Particle-in-Cell Simulations on High-Efficiency Phase-Locking Millimeter-Wave Magnetrons with Unsynchronized High-Voltage Pulses

Abstract

Phase locking is an essential choice for building a coherent array, and a system of phase-locked magnetrons is relatively compact and cheaper than other microwave sources. Previous theoretical and experimental studies on phase locking are conducted using synchronized high-voltage pulses. Here, we investigate the characteristics of two phase-locked magnetrons using particle-in-cell (PIC) simulation software (CST STUDIO SUITE 2020) when two high-voltage pulses have delays. The results show that the magnetrons produced two-level RF signals because the operation could be divided into two stages. The first stage happened when one cathode emitted electrons; then, the electrons formed one spoke, traveling in synchronism with the 0-phase difference mode. Two output ports both produced half the output power of a free-running magnetron. The second stage happened after another cathode started to emit electrons, which were instantly pre-modulated by the electromagnetic field of the 0-phase difference mode produced during the first stage. In the second stage, simulations showed that pre-modulation accelerated the process of electron bunching. Eventually, two magnetrons were phase-locked, and the total output power of the two identical magnetrons nearly doubled the output power of the free-running magnetron, which demonstrated that the magnetrons were phase-locked in the high-efficiency phase-locking regime.

1. Introduction

Humans have discovered various phase-locking phenomena in the natural world for a long time. Phase locking is a complex dynamic process and a field within nonlinear theoretical science. It has numerous applications ranging from engineering to physics and even extends to the realm of biological sciences. The earliest scientific research on phase locking can be traced back to the early year of 1665 when the Dutch mathematician and physicist Christiaan Huygens observed the “Huygens’ pendulum synchronization phenomenon [1]”. He noticed that if he hung two pendulum clocks on a common support, which was a wooden rod suspended between two chairs, regardless of the initial conditions of the pendulums, the two pendulums take about half an hour to achieve anti-phase synchronization (oscillating at the same frequency but in opposite directions). Later on, scientists conducted extensive research into the phenomenon of pendulum synchronization. Gradually, the underlying principles of pendulum synchronization were understood. It was discovered that this phenomenon arises due to the mechanical transfer of energy between pendulums through a resonant coupling rod. When two pendulums achieve anti-phase synchronization, the vibrations imparted by one pendulum onto the resonant coupling rod are precisely canceled out by the vibrations imparted by the other pendulum. This cancellation effect drives the pendulums to achieve anti-phase synchronization [2]. This research result holds significance not only in theoretical physics but also finds broad applications in engineering and various scientific domains. One of the fields where these principles find application is in the realm of microwaves.

People are continuously pursuing higher-power microwave sources because of the demands for scientific research, wireless power transfer, radar, and so on [3,4]. The phase locking of microwave oscillators and building an array of them are promising ways to break the limitation of a single source [5]. A magnetron is one of the optimal choices for an array due to its high efficiency, compactness, and cost-effectiveness. Magnetrons are microwave oscillators, so their phases are random. The phase locking of magnetrons can solve the random phase problem that occurs between different magnetrons; then, the signals produced by magnetrons will become coherent. There has been some progress [6,7,8,9,10,11,12,13,14,15] on the phase locking of magnetrons since phase-locking theories were introduced [16,17,18]. The phase locking of magnetrons can be concisely categorized into three distinct types based on different injection methods and the propagation directions of electromagnetic energy:

