ECE 662 – Microwave Electronics

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

2. 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

3. 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

4. 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

6. Z Z

7. Planar Magnetron Let VA = potential difference between the anode and cathode, and E0=- VA /d. An applied magnetic field is in the x direction (into the paper). The force on the electrons becomes: x z

8. Planar Magnetron

9. Planar Magnetron

10. 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 E0/B0 , independent of vz0 and vy0.

11. uox here is the v0z of our formulation ref: Gerwartowski

12. Planar Magnetron Electrons have dc motion equal to E0/B0, slow wave structure is assumed to be a propagating wave in the direction of the electron flow with a phase velocity equal to E0/B0

13. Planar Magnetron(ref. Hemenway)

14. Planar Magnetron (ref. Hemenway)

15. 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.

16. Circular Magnetron Oscillator ref: Gewartowski

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

18. Brillouin Cloud Note: electrons at the outermost radius of the cloud (r = r0) move faster than those for r < r0. The kinetic energy (of the electrons) increase is due to drop in potential energy.

19. Hull Cutoff Condition For a given B0, the maximum potential difference VA that can be applied between the anode and cathode, for which the Brillouin cloud will fill the space to r = ra is

20. Hull Cutoff Condition B0 < B0min direct current flows to anode and no chance for interaction with rf. B0 > B0min Brillouin cloud has an outer radius r0 < ra and no direct current flows to the anode. For a typical magnetron, B0 > B0min therefore r0 < ra

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

22. Magnetron Fields From Poisson’s equation the charge density: 0falls slightly as r increases from rc (can increase 0 by increasing B0 which follows as electrons spiral in smaller cycloidal orbits about the cathode.

23. Magnetron Fields Outside the Brillouin cloud, r0 < r < ra, in region II, use Gauss’s Theorem:

24. Hartree Relationship The potential difference VA between the cathode and anode to maintain the Brillouin cloud of outer radius r0 is given by:

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

26. 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, vB , as the outer ones in the Brillouin layer.

27. Anode - Cathode Spacing These traveling waves are slow waves with the desired phase velocity, vp ~ vB. Consider the wave equation as follows:

28. Anode - Cathode Spacing The solution of this equation results in hyperbolic trig functions: d/vp  not too large, such that the E at Brillouin layer is insufficient for interaction d/vp  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,

29. 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.

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

31. 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.

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

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

35. Multicavity Circuit - Slow Wave Structure

36. Strapped Cavities

37. Typical Magnetron Cross-Sections (after Collins) • Hole and slot resonators • Rectangular resonators • Sectoral resonators

38. 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

41. 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 loweredor even turned off as appropriate. two two

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

44. General Design Procedures for Multicavity Magnetrons V,I requirements: From Power required may select VA. High VA  keeps current down and strain on cathode, but pulsed high voltage supplies are needed. Note Pin = P0 / efficiency and I A = Pin / VA= P0 / VA. Cathode radius from available current densities for type of cathodes typically used in magnetrons. Typically J0 (A/cm2)  0.1 to 1.0 for continuous, 1 to 10 for pulsed Smaller J0 lower cathode temperature so longer life of tube Too low J0 requires a larger rc

45. General Design Procedures for Multicavity Magnetrons Emitting length of cathode (lc) < anode length, la ; Typically, lc ~ 0.7 to 0.9 la , and la < /2 (prevents higher order modes) Smaller la is consistent with power needs less B0 needed (less weight) Radius r0 (top of Brillouin cloud) from velocity synchronism condition: vp (r = r0) =  r0 / (N/2) = [c r0 /2] [1- (rc2 / r02)]; therefore r0 = rc / [1-( /c)(4/N)]1/2 For an assumed B0, r0 can be calculated for a number of values of N (typically 6 to 16) or 20 to 30 for a small magnetron.

46. General Design Procedures for Multicavity Magnetrons Voltage eVB (r = r0) = (1/2) mvB2 where vB = vp (r = r0) or VB = (vB /5.93x107)2 ; vB in cm/sec ; Hence VB ~ 0.1 to 0.2 VA Note efficiency,  < (1 - VB / VA )*100; hence Smaller VB / VA contributes to improved efficiency Anode radius: ln (ra/ r0) = [VA - VB ] / {[c 2 / 4(e/m)][(r0 4 -rc 4 ) / r0 2 ]} Also Bmin = (45.5 VA) 1/2 [ra /(ra 2 -rc 2 )] << B0 ( /vp)( ra - rc) ~ 4 to 8 N must be even such that N phase shift around the circumference is a whole 2 .

48. Cutaway view of a Coaxial Magnetron

50. NRL Hybrid Inverted Coaxial Magnetron