28 GHz / 10 kW reference point

A nominal frequency of 28 GHz and output power of 10 kW define the reference device. Its operating mode is TE₁,₂ at the first cyclotron harmonic, n = 1. The cavity radius is Rc = 9.05 mm and its uniform section is Zc = 65 mm long. U₀ = 23 kV, anode-to-cathode voltage Ua = 17 kV and beam current Ib = 1 A.

AnGEl report No. 00004196 dated 29.10.2026 gives g = ⟨v⊥⟩/⟨v∥⟩ = 1.6744, transmitted current 1 A and reflected current 0 A. The emitter-center radius is 9 mm and the band is 2 mm long. The emitter center is at z = −157.15 mm; the cavity and main magnetic system are centered at z = 0. B₀ = 1 T and the total emitter-center field is Bk = 0.106 T. Bk is the total field; ΔBk is the cathode-coil contribution. The linear approximation is Bk = 0.106·B₀ + ΔBk, with all fields in T. The coefficient 0.106 is set by the reference point at B₀ = 1 T and ΔBk = 0; separate measured MSS and coil field maps are not provided in the report. MSS saturation, hysteresis and the cathode-coil contribution inside the cavity are not included. Electron-optical system retains its separate teaching reference point.

Electron-optics calibration

The scaling p⊥,cavity ∝ Ua√B₀/Bk^3/2 is anchored to this reference point. The 100 equal-current particles are calibrated to a mean g = 1.6744 and a full Vp spread of 11.371%. The uniform distribution is an assumption, not a reconstruction of the AnGEl trajectories. Space charge, potential depression and electrode fields are not solved.

The effective g = √(⟨q⟩/(1−⟨q⟩)), with q = p⊥²/p², preserves the mean squared transverse momentum of the transmitted particles for the RF model. At the reference point it is 1.668 ≈ 1.67. The reported compression ratio 9.385 uses more precise internal field values; the rounded ratio 1/0.106 is 9.434. The adiabatic compression model is not applicable when Bk ≤ 0 or E⊥/Bk ≥ c. A local planar crossed-field estimate first classifies electrons intercepted by the anode. Only the remaining electrons are checked for magnetic reflection; g is calculated for the beam transmitted to the cavity.

Anode interception uses the same relativistic Hull criterion as the megawatt laboratory, with an illustrative 8 mm gap and ±8% gap variation. These are teaching assumptions, not AnGEl gun dimensions. Ib = Ipassed + Irf + Ia; the reflected and intercepted currents are excluded from RF injection. Orange trajectories terminate on the anode; normal transmitted electrons remain clear of it. This estimates electron interception, not vacuum-arc inception.

Frequency and transferred power

The closed-cylinder equation f₀ = c/(2π)·√[(χ′₁,₂/Rc)² + (π/Zc)²], with χ′₁,₂ = 5.33144277, gives 28.20 GHz. The axial index q = 1 and sinusoidal field profile are teaching assumptions. Open-cavity effects, tapers and frequency pulling are not calculated. At U₀ = 23 kV and B₀ = 1 T, the relativistic cyclotron frequency fc ≈ 26.79 GHz must not be identified with the rounded 28 GHz in the report.

64 initial phases are integrated in a prescribed rotating field at constant p∥. The amplitude, from 0 to 6 MV/m, is selected by the mean energy transfer. P = ηU₀Ib is power transferred to the field, not device output power. The nominal 10 kW corresponds to 43.48% of the 23 kW beam power, but the trajectory report contains no RF calculation or confirmation of this efficiency. Its Wclt = 23 kW is electrical power at the collector without RF interaction. The calculation is not fitted to 10 kW. Losses, self-excitation and mode competition are not calculated.

TRT prototype · Ananichev et al., 2025

Fundamental harmonic; TE₃₃,₁₃ mode. Reference beam: 80 kV, 40 A, calculated pitch factor 1.2. Cavity radius 17.38 mm; beam guiding radius 7.2 mm; LaB₆ emitter diameter 86 mm. Cathode coil sensitivity: 4.56 mT/A. The JASTEC 10T150 magnet provides up to 10 T.

The supply supports up to 100 kV / 50 A. Control ranges are a teaching domain, not a guaranteed operating envelope. The displayed Ua is the anode potential relative to the grounded body. At the reference point Ua = 0 kV and the cathode is at −80 kV, so the gun-forming difference is Uak = Ua + U₀ = 80 kV. The gun solver uses Uak; changing U₀ retains the selected Ua. Both slider scales are fixed. The anode potential spans 0–20 kV and U₀ spans 20–100 kV independently, giving a forming difference Uak of 20–120 kV. All four corners of this control domain are supported. This convention applies to this laboratory only.

Measured results

Generation was observed at 80 kV / 40 A over 9.13–9.29 T. Maximum output: 0.94 MW; reported efficiency: 28%, without energy recovery. The interval 9.13–9.29 T is the observed operating-mode generation range, not a constant peak-power plateau. The rounded beam parameters give P/(U₀Ib) = 29.375%, which differs from the reported 28%; no unsupported loss correction is introduced. Spectrum center: 230.1 GHz at 9.26 T. Pulses: 80 μs at 10 Hz.

A two-channel quasi-optical converter separates RF from the spent beam. High-power RF exits through a side BN vacuum window; the other channel is for an injected locking signal, which is not simulated. Electrons continue to the collector.

Teaching approximations

The anode-interception estimate uses the relativistic planar Hull criterion B_H = (mc/ed)√[(1 + eUa/mc²)² − 1]. Electrons whose local orbit reaches the anode are removed before magnetic compression and RF interaction. Ib = Ipassed + Irf + Ia. The 8 mm gap and ±8% variation are illustrative settings, not data from the TRT paper. Anode bombardment and overcurrent can precede a discharge, but do not by themselves establish a vacuum arc; gas, surface emission and plasma dynamics are not modeled.

Default Zc = 45 mm and cathode-coil contribution Bk = −0.0122 T are selected simulation settings. The paper does not report the cavity length; 45 mm is not a reconstruction of the actual tube. The axial index q = 1 is assumed. Cold-cavity frequency uses χ′₃₃,₁₃ = 83.72322656. The reference emitter field is estimated by flux conservation: Bk,total = 9.26·(7.2/43)² ≈ 0.260 T. Main-field fringe and a signed cathode-coil contribution are superposed. The drawn beam compresses continuously with r²B ≈ const; its Larmor radius follows p⊥²/B ≈ const. The axial field profile is schematic and the electron orbits are enlarged for visibility.

Without gun field maps, the same adiabatic scaling as in the first laboratory is used: q⊥ ∝ Uak²·B₀/Bk,total³ divided by squared momentum. An illustrative ±10% transverse-momentum ensemble models magnetic reflection; it is not the paper’s measured distribution. Passed electrons determine g and RF injection. The reference effective g is 1.2.

64 phases interact with a prescribed sinusoidal RF field, scanned over 0–6 MV/m. Efficiency includes reflected current; P = η·U₀·Ib is power transferred to the RF field before output losses. The displayed efficiency and power are recalculated from the selected parameters; power follows P = ηU₀Ib and is not separately fitted to 0.94 MW. Space charge, self-consistent excitation, mode competition, losses and recovery are not calculated. Searches hold Rc, Zc and their mode frequency fixed and do not prove a global maximum.