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Traveling Wave Tubes, J. Pierce,
Principles of Traveling Wave Tubes, A.S. Gilmour, 1994
The electron cyclotron maser, K.R. Chu, Review of Modern Physics, 2004
Microwave Power Engineering, Vol. 1, E. Okress editor, Academic Press, 1968
Microwave Magnetrons, Collins, MIT Radiation Laboratory Series, V6, 1947
If this were a coax line in TEM mode (not coiled), vp/c = 1, independent of frequency, with unlimited bandwidth and no axial E.
By coiling the conductor, axial velocity is reduced by the amount of the increase in path length, a substantial Ez is created, and vz remains largely independent of frequency. Gain is obtained by synchronizing the beam velocity with the axial wave velocity
(from Principles of Traveling Wave Tubes, A. S. Gilmour)
ω-β diagram showing -1 space harmonic
(from Principles of Traveling Wave Tubes by A. S. Gilmour)
Photograph of ring-loop circuits, L-band through Ku-band
ω-β diagram for connected ring circuit showing -1 space harmonic suppressed
ω-β diagram for TE10 waveguide (from A. S. Gilmour)
ω-β diagram for periodically-loaded waveguide (from A. S. Gilmour)
Coupled-cavity circuit – a practical embodiment of a folded waveguide
Cloverleaf fundamental forward wave circuit produced 5 MW peak power at 10% BW
Basic interaction physics:
In a magnetic field, electrons orbit flux lines at the cyclotron frequency
where e is the electron mass, m0 is the electron mass and B0 is the magnetic field in Tesla
If an RF electric field is applied in the orbit plane, the momentum of the orbiting electrons will be modulated.
Negative mass effect: Electrons which gain energy have increased mass, larger orbit radius, and reduced angular velocity.
Compare beam area of helix, coupled cavity (similar to klystron), and cutoff overmoded waveguide of gyro-amplifier.
Tunnel size in TWTs and klystrons limited by ga, gyro-amplifiers use overmoded waveguide sections. For a gyroklystron, rb is limited by need to have drift tubes cut off to operating mode
Parameter a = v / v║ is the ratio of transverse to parallel velocity
Normal range of a is 1 – 2 making the rotational energy 50% to 80% of total beam energy
At an a of 1, a total efficiency of 30% implies a rotational efficiency of 60%
It is important to minimize energy spread in the beam as the efficiency drops dramatically with increased axial velocity spread.
This is especially important in gyro-amplifiers where axial velocity spread smears the bunch longitudinally as it drifts between cavities.
94 GHz Gyroklystron
100 kW peak power
10 kW average power
Equations of motion for an electron in crossed electrostatic and magnetostatic fields
Non relativistic Lorentz force equation F = ma = q ( E + v x B )
For planar magnetron this separates into three orthogonal components
Assume constant Ez and By
Integrating d2x/dt2, defining initial conditions and substituting into d2z/dt2 yields
Solving for z (and x) gives
Block diagram of CFA amplifier chain at 11 GHz for multi-megawatt system
Driver 10 W
TWT or klystron
30 dB 10 kW
Summary multi-megawatt system
The discussion has covered solid state amplifiers, traveling wave tubes, crossed field amplifiers, and gyro-amplifiers.
Obviously solid state amplifiers and traveling wave tubes do not produce the high peak power required for accelerator sources. They function well as drivers and as low power, wide bandwidth sources.
CFA’s and gyroklystrons can produce the required power but each has significant handicaps in comparison to a high power klystron.
The CFA can produce efficiencies in excess of 70% but has very low gain and requires a multi device amplifier chain to drive the final output CFA.
The gyroklystron has a problem with device efficiency, increasing a to minimize the energy in axial velocity causes increased axial velocity spread in the beam.
Gyro-amplifiers use overmoded RF circuits and therefore have a heat transfer advantage as the frequency increases.