• Injection locking [6,11,14,15,19]: This is the first type, characterized by unidirectional energy flow. Energy is injected into the magnetron using a microwave device. The fundamental structure of injection locking involves injecting a low-power signal of fixed frequency into the output port of a magnetron using a microwave circulator.
• Peer-to-peer locking [12,13,20]: The second type involves a mutual injection of energy through different paths into another magnetron, resulting in frequency locking and phase locking. The limitation of this peer-to-peer locking is that it is challenging to extend the experiments to systems with three or more magnetron array elements [12].
• Mutual coupling locking [5,7,8,9,10,21,22]: The third type comprises magnetrons that mutually inject electromagnetic energy through the same path. This approach is simple and reliable because it does not use microwave circulators with additional volume and weight. The only limitation for increasing the number of magnetrons in a phase-locking array is the lower phase-locking efficiency. In 1990, magnetrons use at least one-third of radiated energy to allow locking, where the power used for coupling is a significant fraction of the total produced. However, The high-efficiency phase-locking mechanism of magnetrons was proposed by the authors of [21]. We prove that adding both proper time delays and carefully designed matching between two magnetrons leads to the establishment of the high-efficiency phase locking of magnetrons. The results imply that the phase-locking efficiency is greater than 99%, which means that little radiated energy is wasted during the phase locking of magnetrons. Considering the high efficiency of magnetrons, the phase-locked arrays based on the mechanism are promising for potential future applications in wireless power beaming or other commercial applications.

However, high-efficiency phase locking still encounters some issues, and one of them is the problem of unsynchronized high-voltage power supplies, which has not been analyzed by anyone so far. In experiments, magnetrons are driven by the high-voltage pulses of power supplies [23,24,25]: If one power supply cannot provide sufficient current, then two or more power supplies are required. The repetition rate of the power supplies is usually above 1000 Hz. It is often difficult to achieve the perfect synchronization of high-voltage pulses between multiple power supplies. In such cases, unsynchronized pulses can have an impact on the high-efficiency phase locking of magnetrons. Therefore, we decided to exploit the knowledge of the mechanism to analyze the dynamical characteristics of phase-locked magnetrons with unsynchronized high-voltage pulses. This work attempts to elucidate the situation where high-voltage pulses between power supplies were triggered at different times. To this end, three-dimensional (3-D), electromagnetic simulation software CST STUDIO SUITE 2020 [26] is used to model and simulate the high-efficiency phase-locked circuit of 22-vane rising-sun magnetrons. Hexahedral mesh cells are utilized in PIC, eigenmode, and time-domain simulations by default.

The first section (Introduction) of the article introduces the history of phase locking and provides a brief classification of magnetron phase locking. Building upon our proposed high-efficiency phase-locking concept, we underscore the innovation of this article: the analysis of the issue arising from unsynchronized high-voltage pulses during the phase-locking process. The second section of the article consists of three parts: firstly, simulation data of a free-running magnetron, primarily intended for comparison with phase-locked magnetron simulations; secondly, simulations of phase-locked magnetrons without time delay; and thirdly, simulations of phase-locked magnetrons with a time delay of 30 ns. The final section contains our Conclusions and Discussions. It summarizes the work presented in this article and discusses potential future experimental endeavors.

2. Simulations and Results

Figure 1 depicts a schematic view of phase-locked magnetrons that are driven by two virtual high-voltage power supplies. The magnetron consists of an anode, cathode, and output. Two magnetrons are connected with a coupling bridge. Magnetron A and B are driven by high-voltage pulses A and B, respectively. Two output ports of magnetrons produce two phase-locked signals.

