 A is a complex amplitude factor which will be determined. in section 8. The fields (35) and (36) show the perturbations 83x and 83y given by, (V 1(AUP sinh (kyo) Ae Vo( +i 8y = id(g-ic) - i n A e i (37) 83x d(g-ic) p osh (kdYo) A e (38) sinh3 (ka) which are to be added to the perturbations given by (31) and (34). We must make a remark here relative to the subsequent calculations of the charge densities: in order to determine the alternating densities to second and third approximations, we make use of equation (31),which expresses the conservation of charge: V-pv = a at We make this equation linear by developing p and v in growing powers of AUp: P = Po P + P P2.*, Y v = vo + v1 + v2 +..... We know, moreover, that p1 is zero. But we must necessarily be careful to express v as function of x, y, t; we knew it up till now only as function of Yo, to, r = t - to. Letting C) = w t - kx, we obtain, according to (27), (29), and (30): y(-d p Jox)a sin~) (59) AUp sinh (ky) yx(a cos - b sin 4)) (39) Yo y -dp'sinh (kd) (gJ) 2 /d 2x 2 a 2+2 b2 w b\ + d ( 4 sinh2 (kd) sinh (2ky) e2r (2 a+2 b ) and, according to (28), (32),and (33):

113 OT x - d AUp cosh (ky) eYx(b cos ~ + a sin ) (40) VoT = x - d Up sinh (kd) 2 - d kd e2Yx(2 ab cos 2c+ (a2-b2)sin 24 UP 4 s inh2 (kd) + aw cosh (2ky)) 7. The Alternating Velocities and the Alternating Densities According to equations (29) and (34), we obtain for the alternating velocities as function of y, to, and T iV O sinh (kd) U sinh (kyO) eXVO re (41) p V2 + v (k) 25Up) sinh (2ky)b e2Vort 2 sinh2 (kd) UP 8(kd) s 3 (ka 3KVo ) eiOe sinrh(kyro) [i(a-ib)2+4ib2+-o (a-ib)] 8 sinh3 (kd) Upo + sinh,(3kyo) Ui(a+ib):2 a] kd _bA sini (kyo)e~i e, ~ih~, [i VO- s(a +ib) vo illnh3(kd),% S) T and kd AUp 2 /AU4 8vx o kd v) cosh(kyo)e)rTe F V (ld) (AI) e2 3r (42) x sinh(kd) -p oU sinh2(kd) Up 08 sinh3(kd) J cosh(2kYo)-a cos 24- b sin 2- o8 si nh3(kd) (U/p )e ei cosh(kyo) 3 a +b +i2ab + oYk(b+ia)] -cosh(j3ky) L (b - ia) (a+ib) I ~ kd 5I - +o sinh(kd) p cosh(kyo)A e e

114 Replacing in equations (41) and (42), yo and r by y, t, and x with the aid of equations(59) and (40), we have the alternating velocities 82vy, 83vy, 82vx, 83vx as function of y, t, and x. We find 82vy = S2Vx = o (43) and kd AU 2; iwt+rx Vy = ivo d.) (I) Asinh(ky)e e - -h ( A cosh(ky)e e sinh3 (kd) UP or for the total velocity, dU UA~ (.iAI ivo.(kd. U1+ (P A e sinh(ky)ei t+X (45) Pp UP1 and kd A7L~p AUp Vx = o+ sin(kd) Up Up e2Yx cosh(ky)e it+rx (46) Up sinh (kd.) We have therefore the following important result: the perturbed velocity of the electrons as function of y and x is proportional to the ratio of the alternating field E ~ = KAU L +(tUp A e e P0 U + siin I (jd) to the continuous field Up Ec = The alternating velocityvy, is proportional to the field Ex, and the alternating velocity Vx, is proportional to the electric field Ey up to the third approximation. However, we can consider that, in a general fashion, in the plane nm gnetron or in the plane magnetron travelling;-wave

