Proceedings of the Institute of Radio EngineersVolume 21, Number 12 December, 1938

NOTE ON THE THEORY OF THE MAGNETRONOSCILLATOR*

BYE. C. S. MEGAW

(Communication from the Research Staff of the M.O. Valve Company, Ltd., Wembley, England)

N A recent note' J. B. Hoag has given a simple argument to showthat the relations

XH= 10,650and

XH= 13,100

between the magnetic field strength H and oscillation wavelength X,should hold approximately in a cylindrical magnetron for the cases ofno space charge and saturated space charge, respectively. It was as-sumed that the half period of the oscillation is equal to the time oftransit of an electron from filament to anode and that the anodevoltage had the critical value for which the electrons just touch theanode surface. The additional assumption was introduced that theelectrons attain their maximum velocity within a negligible distancefrom the filament.

It has been shown elsewhere2 that when the critical relation holdsbetween anode voltage and field strength, the time of transit of anelectron in a cylindrical magnetron is given by

e \-1/2 rra r2 / ro2 2~ -1/2T= (2-Ea) fr; f(r) - -21 - 2) dr

where,Ea -anode voltager =radial distance from electrode axisro=filament radiusra = anode radius

and,f(r) is defined by

Er=f(r) Ea

Okabe's relation3 between X and H, which is identical with Hoag'sfirst result, was obtained by assuming ro infinitely small and putting

* Decimal classification: R133. Original manuscript received by theInstitute, September 8, 1933.

1PROC. I.R.E., vol. 21, p. 1132; August, (1933).2 Megaw, Jour. I.R.E. (London), vol. 72, p. 330; April, (1933).3 Okabe, PROC. I.R.E., vol. 17, p. 652; April, (1929).

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Megaw: Theory of Magnetron Oscillator

f(r) = 1. Now f(r) = 1 means that the whole of the interelectrode spaceis at anode potential so that the electrons will attain their maximumvelocity immediately they leave the filament. This is the additionalassumption made by Hoag in obtaining his first figure for XH, whichshould therefore evidently agree with Okabe's result.

By giving f(r) the forms appropriate to zero and saturated spacecharge the relations

XH = 12,300 (zero space charge)and

XH = 16,700 (saturated space charge)

have been obtained,2 for values of ra/ro of the order of 100, by graphicalintegration of the equation for transit time. These relations agree quitewell with the observed minimum and maximum wavelengths whenthe oscillation amplitude is small.

1.6

U~~~~~~

1.2 _ _ __ A

xB

00 O.i 0.2 0.3 0 4 0.5

RADIAL DISTANCE (r) CMS

Fig. 1. Radial velocity as a function of radial distance from electrode axis.Curve A. f(r) = 1 (constant total velocity)

Curve B. f(r) = (log-) (log -)(actual potential distribution with no space charge)Curve C. f(r) = ( ra 2)(saturated space charge).

In obtaining the relation for saturated space charge it has beenassumed that the potential distribution is not appreciably changedby the presence of the magnetic field. There is experimental evidence4

4 Hull, Phys. Rev., vol. 18, p. 31; July, (1921).

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Megaw: Theory of Magnetron Oscillator

that this is not far from the truth. The last two equations are at leastmore accurate than those obtained on the assumption of constant totalvelocity. The curves shown in Fig. 1 indicate the magnitude of theerror introduced by this assumption in a typical case.

It is found that the observed wavelengths tend to be slightly lowerthan those given by the theory outlined above, particularly when themagnetic field is inclined to the electrode axis so as to obtain a maxi-mum output.2 In this case it appears that the electrons providing theoutput energy describe several loops, with successively decreasing am-plitude, before reaching the anode. The average "transit time" forsuch electrons would probably be smaller than that for electrons whichtouch the anode on their first outward journey.

These curves apply to a magnetron with a 1.0-centimeter diameteranode and a 0.01-centimeter filament when the anode voltage is 400and the magnetic field strength is adjusted to the critical value.

ACKNOWLEDGMENT

The author desires to tender his acknowledgment to the GeneralElectric Company and the Marconi Company on whose behalf thework leading to this publication was done.

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