ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR THEORY OF FREQUENCY MODULATION NOISE IN TUBES EMPLOYING PHASE FOCUSING Technical Report No. 28 Electronic Defense Group Department of Electrical Engineering By: J. L. Stewart Approved by:'__- __ __ Gunnar Hok Project 1970 TASK ORDER NO. EDG-7 CONTRACT NO. DA-36-039 sc-15358 SIGNAL CORPS, DEPARTMENT OF THE ARMY DEPARTMENT OF ARMY PROJECT NO. 3-99-04-042 SIGNAL CORPS PROJECT NO. 29-194B-0 March, 1954

TABLE OF CONTJE2TS Page ABSTRACT iv 1. IJTRODUCTION 1 2. THE MECHANISM OF FM NOISE 2 3. IlDUCED CURRENT IJ TEE MAGINERON 5 4. ELECTRONS IN VOLUME ELE-IENTS 11 5. MEAN-SQUARE DEVIATION OF FM NOISE 12 6. ESTIMAT"E OF NOISE BAISDWIDTH 14 7. HALF BANIEDWIDTH OF THE MACETRON SPECTRUM 17 8. AN APPLICATION TO A VOLTAGE TUNABI3 MAGNETRON 19 REFERENCES 21 DISTRIBUTION LIST 22 iii

ABSTRACT Oscillators employing phase focusing such as carcinotrons and magnetrons have fairly well defined spokes whose rotational speed is proportional to the oscillation frequency. Each spoke is composed of a finite number of electrons having random velocities and consequently is subject to random fluctuations about a mean position. This spoke "jitter" leads to fluctuations in the oscillation frequency which amounts to frequency modulation noise. It is the theoretical evaluation of the parameters of this noise as affecting the power spectrum of the oscillator output that is of interest here. The theoretical results compare favorably with the measured power spectrum of a voltage-tunable magnetron. The application of the formulae require an estimate of electron temperature - no attempt is made to evaluate this temperature. Amplitude fluctuation noise appears to be relatively unimportant in continuous-wave phase-focused oscillators - the evaluation of such noise is not undertaken here. iv

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN THEORY OF FREQUENCY MOIXLATION NOISE IN TUBES EMPLOYING PHASE FOCUSING 1. INTRODUCTION It has been observed that two types of noise exist in the output of ignetrons and other tubes employing phase focusing (Ref. 1). As would be cpected, amplitude-fluctuation noise is present due to shot and flicker effects. 1 addition to this, there exists frequency modulation noise, part of which is )rrelated to the amplitude noise. The study here is restricted to the FM noise, id, in particular, to that component of FM noise introduced because the charge i the tube consists of discrete particles having random velocities. There:ists a second component of FM noise due to flicker and other effects at the.thode which will not be analyzed here. As necessary background, some results of a previous analysis will be ated (Ref. 2). If a carrier is modulated in frequency with rectangular band of ussian noise extending from a radian frequency of zero to B such that the stantaneous frequency deviation of the carrier is directly proportional to the plitude of the noise,then the power spectrum of the resulting modulated Magnetrons will be considered specifically because more data are available.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN carrier will have the asymptotic forms _ (i g> i (1 2tD exp 2D2 w(w ) = t( 1 ). - _ jD2/2 D 1 ( i.[ (D2/2B2)2+ ( wwo)2/B2 B where wu0 is the carrier frequency and D is the rms frequency deviation. The thx decibel half bandwidth of these power spectra are BF = log2 = 1.18D D >> 1 (3 BF = =D2 = 1.57D2/B D << 1 (4 2B B For small D/B, the power spectrum falls off quite slowly with the difference frequency w - w. It should be noted that noise having such a power spectrum might have severe consequences if applicable to the local oscillator tub in a superheterodyne receiver. 2. THE MECHANISM OF FM NOISE In a magnetron, the space charge exists in the form of spokes that rotaat an angular velocity wr 2 If the magnetron operates in the i mode and has N anode segments, there are N/2 spokes and co = Nr (5C 2 where w0 is the generated frequency in radians per second. The current induced in the anode segments is due mainly to the rotation 2. Data related to the analysis of magnetrons, and references to magnetrons in general, will be round in Ref. 2;5. 2 __________

