--- title: "Cross-field cavity amplifier with 90° phase shift" subject: "RF and Microwave Engineering" chapter: 5 key: "semester_7/rf_and_microwave_engineering:crossfield-device-90-degree" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/crossfield-device-90-degree site: BCT Engineering --- # Cross-field cavity amplifier with 90° phase shift RF and Microwave Engineering · Chapter 5: Microwave Generators - Asked in: 2072 Magh, 3+7 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/crossfield-device-90-degree - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-5 ## Question What is cross-field effect? Design and describe the working principle of a cross field cavity device for power amplifier. Consider a 90° phase shift between adjacent cavities. ## Short answer Cross-field effect is electron motion in perpendicular E and B fields, drifting at speed E/B. An 8-cavity anode with n=2 mode gives 90° phase shift, forming two electron spokes. Electrons drift synchronously with the RF wave; phase focusing transfers energy for amplification. A drift section between output and input prevents oscillation. ## Answer #### Cross-field effect The **cross-field effect** is the motion of electrons under a DC electric field E and a DC magnetic field B that are **perpendicular** to each other. The Lorentz force F = −e(E + v × B) bends the electrons into cycloidal paths that drift at speed v = E/B, at right angles to both fields. An RF wave travelling at this drift speed can take **potential energy** from the electrons. This is the basis of M-type tubes (magnetron, CFA, amplitron). #### Cross-field cavity power amplifier with 90° between adjacent cavities **Design choice of cavities and mode** The phase shift between adjacent cavities in mode n of an N-cavity structure is φ = 2πn/N. For φ = 90°: ```text 90° = 360°·n / N -> N = 4n choose N = 8 cavities, n = 2 -> 2 spokes (N = 4, n = 1 also gives 90° with 1 spoke) ``` An 8-cavity anode is chosen. The RF pattern repeats every 4 cavities (0°, 90°, 180°, 270°), and two electron spokes are formed. ```text 8-cavity anode, phases (n = 2) 0° 270° _/ \_ 90° / \ 180° | (cathode) | 180° B0 axial \ 2 spokes/ E radial 90° \_ _/ 270° 0° RF in -> cavity 1 RF out <- cavity 7 (drift section between out and in) ``` **Design steps (outline)** 1. Choose the anode voltage V₀ and anode/cathode radii b, a. 2. Set B₀ above the **Hull cut-off**: B₀c = √(8mV₀/e) / [b(1 − a²/b²)]. 3. Choose V₀, B₀ to satisfy the **Hartree condition**: the electron drift angular velocity equals the RF wave's angular velocity ω/n = ω/2. 4. Size the cavities to resonate (or form a slow-wave band) at the working frequency, and cut out the coupling between output and input with a drift section so the tube cannot oscillate. **Working principle as a power amplifier** 1. The heated cathode emits electrons. They move in the cross fields and drift around the cathode. 2. The input RF is coupled into the first cavity. The cavities, coupled through the interaction space, carry it as a slow wave, with 90° phase step per cavity. 3. Since the drift speed matches the wave speed, electrons stay in a fixed phase relative to the wave. 4. **Phase focusing:** electrons in the retarding RF phase give potential energy to the wave and move to the anode. Those in the accelerating phase gain energy and return to the cathode. Two spokes form and rotate with the wave. 5. The wave grows from cavity to cavity as the spokes give it energy. The amplified signal is coupled out at the output cavity. The drift section stops feedback, so the device amplifies instead of oscillating. **Performance:** efficiency of about 40–70%, gain of about 10–20 dB, about 10% bandwidth, high peak power. Used as radar transmitter output stages. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For 90° between adjacent cavities with N = 8, which mode n and how many spokes result? A. n = 1, one spoke B. n = 2, two spokes C. n = 4, four spokes D. n = 8, eight spokes Answer: B. 90° = 360°·n/N gives N = 4n; N = 8 gives n = 2 and two spokes. (Key point: Cavities for 90 degrees) 2. For the 90° n = 2 design, what angular velocity must the electron drift match? A. w, the RF angular frequency B. w/2, the rotating wave's angular velocity C. 2w D. Zero Answer: B. The wave rotates at ω/n = ω/2, and V0, B0 are chosen so the electron drift equals that. (Key point: Hartree condition) 3. What stops the cross-field amplifier from oscillating? A. A very high magnetic field B. A drift section cutting coupling between output and input C. Using only 4 cavities D. Removing the cathode Answer: B. Without feedback from output to input, the device only amplifies the injected signal. (Key point: Why amplifier not oscillator) 4. In a cross-field device, where does the energy given to the RF wave come from? A. The potential energy of electrons moving to the anode B. The magnetic field directly C. The kinetic energy of a long drift D. The cathode heater Answer: A. Retarding-phase electrons give up potential energy as they move toward the anode, in step with the wave. 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