115 tube the velocity is given by = E Z (!F7) B if the amplitude of relative motion at the entrance is zero, Zo is the unit vector in the direction of the magnetic field.* Equations (45) and (46) determine the kinetic energy with which the electrons reach the anode. The introduction of the alternating velocities from equations (43) and (44) into equation (13) gives the alternating densities P2 and p0. It is found that P2 = P3 = o. (48) Therefore in the magnetron travelling-wave tube the alternating densities are zero up to the third approximation, and we can assume more generally that the density remains constant in the beam. 8. Determination of the Amplitude A of the Third Approximation The value A appearing in equations (35) and (36) is determined by a balance of power. The power -dP given by the current of width Ay is - dP = p vAyh[l + 8y] Ex(yO + Sy)dx, (49) where h is the width of the beam in the z direction. Equation (49) is written in terms of real quantities. The termn (1 + d 8y) gives the variation of the section Ay due to the action of the HF fields. The term Ex(Yo + sy) takes account of the fact that the current is moving in a field variable with y. In taking account of equations (15), (42), (48), and the hypothesis Y < k, equation (49) when we neglect all small terms, becomes * We can demonstrate that equation (47) is also valid if the density of the space charge is continuous and constant (case of the plane magnetron).

- dP = Io (I + y E sinh k(yo +By) (50) Io being the continuous current. With equations (29), (37), and (38), we obtain, for the third approximation term of the power, by using complex quantities: Iokd~x 3 3 A2 e 2 -dP= — k d3 ~ (k.... 1i inh(4lkYo)(b+ia) [-7b27a2i+6b' (51) -8ia- -~8(g+iic) (b +-=v-+ia +sinh(2ky0) F (b+ia) (4bi._-2b2-2a.2+ia_- -8(cr+ic) 2b2+2iab+ia ~oo 2 Up sinh 4 (kd) The power absorbed by the attenuation along dx is + dP1 = -Ex (Y+ ik)dx, (52) with Ex = (kd)[1+A () e sinh (ky) eit+5 E=- Uplp s inh2 (kd) s inh (kd) which gives, for the third approximation, dP k2 sinh (ky) (A+A*) (Y +ik) e dx.(54) aT Rx sinh4(kd) The growth along dxy of the apparent power dP2, twhich is propagated in the direction of the wave, is, for the third approximation 2 4 k _U sinh2(ky) 4Yx [ —-] dP-2 = si c n(+ik)+-(53+ik)A* (55) AU Rx sinh7(kd) According to the principle of conservation of energy we have

117 dP + dP1 + dP2 = o (56) The introduction of the relations (51), (54), and (55) into equation (56) gives the equation which determines A.'We limit ourselves to two particular cases: 1. a = o, Y / o, that is, the forced wave has the same velocity as the free wave and as the electrons. 2. a o, Y = o, that is, the slow wave line has no attenuation. 1. a = o o. Wie obtain according to the equations (51) and (54) to (56) k2d2 r- r A = - [13 cosh (2kyo) - 5]. (57),-2 0 Therefore A is real and positive (Y< o), and the gain increases with am* plitude. WiTe can attempt to calculate the power at the output P = as a function of the power Po, which is transporte.from the input by the amplified wave. We will then have: 2Y x 2 4YX 3 6Yx P = PO e + A2PO e ~ A3P e The calculation made previously gave the coefficient A2, and we have p = [1 0 e2Yx [ kd 13 cosh(2kyo) - 5 po e2Yx (] Quantitatively, the preceding expression is evidentally valid only if the 4Yx -term Po e is small compared with Po e2' x, which allows us to determine the limit of validity of the third approximation. Let us take an example which corresponds to a practical case, kd = 2' kyo = 1.25, Y = 0, voltage between anode and cathode = 2 —, we then deduce