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Lotion of the spokes, their radial motion having only a second-order effect which rill be neglected here. The electrons that comprise the spoke were at one time mitted from the cathode with random velocities. The probability density function f the velocity tangential to the cathode is Gaussian with a variance kT/m (where. is Boltzmann's constant, T is temperature, and m is the mass of the electron).3 The random nature of the tangential emission velocities causes the elocities of the electrons in the spokes to have a Gaussian probability density unction (at least approximately) with a mean related to the spoke rotational elocity r'. Since a space charge at the cathode affects only the normal omponents of velocity, the condition of the spokes with respect to the random otational velocities of the individual electrons will be the same for either pace-charge-limited or temperature-limited emission. Although even the approximate variance of the velocity distribution of lectrons in the cathode-anode region is not known, it is believed that due to the lectron-electron interaction, growing wave amplification, and the relatively igh random velocities of the secondary electrons, the mean-square electron andom velocity between anode and cathode is very much larger than that at the athode. This random motion can be specified in terms of a temperature. Although he temperature of the electrons upon emission is only about 1000~K, in the ateraction space between cathode and anode it may be on the order of millions of agrees Kelvin.4 There are a certain number of electrons in the spokes at any given astant. It is the average rotational velocity of all these electrons, suitably All units are in the M.K.S. system.' Guenard and Huber (Ref.4) report temperatures on the order of 105 degrees Kelvin in non-oscillating tubes. Although no published data seem to be available, it would appear that temperatures in the oscillating magnetron are much larger than this.........5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN weighted in accordance with their relative effectiveness in inducing anode cur - rent, that gives the instantaneous frequency of the magnetron. New electrons a: continually being removed at the anode and entering the spokes at the cathode; from this it can be concluded that the instantaneous frequency will vary in an approximately Gaussian fashion about the average frequency. The correlation function R(T ) of these frequency variations will decrease to zero at r T, where T is the cathode-to-anode transit time of the electrons. Due to the averaging effect of the large number of electrons in the spokes at any instant, "jitter" of the spokes will be much smaller than that accountable to a single electron. However, the small transit time indicates thE the rate of change of instantaneous frequency is rather high. Consequently, it is apparent that the output spectrum of the magnetron will have the form of Eq (for D/B very small). The calculation of the equivalent D and B, and especially of the half bandwidth BF, for the type of noise described above are of primary concern here. There appears to be no reason why any correlation should be expected between this type of FM noise and amplitude-fluctuation noise. Flicker effect at the cathode and amplitude-fluctuation noise impress ed upon the various power supply voltages introduces a second kind of FM noise. Variations in the conditions of the electrons in the magnetron in this case occur rather slowly; hence, the equivalent modulator bandwidth B is quite small. On the other hand, the magnitude of these disturbances may be relatively large; therefore, it would be expected that the equivalent deviation D is large. Thus, D/B is large and the shape of the power spectrum would appear to be that of Eq 1 It is this spectrum that would be observed on the presentation of a high resolution spectrum analyzer tuned very near to the carrier frequency. This type of l —------------ ^ --------------— 4

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN.oise is relatively unimportant for magnetrons used as local oscillators lecause of the rapidity with which the spectrum falls off with the difference requency c- C. As contrasted to the other type of FM noise, there is every eason to expect a relatively high correlation between this type of noise and mplitude -fluctuation noise. 3. DIDUCED CURRENT IN THE MAGNETR.ON Consider Fig. 1 which is a diagram of spokes in a magnetron. An mpedance Z is connected between the positive and negative anode segments, and Iternate segments are connected together. The induced current flows through the mpedance Z resulting in a voltage between adjacent anode segments. The magnetrcn ill be assumed axially symmetric and of length L. Let it be assumed that the total charge in the spokes is constant and hat all induced current is caused by rotation of the spokes rather than by the idial motion of the electrons. Then, the space charge density function of the,pokes and the function 4k, giving indirectly the potential of the field in the:athode-anode region, can be expanded into Fourier series. i, is something ike a Greens function, being unity at the electrode through which a flux alculation is desired, and zero on all other electrodes. = k Akcos (6) p= Z [B cos iN__ + Cj sin jNQ (7) j 2 (7) The angle 9 is the angle in a fixed coordinate system, and 9' is the ngle in a system rotating with the spokes. For convenience, 9 = 0 is assumed o correspond to the center line of a spoke. Also, it is assumed that at the 5 -