2.1. PIC Results of a Free-Running Magnetron

A free-running magnetron is a single oscillating magnetron without phase locking. Typically, during PIC simulations, the rise time is set to 1 ns or even shorter. Thus, considering practicable configurations to simulate real pulses [27,28,29,30], the rise time is set to 20 ns, the hold time is set to 60 ns, and the fall time is set to 20 ns, as shown in Figure 2a. Once the anode voltage rises to the Buneman–Hartree voltage, which is 10 kV at 16 ns, the magnetron starts to oscillate, and then the voltage increases to the operating voltage of 12.4 kV at 20 ns. After about 8 ns, the output power reaches 90% of the maximum value of full output power. Because the competition of other modes is weak, spokes thus form fast and travel in synchronism with the π mode; then, the output power becomes saturated. Although this is a simplification of an actual pulse, it still manages to illustrate the characteristics of magnetron oscillation. For the free-running magnetron, about 1.8 million mesh cells are used and 250 thousand macroparticles exist after oscillation becomes stable. The cathode emission model is DC. In the DC Emission model, the particle source emission current is fixed. Cathode priming is used for fast startup. For the cathode priming of a 22-cavity magnetron operating in the π mode (11 electron spokes), the cathode has 11 azimuthally periodic, emitting regions around the cathode [31]. An absolute current of 50 A ensures that the cathode possesses sufficient emission capability, and the current rise time is 0.1 ns. The operation voltage is set at 12.4 kV at 1.1 T, as shown in Figure 2b; the output power reaches a peak value of 46.5 kW at 36.1 GHz, and the anode current is 6.8 A, which results in an electronic efficiency of 55.1%. Notably, because there is no significant influence on the simulation of the free-running magnetron and the phase-locking mechanism, except for an increase in simulation time, a decrease in output power, and an increase in anode currents [32], the ohmic loss is temporarily ignored by using PEC as the anode material.

2.2. Eigenmode and Time-Domain Analysis of the 0-Phase Difference Mode

Phase locking occurred at a bridge length of L_b ≈ nλ_b/2, where λ_b is the wavelength in the coupling bridge corresponding to the dominant mode at the operating frequency. The 0-phase difference mode means that there are zero phase differences with respect to the π mode between two magnetrons, and the π-phase difference mode means there are 180-degree phase differences with respect to the π mode between two magnetrons. Because the two modes are based on the π mode that is operated in free-running magnetrons, they would compete with each other once their operating frequencies are close. To eliminate competition, it is necessary to increase the frequency separation between an operating mode and a competition mode.

The operating mode of our simulations is the 0-phase difference mode, as shown in Figure 3a. Figure 3b depicted the magnitude of two normalized tangential electric fields along the two circles with radius r. Eleven periods confirm that there is the π mode in the 22-vane rising-sun magnetron. As shown in Figure 3c,d, the phase difference of the π mode between two magnetrons is 180 degrees; thus, they comprise the π-phase difference mode. Because the frequency of the π-phase difference mode in Figure 3c is lower than the operating 0-phase difference mode, it is referred to as the (π−)-phase difference mode. Similarly, the frequency of the π-phase difference mode in Figure 3c is higher, and it is referred to as the (π+)-phase difference mode. Once the coupling bridge is perfectly matched, the (π−)- and (π+)-phase difference modes will not compete with the 0-phase difference mode because of the sufficient frequency separation between them, as shown in Figure 3e, which was obtained using the time-domain solver in CST. With a perfectly matched coupling bridge, two magnetrons lost less power during phase locking, and they are phase-locked in the high-efficiency regime.

The coupling bridge brings a time-varying delay between two magnetrons because the finite speed of signal propagation leads to some new characteristics of system dynamics. The 0-phase difference mode is not simply two separated π modes but two π modes that are coupled tightly as a whole. The phase-locked magnetrons operating at the 0-phase difference mode can be observed as the entire standing wave is excited. No matter which cathode starts to emit electrons, interactions produce oscillation in the 0-phase difference mode rather than the two separated π modes. If another cathode emits electrons afterward, the electrons of another magnetron will be phase sorted by the 0-phase difference mode. Thus, we predict that time delays between two power supplies would not change the process of phase locking.

Two situations are simulated and compared. The first is the situation where the time delay between two high-voltage supplies equals 0 nanoseconds; namely, there is no delay between the operation of two magnetrons. The second situation is that the time delay equals 30 nanoseconds. The time delay between two high-voltage pulses makes the emissions of two cathodes of magnetrons occur at different times, and the new characteristics of the phase-locked magnetrons are observed.