118l = S e 1 - 0.78 where Uc Io is the applied powrer. For an efficiency of 50 per cent the second term, therefore, gives a correction of 25 per cent. We can note, in passing, an essential difference between the nonlinearities in the magnetron travelling-wave t'ne and- the phenomena of P eyx the same nature in the travelling-wave tube, namely, that the factor. ~is generally negative and inversely proportional to the optimum gainYOpt while here the factor is positive and independent of the optimumzx gain. 2 a _0, Y= 0. Starting from equation (51) and (514) to (56), and the relation (20)) which gives R,=, we obtain the real part A1 from A = A11 ~ j B (in fact, only the real part occurs in P). If 2kyo is sufficiently large, A1 is reduced to kd 9 - 8 ap 3 b 1 a 2 cosh (2iky) (59) 2l8, 1 -ap 9 b2 + a2 with 3b 81 - a e2 (3b2 + a2) [7(a2 + b) +, 8 b w+ )-16 ab [- + ga +c b + Vo Fig. 3 gives the relative value A- (Aol being the value of A1 for a = 0) Aol as a function of u. u represents the normalized difference between the velocity of the free wave and the velocity of the electrons u = _ _ V Yopt. Yopt is the optimum value of gain, which is that which we have for a tube without attenuation. It is seen that tvl increases rapidly with u. It follows

119 that we must expect hysteresis effects if the velocity of the electrons is not equal to the velocity of the free w.aves aes we have foad experimentally in magnetrons for the region of oscillations of resonance. A/Ao 2,5 2 1,5 0,5 -15 -1 -0,5 0 0,5 1 1,5 Fig. 3 9. Electron Efficiency In section 8, we have seen that the third approximation leads to a gain higher than the first and that the nonlinearities give only corrections in the neighborhood of 25 per cent for efficiencies of the order of 50 per cent. It istherefore, not, as in the travelling-wave tube, the mechanism of energy exchange which determines the amplitude, but, to the contrary, the absorption by the anode. In order to determine the electronic efficiency, it is therefore necessary to determine the velocity with which the electrons are captured by the electrodes. In [2], we have evaluated the electronic efficiency by assuming that all of the electrons would be absorbed by the anode with a velocity vo = E/B. Now, in section 7 we have determined the velocity of the electrons under the influence of the HI fields and we have found [see equations (45) and (46)] that the velocity of the electrons is, following Ox: v = v (- o), (60)

120 (7y = alternating field at point y, x at time t) and foll1rin O0y: v V= -v L (61) The kinetic energy is found inmlediately: 2 2 m 2 2 [ (+ ) () ] (62) x'( y 2 Eo / Wae can show that y |< Eo |(EE) | < 0 for y = d (63) and the original evaluation is found to be justified. According to the definition of Rx, we have E = J2T~ cosh (kyl) 2 Rx PS sinh (ky Ps = 7 Uc Io = power at the output. Replacing Rx by equation (20), we obtain 2kd7 cosh (ky) (6) FixJ| k sinh (ky)7 k (64) In practical cases: kd 2, y d, < 1 and U < 1 we have | |X2 <<l, k where is of the order of. We do not commit appreciable error in writing equation (62) E m v2 kinetic 2 o It is necessary, therefore, to demonstrate that all the electrons which enter for different phases to into the space of the discharge reach the anode. We will restrict ourselves to the case where the velocity of

121 the electrons is equal to that of the free wrave, that is to say, we will let a = O and -where the attenuation of the slow line is negliFgible. IWe calculate the distance y of the electrons as a function of the phase of entrance Wto and we obtain, in fourth approximation, =,-d Up e X sinl(1y ) kd / AUp x\ sinh knyo) =Y =o d- e sinh sine to + dt esi (kd) k22 dkd AUl.erX sinh(31 o) - sinh kyo) sinw t 12d Up sinh (kd) — co t0 2 sinh Q4ky0) - sih (2y) (6(23 sinh inh (kd k2 d2e 2 15 sinh 3 ) 2(4 ) - sih (ky) sin 6l-8 dyi. Up. sinh (kd) (kd) d k d (Up3e i 13 sinh (3kyo) - 2310 sinh (2o) + d 12.128 j-v W' - U~ / sinhs (kd) The last two terms in equation (65) occur because of the nonlinearities of the initial equations. In equation (65), we have neglected the absorption by the anode. This absorption entails a diminution of gain and, therefore, in the last two terms of equation (65), of the other coefficients. Ie shall neglect these two terms in the following considerations because they do not chfange appreciably the motions of the electrons in the y direction.

