ECE 662 – Microwave Electronics Cross-Field Devices: Magnetrons April 7, 14, 2005

Magnetrons Early microwave device –Concept invented by Hull in 1913 –Initial devices in 1920’s and 30’s Cavity Magnetron (UK) – 10 kW –Rapid engineering and Production –Radiation Lab (MIT) established Relativistic Cavity Magnetron (1975) – 900MW Advanced Relativistic Magnetrons (1986) - 8 GW Commercial Magnetrons (2003) - 5 MW

Magnetrons Inherently efficient Delivers large powers (up to GW pulsed power and MW cw) Limited electronic tuning, i.e., BW limited Low cost Industrial uses –microwave ovens –industrial heating –drying wood –processing and bonding materials

Magnetrons B no longer used to confine electron beam as in a Klystron - B is an integral part of rf interation. Multicavity block Coaxial cathode Coupling - I/O- loop or Waveguide

Z Z

Planar Magnetron Let V A = potential difference between the anode and cathode, and E 0 =- V A /d. An applied magnetic field is in the x direction (into the paper). The force on the electrons becomes: x z

Planar Magnetron

This neglects space charge - tends to make trajectory more “straight”. Result - frequency of cycloidal motion is  c  f  B and (e/m) KEY: average drift velocity of electrons in z direction is E 0 /B 0, independent of v z0 and v y0.

u ox here is the v 0z of our formulation ref: Gerwartowski

Planar Magnetron Electrons have dc motion equal to E 0 /B 0, slow wave structure is assumed to be a propagating wave in the direction of the electron flow with a phase velocity equal to E 0 /B 0

Planar Magnetron (ref. Hemenway)

Circular Magnetron (conventional geometry) Electrons tend to move parallel to the cathode. After a few periods in the cylindrical geometry the electron cloud so formed is known as the Brillouin cloud. A ring forms around the cathode.

Circular Magnetron Oscillator ref: Gewartowski

Brillouin Cloud Next, compute the electron angular velocity d  /dt for actual geometry. Note region I inside the Brillouin cloud and region II outside.

Brillouin Cloud Note: electrons at the outermost radius of the cloud (r = r 0 ) move faster than those for r < r 0. The kinetic energy (of the electrons) increase is due to drop in potential energy.

Hull Cutoff Condition For a given B 0, the maximum potential difference V A that can be applied between the anode and cathode, for which the Brillouin cloud will fill the space to r = r a is

Hull Cutoff Condition B 0 < B 0min direct current flows to anode and no chance for interaction with rf. B 0 > B 0min Brillouin cloud has an outer radius r 0 < r a and no direct current flows to the anode. For a typical magnetron, B 0 > B 0min therefore r 0 < r a

Magnetron Fields From radial force equation (1), consider electrons following circular trajectory in Brillouin cloud. Assume that

Magnetron Fields From Poisson’s equation the charge density:  0 falls slightly as r increases from r c (can increase  0 by increasing B 0 which follows as electrons spiral in smaller cycloidal orbits about the cathode.

Magnetron Fields Outside the Brillouin cloud, r 0 < r < r a, in region II, use Gauss’s Theorem:

Hartree Relationship The potential difference V A between the cathode and anode to maintain the Brillouin cloud of outer radius r 0 is given by:

Hartree Relationship This v B is important since it gives the velocity of the electrons at the outer radius of the Brillouin cloud. It is this velocity v B that is to match the velocity of the traveling waves on the multicavity structure.

Anode - Cathode Spacing Desire microwave field repetition with spatial periodicity of the structure. This field will have traveling wave components the most important of which is a component traveling in the same direction with about the same velocity, v B, as the outer ones in the Brillouin layer.

Anode - Cathode Spacing These traveling waves are slow waves with the desired phase velocity, v p ~ v B. Consider the wave equation as follows:

Anode - Cathode Spacing The solution of this equation results in hyperbolic trig functions:  d/v p  not too large, such that the E at Brillouin layer is insufficient for interaction  d/v p  not too small such that the E is so large that fields exert large force on electrons and cause rapid loss to the anode thereby reducing efficiency. Typically,

Multicavity Circuit - Slow Wave Structure Equivalent circuit of multicavity structure - here each cavity has been replaced by its LC equivalent. This circuit is like a transmission line filter “T” equivalent.

Multicavity Circuit - Slow Wave Structure The circuit acts like low-loss filter interactive impedance = input impedance of an infinite series of identical networks.

Multicavity Circuit - Slow Wave Structure Rf field repeats with periodicity p (spacing of adjacent cavities). Field at distance z+np is same as z.  = phase shift per unit length of phase constant of wave propagating down the structure. For a circular reentrant structure anode with N cavities, fields are indistinguishable for Z as for Z + np.

Fields and Charge Distributions for two Principal Modes of an Eight-Oscillator Magnetron

Multicavity Circuit - Slow Wave Structure

Strapped Cavities

Typical Magnetron Cross-Sections (after Collins) (a) Hole and slot resonators (b) Rectangular resonators (c) Sectoral resonators

Typical Magnetron Cross-Sections (after Collins) (d) Single ring strap connecting alternate vanes (e) Rising sun anode with alternate resonators of different shapes (f) Inverted magnetron with the cathode exterior to the anode

The unfavorable electrons hit the cathode and give up as heat excess energy picked up from the field. As a result, the cathode heater can be lowered or even turned off as appropriate. two

Rotating wheel formed by the favorable electrons in a magnetron oscillating in the  mode ref: Ghandi

General Design Procedures for Multicavity Magnetrons V,I requirements: From Power required may select V A. High V A  keeps current down and strain on cathode, but pulsed high voltage supplies are needed. Note P in = P 0 / efficiency and I A = P in / V A = P 0 /  V A. Cathode radius from available current densities for type of cathodes typically used in magnetrons. Typically J 0 (A/cm 2 )  0.1 to 1.0 for continuous, 1 to 10 for pulsed Smaller J 0  lower cathode temperature so longer life of tube Too low J 0  requires a larger r c

General Design Procedures for Multicavity Magnetrons Emitting length of cathode (l c ) < anode length, l a ; Typically, l c ~ 0.7 to 0.9 l a, and l a < /2 (prevents higher order modes) Smaller l a is consistent with power needs less B 0 needed (less weight) Radius r 0 (top of Brillouin cloud) from velocity synchronism condition: v p (r = r 0 ) =   r 0 / (N/2) = [  c r 0 /2] [1- (r c 2 / r 0 2) ]; therefore r 0 = r c / [1-(   /  c )(4/N)] 1/2 For an assumed B 0, r 0 can be calculated for a number of values of N (typically 6 to 16) or 20 to 30 for a small magnetron.

General Design Procedures for Multicavity Magnetrons Voltage eV B (r = r 0 ) = (1/2) mv B 2 where v B = v p (r = r 0 ) or V B = (v B /5.93x10 7 ) 2 ; v B in cm/sec ; Hence V B ~ 0.1 to 0.2 V A Note efficiency,  < (1 - V B / V A )*100; hence Smaller V B / V A contributes to improved efficiency Anode radius: ln (r a / r 0 ) = [V A - V B ] / {[  c 2 / 4(e/m)][(r 0 4 -r c 4 ) / r 0 2 ]} Also B min = (45.5 V A ) 1/2 [r a /(r a 2 -r c 2 )] << B 0 (  /v p )( r a - r c ) ~ 4 to 8 N must be even such that N  phase shift around the circumference is a whole 2 .

Cutaway view of a Coaxial Magnetron

NRL Hybrid Inverted Coaxial Magnetron