2.3. Phase Locking of Two Magentrons at Pulse Time Delay = 0 ns

Figure 4a depicts two normalized high-voltage pulses with no delay, and the situation is ideal because no two pulses are the same in an actual engineering experiment. The applied voltage and magnetic field are not changed, and two phase-locked magnetrons operate at 12.4 kV at 1.1 T. About 5.1 million mesh cells and 500 thousand macroparticles are used. For the phase-locked magnetrons, cathode priming is still used for fast startup [31]. The absolute currents of two cathodes are 50 A to ensure the sufficient capability of cathode emission [33]. Figure 4b shows that two magnetrons are phase-locked and reach a peak value of ~46.5 kW, nearly the same as the output power of the free-running magnetron. In Figure 4b, three periods from 50 ns to 50.09 ns are zoomed in to demonstrate that the phase difference between two signals is less than 1°. The anode currents of magnetrons A and B are stable at around 6.8 A, which is almost the same value as that of the free-running magnetron, as shown in Figure 4c.

Figure 5 shows that the magnetrons start to oscillate when the two high-voltage signals have increased to an operating voltage of 12.4 kV at 20 ns. After about 8 ns, the output power reaches 90% of the maximum value of the full output power. We defined it as the oscillation build-up time; that is, the oscillation build-up time is the time interval between the time when the high-voltage pulse increases to the operating voltage and the time when the output power reaches 90% of the maximum value.

Two spoke-like electron clouds gradually formed after 20 ns because of the synchronous interaction of two rotating electron clouds with the alternating RF electric field of the 0-phase difference mode. It is depicted in Figure 6 that the rotating electron cloud at 20 ns is distorted into a spoke-like pattern observed at 40 ns. In Figure 6b, two spokes in magnetrons A and B are almost the same because the 0-phase difference mode shown in Figure 3a maintains synchronism with two spokes, which are consistent with the synchronous output signals in Figure 4b.

2.4. Phase Locking of Two Magentrons at Pulse Time Delay = 30 ns

Figure 7a shows that pulse B has 30 ns delays compared to pulse A. The applied voltage and magnetic field have not changed, and two phase-locked magnetrons operate at 12.4 kV at 1.1 T. About 5.1 million mesh cells and 500 thousand macroparticles are used. The absolute currents of two cathodes are 50 A, so the capability of cathode emission is sufficient by default [33]. Figure 7b shows the output signals from magnetrons A and B. Figure 7c shows the anode currents from magnetrons A and B. Figure 8 is three zoomed-in signal pictures of Figure 7b at three different time. The normalized power depicted in Figure 9 exhibits a two-level waveform.

The first level is between 20 ns and 50 ns, as shown in Figure 9. At 20 ns, the high-voltage pulse of magnetron A reaches an operating voltage of 12.4 kV, and high-voltage pulse A leads to the emission of electrons from the cathode of magnetron A; an electron cloud in magnetron A is shown in Figure 10a. After 10 ns of the oscillation build-up time, magnetron A produces two signals from output A and output B, and the spoke-like electron cloud that is only in magnetron A travels in synchronism with the 0-phase difference mode with lower field intensities. Both output powers from outputs A and B are ~23.1 kW, only half as much as the output of the free-running magnetron. The free-running magnetron outputs a signal of 46.5 kW from one port, whereas two signals from two ports of the phase-locked magnetrons reach ~23.1 kW. The anode current is about 5.5 A in total, and the corresponding electronic efficiency is 67.7%. There is a 12.6% increase in electronic efficiency compared with that of the free-running magnetron. As shown in Figure 8a, the phase angle of signals from outputs A and B is ~−3°. This might be caused by the electronic admittance of the rotating electron cloud in magnetron B.

In the first stage, only one spoke-like electron cloud in magnetron A rotates with the alternating RF electric field of the 0-phase difference mode, as shown in Figure 10b. Between 30 ns and 40 ns, the anode current of magnetron A is 5.5 A and that of magnetron B is 0 A. During this time, the total power of two magnetrons is a little lower than the output power of the free-running magnetron but exhibits higher electronic efficiency.