122 TWe see that, according to equation (65), in the first, third, etc., approximations, the electrons which enter in a phase ir<wto < 2r are moved toward the anode whereas the electrons in the phase 0 < to < <t are mroved toward the negative electrode. In the second and the fourth approximation, all of the electrons have a component toward the anode. If we take account of the two last terms of equation (65), the behavior of the motion in the y direction is the same; only the coefficients change. Replacing ATUp by Ex (6) and deducing the coupling resistance of equation (20),.we obtain _U'_ e \_ sinh (kd) sinh /.To r A, p ) - sinh (vsyo) v kd U I cosh (1:,ro) sinh (kd) / U tnh(ky), sinh (kyo) kd U where P I E power in first approximation. Pi - E Ex e2 x c Io 0o UC Io In Figs. 4I and 5, we have drawn y/d as a function of the entering phase wto for different values of kd, Lyo, and o0 Uc/Up, according to equation (65), by neglecting the two last terms (Figs. 1 - and 5). We see that the rate at which the electrons are captured by the anode increases as 7o Uc/Up and kd increase and kyo decreases. Let us take a practical example: kd = 2, Uc/Up = 0.5. For kyO = 1.25, 7o cannot exceed about 0.75, the value of the ideal efficiency that vwe calculated if all of the electrons reach the anode. In this case, o Uc,/Up is of the order of 0.3 or 0.4; therefore, only about 50 per cent of the electrons reach the anode, and nearly 80 per cent of the electrons give

123 Jo- I,0 0 0, 0 0,3 0'a, 0/ 8'-0 =' 0,2 = ~0,0,4 0,2 0,260 120 180 240 300 wt0 60 120 180 240 300 w1o,Ad=2,O kYo=0,625 hd-2,0 hAYo 1,25 Fig. 4. Y "0,7 Y o,,2 0,6 0,8,4 0,8 0,2 ~ 0,0 0,6 016 0,2, X _ 60 120 180 240 300 uto 60 120 180 240 300 w t:d=:4,,0 AYo=0,625 hd=4,0 Yo- =1,25 Fig. 5.

energy to the El' field, while 20 per cent gain energy. The efficiency calculated according to the method given in [2] is much larger than the practical efficiency. In the case of Fig. 5, kd = 4, kyo = 1.25, Uc/Up = 1, the efficiency calculated according to [2] is 69 per cent. Therefore, o7 o Uc/Up is of the order of 0.6 or 0.7. In this case, almost all the electrons reach the anode. The evaluation of the efficiency, according to [2], is therefore justified. This is the same in the case of Fig. 4, with kd = 2.0, kyo = 0.625 and U_ = 1. To sum up: If the electrons move sufficiently far from the anode, almost all reach the anode and the efficiency is high. If the electrons move in the vicinity of the anode, the electrons with unfavorable entering phase do not reach the anode. The efficiency is smaller than the result of the calculation according to the method given in [2]. The error which we commit in this case is smaller than 0.5. This effect is understandable if we consider the physical behavior of the tube. The field Ey gives a focussing action in favorable phase. The field Ex of unfavorable phase guides the electrons toward the negative electrode. The fielcd Ld of favorable phase guides the electrons toward the anode. Therefore, if the electrons move in the vicinity of the negative electrode, that is, in a large field Ey, compared to Ex, there is a focussing action in favorable phase, and the beam is guided slowly toward the anode. If the electrons move in the vicinity of the anode, Ex is of the same order of magnitude as EyY and the electrons of unfavorable phase move toward the negative electrode. They arrive in a smnller Ly field and the focussing is not sufficient. For a high efficiency it is therefore favorable to use the same voltage on the cathode and the negative electrode.