VS-91-Vb w3r tp-ZS-V OZ61 r7 ELECTRICAL DEGREES *''' CATHODE FIG. I CROSS SECTION OF THE MAGNETRON. 6 ~7 1~,./ ~." i.~: ~~~~~~~.~:,, ~r~ ~~~~-,\~~~m ~~~~~~~~~~~

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN *eference time t = 0, G = 9'. Thus, 9 = 9- W rt (8) The displacement current induced in the anode circuit is given by'eich (Ref. 3).) Tr r^ i = - k P Lrdrd@ (9) it5 rc Substituting Equations 6 and 7 in Eq 9 (observing that because the oefficients are independent of 9@, orthogonality relations exist, causing all erriis for k ~ j to vanish), there is obtained L Ct k [ cos -r cos ra A rdr i=L~ k0 t k 2 2 rc AkBk rc cos kN kNot ],Ardr (10) + f co sin J AkC. -It rc Simplifying, substituting for 9', and integrating over 9 results in d v S rkN w ra kNWrt ra i = L~r C tr a iC o Amrdr - sin A rdr (11) k r rc L C c The generated frequency is related to the rotation frequency according o Eq 5. Thus, the argument of the sine and cosine terms is k ot. The most nportant term of Eq 11 is the fundamental. Thus, for k = 1, and after differeniating with respect to time, ra r = -Ligw) sin w0t r A1Bjrdr + cos W -t f ACrdr (12) _________ - rc ~ ^~~~~~r''The lower limit r can be interpreted as either the radius of the cathode or the radius of the sub-synchronous swarm, whichever is applicable. --------------------— 7 —---------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The magnitude of il is given by the square root of the sum of the squar of the quadrature components, or 2gw [(i rra Aa Lrdr"'g 2 l1/2 i 2qo i c AltBo I C l i^ = _c~ LJ2 1 1 Lrdr + ( J Lrdr (15) x \rc 2<1 / \ q rc where the total charge in the spokes q has been introduced and where the equation has been written in this form because it has been proven that the maximum possible value of the square root of Eq 15 is unity (Ref. 5). Thus, 2qw 0/t is the maximum fundamental current that can be induced in the magnetron under any circumstances. When the spokes are symmetrically located around the cathode, all the Ck are zero and Eq 13 takes the simple form ra il = L wo J A1Blrdr (14) Pc If it is assumed that the space charge has a negligible effect upon the potential in the anode-cathode region by comparison with that caused by the voltage between anode segments, then the coefficients Ak are functions only of the radius r, and are obtainable through a solution of Laplace's equation in two dimensions. Consequently, it is permissible to assume a normalized function A(u) such that A1 = AA (u) A(1) = 1 (1) in which Am is the value of Al at r = ra (which is generally the maximum value of A1) and where the variable u is the normalized radius r/ra. The coefficients Bk and Ck are defined by the usual Fourier series formula as (16) Bk = &L f p (r,i') cos0 - d9' -27 l N_______________- 8N

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN here p(r,@) is the space-charge function. Equation 7 gives the function p(r,9') n terms of the Bk and Ck (if they art not zero). Let it be assumed that each spoke has a uniform charge density in 3 < 9' < 1 and zero charge density for 9|' > 1; then, p (r,') = p(r) p('() (17) nd 2ir fi p (9')d9' = p(e')dG' = 2p (18) w- I The above assumption may be highly unrealistic in some cases. However, one equivalent rectangular spoke can usually be hypothesized and some equivalent obtained. Using Equations 15, 16, 17, and 18 in Eq 14, and integrating, there is;tained ra, -sin HP 1 i IiPL co 2 AmA(u)p (r)rdr (19) 2 /rc It will be assumed that p is the charge density of one spoke; hence, Sf V is the volume of one spoke and q is the total charge in the magnetron f p(r,) dV = q(4) (20) V Since dV = L r dr d9', Eq 20 becomes 2i r r Tr- a a J J p (r,' ) LrdrdG' = 2Lp p(r)rdr = q2 (21) 2r rc -----— 9 C