The second level is between 50 ns and 80 ns, as shown in Figure 9. The operation voltage of magnetron B reaches 12.4 kV at 50 ns. After only 4 ns of the oscillation build-up time, two magnetrons are phase-locked, and output power increases from 50% to 90% of the maximum value of the free-running magnetron. As shown in Figure 8b, the phase angle of signals from ports A and B is ~0°. During this time, the total power of two magnetrons is nearly twice as much as the output power of a free-running magnetron, and their electronic efficiencies are kept nearly the same; the two signals are phase-locked.

Notably, the oscillation build-up time of the second stage is only about half of the oscillation build-up time of the free-running magnetron. After magnetron A operated, the oscillation build-up time of magnetron B is significantly shorted. The reasons are listed as follows

Figure 10a shows that a cathode of magnetron A starts to emit electrons by increasing high-voltage pulse A; the spokes rotating in magnetron A do not travel in synchronism with the single π mode, but they travel in synchronism with the 0-phase difference mode shown in Figure 3a; thus, microwave signals are outputted from port A and B simultaneously. After cathode B starts being emitted due to an increase in high-voltage pulse B, the 0-phase difference mode operated at resonant frequency begins to pre-modulate and phase sort the random motions of the rotating electron cloud in magnetron B into a spoke formation for the 0-phase difference mode [34]. The pre-modulation and phase sorting accelerate the space charge bunching process of magnetron B, which is most susceptible to external influences. Figure 10b shows the above pre-modulation and phase-sorting processing. Thus, the oscillation build-up time of the second magnetron, magnetron B, reduces the build-up time by nearly half (4 ns/8 ns = 1/2). Then, the spoke-like electron cloud in magnetron B forms and is phase-locked with that in magnetron A, while the 0-phase difference mode travels in synchronism with two spoke-like electron clouds, as shown in Figure 10c.

At 100 ns, when only magnetron B oscillates, the corresponding output power produced from two ports is nearly half the output power of the free-running magnetron. As the entire circuit is reciprocal, the anode current of magnetron B is 5.5 A and that of magnetron A is 0 A at this time. Because of the electronic admittance of the rotating electron cloud in magnetron A, as shown in Figure 8b, the phase angle of signals produced from magnetrons A and B is ~3°. Figure 10d shows that there is one spoke-like electron in magnetron B. This is similar to only one spoke in magnetron A when only magnetron A oscillates. At this time, the spokes that are rotated in magnetron B travel in the 0-phase difference mode, so microwave signals are still outputted from Ports A and B. Due to the decrease in high-voltage pulse A, cathode A no longer emits electrons, and the space charge in magnetron A fade away.

The above analysis of 30 ns delays between two high-voltage pulses of magnetron power supplies implies that the magnetrons can still be phase locked in combining two pulses.

3. Conclusions and Discussions

The results of the numerical simulations of two phase-locking magnetrons are encouraging when the high-voltage pulses of different power supplies have a certain delay. The output signals from magnetron A and B could be divided into two stages. The first stage happened when one cathode emitted electrons and then the electrons formed one spoke. Two ports both produced half the output power of a free-running magnetron. The second stage happened after another cathode started to emit electrons that were instantly pre-modulated by the electromagnetic field of the 0-phase difference mode produced during the first stage. In the second stage, simulations showed that pre-modulation accelerated the process of electron bunching. Eventually, two magnetrons were phase-locked, and the total output power of the two identical magnetrons nearly doubled the output power of the free-running magnetron; their efficiencies were nearly the same as that of the free-running magnetrons. The oscillation build-up time of the second level was only about 1/2 of the oscillation build-up time of a free-running magnetron, which provided a potential solution for the rapid start-up of magnetrons. Simulations and analysis proved that the phase-locked circuit and carefully designed coupling bridge have good robustness, and they will be beneficial to the ongoing experiment that is currently taking place at the University of Electronic Science and Technology of China.