125 10. Calcul.ation of the -unjarental Term of th? Current The current is given by Io = Po ( V0o + Vx) (1 + (66);hich is rract icall J ~ai a Lettinr-, -1, hv s Lett;ing i = i — i+, we have, assuming, for simpr.licity, that r- 0 and a. - G: i- id Anp x il = i _ - cosbh (kyo) e, (6 ) {k(ha));- ( AU.T-) e A K~ -osh (kyi) Io sinh3 ((k kd) ) 2CL -. + 2.-1 cosh (3k-Yo)]. (68) From which, for the amplitude, i(I' I I'1' I i 3~ Cosh (lr2ky() ~ 22( IIOI il I Il 5o.1283 cosh' (ky,0o) Thus the maxirrmum value of j is 0 1 16 g coV11 + 15 cosh (2o) kYo For e >> 0, we obtain L| T | 1.38, L Ill.X and for ky, << 1 ii A

e26,Then we inject the beam in the vicinity of the negative electrode, tle moddulation of current is greater tha-n if we inject near the anode. This is understandable because the field LE is zero on the negative electrode, and the electrons therefore are not subjected, at the moment of their entrance into the space of the discharge, to the focussing action of the field Ey; on the contral-ry, when they enter in the vicinity of the anode, they are subjected at the same time to the action of flx and Eyn which entails a smallor modulation of current. 11. Conclusions In this article we have calculated more exactly the gain for small signals, neglecting the hypothesis expressed by equation (1). We have found four waves iwhich fulfill the initial conditions. The gain is increased by a factors because the beam moves in a field variable with y. On the other hand, in the appendix, we hve calculatec the resistance of coupling for a helical plate and we found that the resistance of coupling is smaller by a.pproximmately a factor of two. Therefore, the numerical values calculated according to (3) remain almost the same. The influence of the nonline.arities of the initial equations [equation (3)] entail an increase of the gain with amplitude. The increase of the gain is larger if the velocity of the electrons is not equal to the velocity of the free wave. The attenuating densities are zero up to the third apIpro;xi-mation, and the effect of the space charge are accordingly small. The velocity of the electrons at the place x, y, t, is given by F xB - A-.., nwhere i is the electric field at the point x, y, t perpendicular to the velocity. Therefore, the electrons are captured with their continuous velocity, since, on the other hand, the HT' field is small compared with the continuous ficeld.

1-7 The late of electron capture by the anode is greater than 50 rer cent. If the electrons enter in the vicinity of- the ncative electrode, i'rle rate is 100 per cent. For a high effi cincy, it is th -eref'ore rfavorable to use the s.me voltage on the cathode find on t.he negative electrode. There remain still three rroblemr, to be discussed: 1. To calculate the rate of capturre of the electrons bv the anode, it is necessary to knowJ the influence of the absorption Tby the 2.noale on thoe;gain. 2. In this article, the calculation is rmide urnder the hypothesis of an ideal optical system. In practice, there always exists a relative motion. The influence of this will be studied in a-3 future article. 3- Up to the present we have assum.ed that.t the effects of the sp'ace charge are small because the electronic density is constant. But it is well Tmoowr from the work of Brillouin that the continuous density of space charge entails a continuous velocity variable in the section of the beam. This effect must influence the gain of the mZagnetron travelling-wave tube, Up to the present, we have studied only an electron flow with different. velocities without a slow line [8,9]. On the other hand, the alternating density must influence the behavior of the magnetron travell.ing-wave tube. WqAe will treat these effects in another article. APPEND IX Wle are going to calculate here the coupling resistance of a helical plate following a method given by Pierce. Since the velocity of the wave which is propagated in tihe direction of the beam is small comnpared with the velocity of light, the fields can be derived simply from a. scalar notential.

