Q2Microwave Theory And Techniques

Structure: a cylindrical magnetron consists of a central cylindrical cathode surrounded coaxially by an anode block containing 8 resonant cavities arranged symmetrically around its inner circumference, all immersed in a strong axial static magnetic field B0 (parallel to the cathode axis) while a DC anode-cathode voltage Va establishes a radial static electric field E0 between cathode and anode.

Electron motion — crossed-field principle: electrons emitted from the cathode experience the combination of the radial electric force (−eE0, pulling them outward toward the anode) and the magnetic force (−ev×B0, deflecting their motion sideways) — in the crossed-field (E⊥B) configuration of the magnetron, this combination produces a cycloidal electron trajectory that drifts azimuthally around the cathode-anode gap rather than moving directly across it, with the electron's average drift velocity given by the E×B drift, vd=E0/B0.

RF interaction and the π-mode: each of the 8 cavities acts as a resonant circuit with its own gap field; when adjacent cavities' gap voltages alternate in polarity (the π-mode, in which the RF phase difference between adjacent cavities is exactly π radians, the mode normally selected for magnetron operation via strapping of alternate anode segments), the resulting RF field pattern rotates around the anode structure at the correct angular velocity to stay synchronized with the drifting electron cloud. Electrons that arrive at each cavity gap at the correct RF phase are decelerated (giving up potential energy to the RF field, sustaining oscillation) while electrons at the wrong phase are re-accelerated back toward the cathode; the collective, self-sorting effect of this interaction (analogous to bunching in a klystron, but occurring azimuthally around the magnetron rather than axially along a drift tube) concentrates electrons into spoke-like bunches that continuously transfer their kinetic energy into the RF cavity fields, sustaining strong oscillation at the resonant frequency set by the cavity geometry.

Hartree voltage derivation (linear magnetron model): the Hartree condition gives the minimum DC anode voltage Va at which oscillation can be sustained for a given magnetic field B0, derived by requiring that an electron just reaching the anode (at the edge of the interaction space) does so with exactly zero radial velocity while simultaneously matching the synchronism condition with the RF wave. Using the simplified planar (linear) magnetron model, with cathode-anode spacing d and electron charge-to-mass ratio e/m, the Hartree voltage is obtained by combining the energy equation (from the work done by the DC field, ½mv²=eVa at the anode with v the tangential/drift velocity) with the synchronism condition (that the electron's azimuthal — here, linear — drift velocity vd = E0/B0 = Va/(B0d) must match the phase velocity of the RF wave along the anode structure, vph = ωd/π for the π-mode with N cavities, ω the operating angular frequency):

Combining this synchronism condition with the cycloidal-motion energy relation for the crossed-field electron (accounting for the magnetic field's contribution to the electron's total kinetic energy at the anode) yields the standard Hartree voltage formula:

This Hartree voltage represents the threshold anode voltage below which no oscillation can be sustained for a given B0 (the electron cloud fails to reach synchronism with the RF wave); operating the magnetron above the Hartree voltage (but below the higher 'Hull cutoff' voltage at which electrons would reach the anode even with zero applied RF field, corresponding to a DC short-circuit condition) is the necessary operating regime for stable, self-sustained magnetron oscillation, and the Hartree condition is the standard design equation relating required anode voltage to magnetic field strength for a magnetron operating at a specified frequency and geometry.