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN from which ra 1 p(r)rdr = q = r 2 u p (u) du (22) "r~c araL r a rc rc/ra The function p (r) gives the manner in which the electron density varies with r. This can be normalized to a function B(r) and B(r/ra) as (r) B(r) (23) p () - 2i3 LgNra2 such that ra r0 rB(r)dr = uB(u)du = 1 (24) -c c/ra If Eq 23 is substituted in an integral and integration is performed over B(u), only the volume of one spoke need be considered in order to include all N/2 spokes. Using Equation 23 in Equation 19, there is obtained the final expression for induced curent, 1| Aw [(q - 2 uA(u)B(u)du (25) rC/ra The maximum (but unrealizable) current is given when the spokes are very narrow and the total charge is located very close to the anode. Then A(u) = 1 and the integral over B(u) is unity giving I11, I = A moq (26) ------------------- 10

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN As a specific approximation, assume that B(u) varies as l/u; the onstant of proportionality is found from Eq 24 as 1/(1 - rc/ra). Also, assume hat A(u) varies as uk where k is some positive number. Then, Eq 25 becomes il A Woq (s in 2 - (rc/ra) k+l oq 2 c(/ (27) 1 k+l N3 2 1-'rc/ra hich is approximately Am! ic |o q (28) 4. ELECTRONS IN VOLUME ELEMENTS The charge dq in the volume element dV (which includes the corresponding element in every spoke) is found from Eq 21 as N p(r,') dV = dq (29) But p(r,G') = p(r) as given by Eq 23 and dq = n'e where n' is the lumber of electrons in dV. Thus, n' = qB (u)dV qB(r)dV 2Lpera 2LPe (50) The partial current ip due to the charge in dV can be found via a ubstitution for A1 and B1 in Eq 14 as ra c i = 0nLwo J ArA.(u) E J p(r) cos - dG' rdr (51) rC -1

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Substituting for p(r) and considering only the volume element, ip = Am q (cos Ni ) A(u)B(r)dV (32) 2LP - 2 Substituting for B(r) in Eq 32 results in i.p N 2 A (u)n' = ipl A(u) cos l' (55) where il is the value of Eq 33 at 9' = 0 and r = ra. Thus, the relative effectiveness of any given electron in inducing current in the anode is proportional to A(u) and to cos(NG'/2) as expected.. MEA-SQUARE DEVIATION OF FM NOISE The probability density function of the transverse velocity of one electron is (at least approximately) Gaussian with a variance kT/m. At a radius r, the velocity v is wr = v/r. Thus, the variance of the density function for Wr is kT/(mr2). In one revolution of the spoke, there are N/2 cycles of the carrier; hence, the variance of the generated frequency due to a single electron is / N 2 kT Ae N2 2 (34) which can be considered to refer to an electron in the most favorable position for inducing current in the anode. For any other electron, Eq 34 must be multiplied by the square of the relative effectiveness of the electron. The variance due to the n' electrons in the volume element dV is less than that of a single electron. In order to obtain the appropriate average, 12

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN q 34 must be divided by n'.'Tus, 2 C08~~~~2 I fJ2 kT [A(r) c s N5' nACw, = mr- n' (35) Substituting for n' from Eq 30, 1 (2 2 r2 qB(r) dV (56) ( kN ( kT / [A(r) cos NG'j ]2LPe n[L 2J The average for all electrons in the magnetron is obtained by summing q 36 for the n' electrons in each volume element dV. This integration yields he reciprocal of the variance D2 of the FM noise as 2 13 ra (21 mq dat rB(r) m (K ) 2pekT / 2N90J' J A (r) | cop 2 \c / (57) ra 3 2 mq r3B( dr -VN I BekT tan 2 - A-(r) rc Inverting and substituting for q from Eq 25, D2 T2 ekTw cos 2 r (38) D = 1 5-. Q cos -- F -c,here F(rc/ra) is a geometrical constant given by F( rc /a (39) 13 e 3Bru) c/ra (u) In order to have some (albeit rough) estimate of F(r/r), assume B(u) ----— 5~____