128,Te shall assume that the helix is formed of twTo plane conductors separa.ted by an interval 2E, the negative electrode being located in the plane y = - D (see Fig. 6). Y He/lice i+ E Hel/ ce id I Electrode negat've h Fig. 6 On the helix we have, in the x direction, a sinusoidal force which is derived from a potential of the form ~= (y) ei(wt - ca) (1') We have for - D y. - E: = B sinh Cz(y + D)] ei(wt (2') for - E y < + E i({ ct - ax) = A cosh cay e t -x) for + E y oo = C e C7 ei(wt - ax) The fields are derived from S by EY a y Ex = - y 0 X o X

129 where: for - D y - E: Fl = - 0t A cosh (oEf) cash 2 _~ ).i(wt - a~) ) sinh c: (x + D) i(w t - cz) Exl = + i a: A cosh (cE) si ( —h- e for - y + E yi F - cz A sinh (c.;r) ei( t - C x), YII (61) ExII = + i c A cosh (Cty) ei(wt - for + E Y + o: EyIII c= A cosh (cE) e-(Y - E) ei(w t - ox) yIII (7') ExIII =+ i a A cosh (SE) e ei(wt - c) 7 The power W transported by a length, h, of helix is W = J VE, where Q is the electromagnetic energy stored per unit length, and V g is the group velocity of the wave. Now, Q is given by OD = 8^h (EXEX + EB ) dE y (9') -D where W - 2 Eoh sinh (2 a) + cosh (cE) 1 + cot cz(D-E} (10') g 2

rphe resistance of couplingr is therefore ExLx _io c cx sinh[ch (p + y) _.x - 2~ E- 0 v h x 2 h sin[laq(D-E)] [2 tanh oP sinh c;(D-l) + eC(D-E)] (l1') The value of Rx given by (15') in [31 is identical to the value found here in (11') when we put E = 0. On the other hand, we have in practice: anda (D - cE) i 2 Cand in this case} Rx is practically half of RX (It = 9).

131 BIBLIOGRRIHY 1. R. Warnecke and P. Gutnard, "Sur l'aide que peuvent apporter en tkldvision quelques recentes conceptions concernant les tubes 6lectroniques pour ultrahautes frequences", Annales de Radiodlectricite, October, 1948, p. 259. 2. P. Warnecke, W. Kleen, A. Lerbs, 0. Doehler and H. Huber, "The magnetrontype traveling-wave amplifier, P. I. R. E., vol. 38, No. 5, May 1950. 5. J. Brossart and 0. Doehler, "Les tubes h propagation d'onde a champ magngtique", Annales de Radiodlectricit6, October 1948, p. 328. 4. J. R. Pierce, "Transverse field in traveling-wave tubes", Bell. Syst. Tech. Journ.,vol. 27, 1948, pp. 732-746. 5. 0. Doehler and W. Kleen, "Influence du vecteur dlectrique transversal dans la ligne t retard du tube a propagation don d'onde", Annales de Radio6lectricit&, January 1949, P. 76. 6. J R.. Pierce, "Circuits for traveling-wave tubes", P. I. R. E., vol. 37, 1949, pp. 510-515. 7. 0. Doehler and W. Kleen, "Phenomenes non lingaires dans les tubes a propagation d'onde", Annales de Radiodlectricit4, April 1948, p. 124. 8. R. Warnecke, 0. Doehler and W. Kleen, "Amplification d'ondes glectromar.gnetice s par interaction entre des flux dlectroniques se d4placant dans des champs glectrique et magngtique croises", C. R. Acad. Sc., vol. 229, 1949, p. 709. 9. O. Buneman, "Generation and amplification of waves in dense charged beams under crossed fields", Nature, vol. 165, 1950, p. 274.

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