Q7Microwave Engineering 2

(a) Operation and Applications of Magnetron

A magnetron is a high-power microwave oscillator tube consisting of a central cylindrical cathode surrounded by a coaxial anode block containing a ring of resonant cavities (slots) machined into its inner surface, with the entire structure placed within a strong axial magnetic field (parallel to the tube's axis, and hence perpendicular to the radial DC electric field established between the cathode and anode).

Operation: electrons emitted (thermionically) from the heated central cathode are accelerated radially outward by the DC electric field toward the anode, but the strong axial magnetic field simultaneously exerts a Lorentz force on these moving electrons, causing them to curve rather than travel in a straight radial path. Under appropriately designed operating conditions (specifically, an applied magnetic field near, but somewhat above, the Hull cutoff value discussed in a related answer), electrons follow complex cycloidal (looping) paths through the interaction space between the cathode and the anode's resonant cavities, rather than either striking the anode directly (which would happen at zero magnetic field) or immediately returning to the cathode without gaining useful energy (which would happen at very strong magnetic field, below the Hull cutoff voltage for a given field). As these circulating electrons pass close to the gaps of the anode's resonant cavities, they interact with the RF electric fields present at each cavity gap (initially arising from small random noise fluctuations, then rapidly building up through positive feedback), and if the electrons' rotational velocity is properly synchronized with the RF field's phase progression around the anode structure (the electrons should encounter each successive cavity gap when its RF field is oriented to decelerate the electron, converting the electron's kinetic energy into RF electromagnetic energy delivered to the cavity), a large population of electrons becomes organized into rotating 'spokes' of concentrated charge, continuously delivering energy to the cavity fields and sustaining a strong, continuous microwave oscillation extracted from one of the cavities via a suitable output coupling (such as a small coupling loop or aperture connected to an external waveguide).

Applications: the magnetron's ability to generate very high pulsed or continuous-wave microwave power (from a few hundred watts up to several megawatts in pulsed radar applications) with relatively high efficiency and comparatively simple, compact, and rugged construction has made it the traditional workhorse microwave source for high-power radar transmitters (particularly in older and many current marine, air-traffic-control, and weather radar systems), and, at much lower power levels, the magnetron remains the standard microwave power source used in virtually all domestic and commercial microwave ovens, where its output at approximately 2.45 GHz is specifically chosen to efficiently couple energy into water molecules within food via dielectric heating.

(b) Bunching Process and Efficiency of Reflex Klystron

The reflex klystron is a low-power microwave oscillator tube using a single resonant cavity through which an electron beam passes twice (once outbound toward a repeller electrode, and once again on its return path), with the interaction between the beam and the cavity field on both passes together sustaining oscillation.

Bunching process: electrons emitted from the cathode are accelerated toward and pass through the gap of a single resonant cavity, where a small initial RF voltage across the gap (arising from noise, then building up through positive feedback once oscillation is established) velocity-modulates the beam — electrons passing through the gap when the RF field is accelerating gain extra velocity, while those passing through when the field is decelerating lose velocity, and electrons passing through at the zero-crossing of the RF field pass through essentially unaffected. Beyond the cavity gap, this velocity-modulated beam travels into a field-free drift/repeller space toward a negatively-biased repeller electrode, which repels all the electrons back the way they came (since none can reach the negative repeller and continue past it) — during this return journey, the faster (accelerated) electrons catch up with the slower (unaffected or decelerated) electrons that left the cavity slightly earlier, causing the beam to progressively 'bunch' (group together in space) as it returns back toward the cavity gap. By appropriately designing the repeller voltage (and hence the transit time electrons spend in the repeller space), the returning bunches can be made to arrive back at the cavity gap at the optimum phase to deliver maximum energy to the cavity's RF field (i.e., arriving when the gap field is now in its decelerating phase relative to the returning electrons, so that the bunched electrons are slowed down by the field, giving up their kinetic energy to reinforce and sustain the cavity's oscillation), closing the feedback loop that sustains continuous oscillation.

Derivation of Efficiency Equation

The electronic efficiency of a reflex klystron, defined as the ratio of AC (RF) power delivered to the cavity to the DC beam power supplied by the accelerating voltage, can be derived by considering the energy given up by the bunched returning electron beam to the cavity field. The DC beam power is Pdc=V0 x I0 (accelerating voltage times average beam current), while the AC power delivered to the cavity is Pac=(1/2) x V1 x I2 x beta_i, where V1 is the RF gap voltage amplitude, I2 is the fundamental (first-harmonic) component of the bunched beam current induced by the velocity modulation and drift-space bunching process (itself expressible via the beam-coupling coefficient and a Bessel-function bunching parameter X, as I2=2 x I0 x J1(X), where J1 is the first-order Bessel function), and beta_i is the gap beam-coupling coefficient accounting for the finite electron transit time across the gap itself.

Substituting I2=2 I0 J1(X) and expressing the RF gap voltage in terms of the normalized bunching parameter X (which itself depends on V1, the DC voltage, and the drift-space transit angle), this expression can be reduced to a form showing that the maximum achievable electronic efficiency of a reflex klystron, optimized over the bunching parameter X, works out to a theoretical maximum of approximately 22.7% (occurring at the optimum value of the bunching parameter, X approximately equal to 2.408, radians, corresponding to the first maximum of the relevant Bessel-function-based efficiency expression) — in practice, actual reflex klystron efficiency is typically somewhat lower than this theoretical maximum (commonly in the range of 10-20%) due to additional losses (cavity wall losses, beam interception, and non-ideal bunching), but this Bessel-function-derived theoretical efficiency ceiling is a well-known, characteristic performance limit of the reflex klystron's basic single-cavity, single-beam-pass bunching mechanism, and is one of the key reasons reflex klystrons are generally used only as relatively low-power, though conveniently simple and mechanically/electronically tunable, microwave sources (such as laboratory signal generators and local oscillators) rather than as high-power microwave amplifiers or transmitters.