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN varies as 1/u as before and that A(u) = uk. Then, 5-2k 1 -(r k+ k+1 1 - (r/a) 3-2k k z.5 (40) F (rc/ra)- r /ra 2(5 1- ( rc/ra)2'5......(........ k = 1.5 2.5 log (ra/rc) which is plotted as a function of rc/ra for several values of k in Fig. 2. Except for the induced current, all quantities in Eq 39 can be specified for a given magnetron. However, if the power of the oscillations of the tube is known, the quantity [il can at least be estimated. Let R be the shunt resistance of the loaded cavity. Then, the power P is given approximately by P = li1 (41) 2 from which i[ i = (2P/R)1/2 (42) If the induced current is not in phase with the voltage between spokes, some inaccuracies in using the above relation will result. However, the error will not be large and can be taken into account (approximately) in many cases. 6. ESTIMATE OF NOISE BANDWIDTH It would appear feasible to obtain an analytic expression for the bandwidth of the noise by setting up a picture of the electrons moving from cathode to anode and computing a correlation function for the instantaneous frequency. However, many of the parameters in the magnetron are known only vaguely; in particular, the transit time. Thus, rather than obtaining an _-.....- 14.........

frfr-91-tr wr ZZ-L9-V OL61 1.0 0.8 L. 0.6 0 0.2 0.4 0.6 0.8 1.0 c/ ra FIG. 2 GEOMETRIC CONSTANT FOR CALCULATING DEVIATION. 1.0 -— ____ — 0.8 04 2 5 0.2 0,, 0 0.2 0.4 0.6 0.8 1.0 rc/ ra FIG. 3 GEOMETRIC CONSTANT FOR CALCULATING SPECTRUM WIDTH. 15

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN analytical solution, an estimate will be made. If T' is the transit time, the correlation function R( ) is zero at r= T'. In order to get a simple estimate> it will be assumed that 1- — T 1 I tTO v Tv R (T) = { fV~T (43) -- >1 Tt The power spectrum of the equivalent low-pass modulator producing the F! noise can be calculated from this assumption as sin coT' W( c ) = 4 f (r ) cos r d T = T'- T' (44) o 2 The half-power radian bandwidth of this power spectrum is B = 2.74 = CR T' T' (45) Thus, for the approximate correlation function assumed, a calculation of the bandwidth has been reduced to a calculation of the transit time. Because the electrons most effective in inducing current are fairly close to the anode, the bandwidth given above is probably too small. If one asumes a CB of about three or four the error should not be excessive. Let it be assumed that the radial velocity of the electrons is constant. Then, it is possible to relate the transit time to the dc anode current Io as Electrons/second = 2r x 1018 = n = (46) 0; t — e v'I inT'e where n is the number of electrons in the magnetron at any one instant. ____________ — ~~~~~16

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN?hus q ml T 2= 2x 101T Ioe (47) The value of the total charge q is substituted into this expression'ro Eq 25 to give cB =.,C = j.lo:eLB (r o: uA(u)B(u)du (48) B. Cg 2.,x 10i1 I ^CB \SNr /A )( )d () I2 rc/ra Assuming that B(u) varies as 1/u and that A(u) = uk an approximate xpression can be obtained as 2tn x 1018 Ioe oCB S 1 - (rc/r)k+l (k+l) i \ 1 - (rC/ra) hich is very approximately 2rn x 1018 I e oCB (k+1) |il (50) 7. HALF BAMWIDTH OF THE MAGNETRON SPECTRUM Relative to unity at the center frequency, the power spectrum of the agnetron output is 2 W (A W ) = (51) M B('2 + Aw 2 here BF is the half bandwidth given by B = 2DB (52) 2B 17 -

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN If D2 from Eq 38 and B from Eq 48 are substituted into Eq 52, the bandwidth BF will be neither a function of the induced current il nor the generated frequency co. 2kT~~ 2 F (2) 4 X 101'IomCBra ( /an G(rc/ra) (53) The geometric constant G(rc/ra) is given by G(rc/ra) = 1 (54) -(U) d f A'e (U) Using the approximations often used before, f(1 - rc/ra)(3 - 2k) k 1.5 1 - (rc/ra)3-2k G(rc/ra) = (55) 1 - r/ra log (ra/r) k = 1.5 which is plotted in Fig. 3 as a function of rc/ra for various values of k. For the special value k = 2, G(rc/ra) = rc/ra. The function of the angle P in Eq 553 will usually be quite close to unity and can be approximated as such. Assuming the angle factor is unity and k = 2, Eq 54 takes the particularly simple form (1/2)2 (rc/ra)kT BF' 4 x'lO8i IomCBra2 ___________- 18........

