Q8Microwave Engineering 2

(a) Cutoff Magnetic Field for Cylindrical Magnetron

In a cylindrical magnetron, electrons emitted from the central cathode (radius a) are accelerated toward the surrounding cylindrical anode (radius b) by the applied DC voltage V0, while a uniform axial magnetic flux density B exerts a Lorentz force on the moving electrons, causing their paths to curve. The cutoff magnetic field, Bc, is defined as the specific critical value of B at which an electron emitted from the cathode (with zero initial velocity) just barely fails to reach the anode, grazing it tangentially with zero radial velocity component at the point of closest approach to the anode — for magnetic field values above this cutoff value, electrons can no longer reach the anode at all (they curve back toward the cathode before arriving), while for field values below cutoff, electrons do reach the anode.

Derivation: applying the Lorentz force equation and Newton's second law to an electron's motion in the combined radial electric field and axial magnetic field of the cylindrical magnetron geometry, and using the conservation of angular momentum (which arises naturally from the specific form of the magnetic force term in cylindrical coordinates) together with the work-energy theorem (relating the electron's kinetic energy at any radius to the potential difference through which it has been accelerated from the cathode), the electron's tangential velocity at the anode radius b, for the critical grazing condition (zero radial velocity at r=b), can be shown to satisfy:

and by the work-energy theorem, the electron's total kinetic energy at radius b (having been accelerated through the full potential difference V0 from cathode to anode) must equal (1/2) m v_theta(b)^2 = e V0 (since at the critical grazing condition, all velocity at the anode is purely tangential, with zero radial component). Equating these two expressions for the electron's kinetic energy/velocity at the anode and solving for B gives the cutoff magnetic field:

where m and e are the electron's mass and charge magnitude respectively, V0 is the applied anode-cathode DC voltage, and a and b are the cathode and anode radii respectively. This cutoff-field expression is of central practical importance in magnetron design and operation, since a magnetron is normally operated with an applied magnetic field somewhat above this cutoff value (ensuring that, in the absence of any RF cavity interaction, electrons would not directly reach the anode via a simple radial path), so that the actual electron trajectories reaching the anode (and hence contributing to anode current and RF power generation) occur only through the cavity-field-assisted spoke/bunching mechanism described in the corresponding magnetron operation answer, rather than through simple, uncontrolled DC conduction directly across the interaction space.

(b) Problems with Conventional Tubes at Microwave Frequencies and Two-Cavity Klystron Operation

Conventional vacuum tubes (such as ordinary triodes, tetrodes, and pentodes), which perform very satisfactorily as amplifying and oscillating devices at radio and lower frequencies, suffer from several fundamental performance limitations that become progressively more severe as operating frequency is increased into the microwave range.

• Interelectrode capacitance: the physical capacitance existing between a conventional tube's closely-spaced electrodes (particularly grid-to-cathode and grid-to-plate capacitance) presents an increasingly low reactive impedance as frequency rises, progressively short-circuiting the tube's intended signal path and severely degrading gain at microwave frequencies.
• Lead inductance: the physical lead wires connecting a conventional tube's internal electrodes to its external terminals possess a small but non-negligible series inductance, which similarly presents an increasingly significant reactive impedance at microwave frequencies, further degrading circuit performance and making it difficult to realize the intended resonant circuit characteristics at the desired microwave frequency.
• Electron transit-time effects: at microwave frequencies, the finite time an electron takes to physically transit the space between a conventional tube's cathode and plate becomes an appreciable fraction of (or even comparable to or longer than) the RF signal's own period, meaning the applied grid voltage changes significantly during the electron's transit, causing the electron stream's response to lag and become out of phase with the applied signal, fundamentally degrading gain, introducing excessive noise, and in severe cases causing the tube's input impedance to become resistive (dissipative) and load down the driving signal source, further compounding the frequency-limitation problem.

These combined limitations mean conventional grid-controlled vacuum tubes become essentially unusable as efficient amplifying or oscillating devices much above the lower UHF range, motivating the development of specialized microwave tube designs (klystrons, magnetrons, traveling-wave tubes, and similar velocity-modulation/transit-time-based devices) that work with, rather than against, electron transit-time effects.

Two-Cavity Klystron Operation

The two-cavity klystron amplifier consists of an electron gun (cathode and accelerating anode) producing a high-velocity electron beam, which first passes through the gap of an input (buncher) resonant cavity, then travels through a field-free drift space, and finally passes through the gap of a second, output (catcher) resonant cavity, before being collected at a collector electrode. The RF input signal to be amplified is applied to the buncher cavity, velocity-modulating the electron beam exactly as described for the reflex klystron in an earlier answer (electrons passing through the buncher gap during the accelerating half-cycle speed up, those passing during the decelerating half-cycle slow down). As this velocity-modulated beam travels through the drift space, faster electrons catch up with slower ones ahead of them (that left the buncher gap slightly earlier), causing the beam to bunch (group together) by the time it reaches the catcher cavity gap, exactly analogous to the bunching mechanism in the reflex klystron, but here occurring during a single, one-way transit through a separate drift tube rather than during a round-trip transit to and from a repeller electrode. The bunched beam, now carrying a strong RF current component at the input signal's frequency, induces a corresponding RF voltage across the catcher cavity gap as it passes through, and since the catcher cavity gap is oriented to extract kinetic energy from the bunched electrons (decelerating them as they pass through), this delivers amplified RF power to the catcher cavity, which is then coupled out to an external load via a suitable output coupling structure — because the buncher and catcher cavities are physically separate, distinct resonant structures (unlike the single shared cavity of the reflex klystron), the two-cavity klystron functions as a true amplifier (with independent input and output ports and unidirectional signal flow from buncher to catcher), rather than purely as an oscillator, making the two-cavity (and, in higher-power/higher-gain designs, multi-cavity) klystron a foundational and still widely-used high-power microwave amplifier technology, particularly for radar transmitters, satellite communication ground stations, and particle accelerator RF power sources.