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 8. ANT APPLICATION TO A VOLTAGE TUnABLE,IAGE-IRO1U The application of the foregoing theory can be made to a small voltage mnable magnetron of the type developed over recent years at the University of ichigan Tube Laboratory. Approximated dimensions and operating conditions are Lken as follows: N/2 = 6 ra/rc 0.7 r = 0.005 meters I ~ = 0.01 amperes It will be assumed that T = 2.2x106 ~K (200 electron volts) CB = 3 G(rc/ra) = 0.8 Angle factor = 1.0 Using Eq 53, Bp is found to be 517 radians per second, or 50.6 cycles r second. At difference frequencies of 0.1x10 5 106, and 10x106 cycles per Pcond, the spectrum is down about 66, 86, and 106 decibels respectively, from iat at Aw = 0. These figures compare favorably with observed values. The Lape of the spectrum has repeatedly been observed to follow the "single-tuned" tsponse pattern. It may be of interest to calculate the deviation D2 and bandwidth B for.e example. For this, Eq 38 can be used. It will be assumed that P = 0.2 watts R = 50 ohms ~o = l8x0l9 radians per second Assume cos (INp/2)F(rc/ra) = 0.4 19

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The value of induced current f il is found to be 0.09 amperes. Then, D2 = 61x1010 or D = 780,000 radians per second which is 124 kilocycles per second. Since B is related to D2 and B by Eq 4, the values of BX and D' calculated for the examznple can be used to find B = 3020x100 radians per second, or 481 megacycles per second. 20

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN REFERENCES -. Microwave Noise Study", Quarterly Progress Report Nos 1, 2, and 3, February 1, 1953 to November 1, 1953, Raytheon Manufacturing Co., Contract AF 19(604)-636.. J. L. Stewart, "The Pmoer Spectra of Signals Phase and Frequency Modulated by Gaussian Noise," Technical Report No. 23, Electronics Defense Group, Department of Electrical Engineering, University of Michigan, November 1953. i. H. W. Welch, Jr., "Prediction of Traveling-Wave Magnetron Frequency Characteristics: Frequency Pushing and Voltage Tuning," Proceedings Of I.R.E. vol. 41, No. 11, pp 1631-53, November 1953.. P. Guenard and H. Huber, "Etude Experimentale de L'interaction Par Ondes De Charge D' espace Au Sein D'un Faisceau Electronique se Deplacant Dans De Champs Electrique Et Magnetique Croises." Annales De Radioelectricite vol 7, No. 30, October 1952,pp 252-78. H. W. Welch, Jr., S. Ruthberg,H. W. Batten, and W. Peterson, "Analysis of Dynamic Characteristics of the Magnetron Space Charge —Preliminary Results." University of Michigan, Department of Electrical Engineering, Electron Tube Laboratory, Technical Report No. 5, January 1951........._ ~21

DISTIIBUTION LIST 1 copy Director, Electronic Research Laboratory Stanford University Stanford, California Attn: Dean Fred Terman 1 copy Commanding Officer Signal Corps Electronic Warfare Center Fort Monmouth, New Jersey 1 copy Chief, Engineering and Technical Division Office of the Chief Signal Officer Department of the Army Washington 25, D. C. Attn: SIGGE-C 1 copy Chief, Plans and Operations Division Office of the Chief Signal Officer Washington 25, D. C. Attn: SIGOP-5 1 copy Countermeasures Laboratory Gilfillan Brothers, Inc. 1815 Venice Blvd. Los Angeles 6, California 1 copy Commanding Officer White Sands Signal Corps Agency White Sands Proving Ground Las Cruces, New Mexico Attn: SIGWS-CM 1 copy Commanding Officer Signal Corps Electronics Research Unit 9560th TSU Mountain View, California 75 copies Transportation Officer, SCEL Evans Signal Laboratory Building No. 42, Belmar, New Jersey For - Signal Property Officer Inspect at Destination File No. 25052-PH-51-91(1443) 22

1 copy H. W. Welch, Jr. Engineering Research Institute University of Michigan Ann Arbor, Michigan 1 copy Document Room Willow Run Research Center University of Michigan Willow Run, Michigan 11 copies Electronic Defense Group Project File University of Michigan Ann Arbor, Michigan 1 copy Engineering Research Institute Project File University of Michigan Ann Arbor, Michigan 23

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