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Mengfan ZHANG, Dingguo LAI, Qifu XU, Mengtong QIU. Impedance control method of the cylindrical virtual cathode reflex triode based on multi-ring cathode[J]. Plasma Science and Technology, 2025, 27(4): 044011. DOI: 10.1088/2058-6272/adbc35
Citation: Mengfan ZHANG, Dingguo LAI, Qifu XU, Mengtong QIU. Impedance control method of the cylindrical virtual cathode reflex triode based on multi-ring cathode[J]. Plasma Science and Technology, 2025, 27(4): 044011. DOI: 10.1088/2058-6272/adbc35

Impedance control method of the cylindrical virtual cathode reflex triode based on multi-ring cathode

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  • Author Bio:

    Dingguo LAI: laidingguo@nint.ac.cn

  • Corresponding author:

    Dingguo LAI, laidingguo@nint.ac.cn

  • Received Date: October 23, 2024
  • Revised Date: February 26, 2025
  • Accepted Date: March 02, 2025
  • Available Online: March 03, 2025
  • Published Date: April 01, 2025
  • The cylindrical virtual cathode reflex triode is a new type of pulsed hard X-ray load, which has the advantages of simple structure, high radiation conversion efficiency, and simplicity in series-parallel operation. This paper presents a method to reduce the impedance of the triode using a multi-ring cathode. The average electric field on the ring-cathode emission surface is enhanced due to edge effect, and the beam intensity is greatly increased in proportion to the square of the electric field strength. Multi-ring cathode is used to enlarge the emission area. Therefore, the reflex triode can work at lower impedance and generate a stronger beam under the same anode-cathode gap. In addition, the electric field enhancement of the cathode reduces the cathode emission stabilization time and enhances the operation stability of the triode. The effects of parameters such as ring width and ring gap on the triode impedance are simulated and studied. The cathode emission stabilization time and the X-ray conversion efficiency are compared. The design basis of cathode structure parameters and the impedance control method of the cylindrical virtual cathode reflex triode are given according to the simulations.

  • The utilization of pulsed hard X-rays within the 10‒100 keV range is essential for investigating the system-generated electromagnetic pulse (SGEMP) effect. Currently, high-energy density pulsed hard X-rays are generated by bombarding high-Z targets with high-current pulsed electron beams to induce bremsstrahlung [1]. To meet the fidelity requirements of the X-ray spectrum for SGEMP studies, the electron energy must be kept below 300 keV. However, achieving high-intensity and large-area pulsed hard X-rays poses challenges due to the low conversion efficiency of low-energy electron-generated X-rays [2]. In recent years, advancements such as series diodes [35], planar reflection triodes [69], cylindrical reflection triodes (CRT) [1013], and their series-parallel technologies [14, 15] have been developed to enhance X-ray conversion efficiency and address load series-parallel operation issues. However, the external X-ray of the CRT target cannot be used effectively.

    To tackle these challenges, the research team at the Northwest Institute of Nuclear Technology has proposed a cylindrical virtual cathode reflex triode (CVCRT) load. This innovative structure simplifies design, enables modular series-parallel operation, reduces X-ray loss outside the target, and enhances X-ray intensity over the effective radiation area by approximately 2 times under similar voltage and current conditions [1, 16]. While improving load conversion efficiency is crucial for generating a pulsed hard X-ray radiation field with a fluence exceeding 1 J/cm2 and a radiation area of thousands of cm2, reducing impedance is also vital to produce a stronger beam within limited voltage constraints.

    In this paper, we propose an impedance control method based on a multi-ring cathode for CVCRTs. We investigate the impact of cathode geometric parameters on triode impedance both theoretically and numerically, providing a foundation for designing low-impedance triodes.

    The CVCRT operates by utilizing a cylindrical anode foil, a grounded cathode, and the vacuum anode-cathode (AK) gap between them. When a positive high-voltage pulse is applied to the anode, electrons are emitted from the cathode surface, moving towards the anode under the gap electric field and interacting with the anode. Part of the kinetic energy of the electrons is converted into thermal energy and deposited in the anode, and a small part is converted into photon energy to generate X-rays via bremsstrahlung. The electrons penetrate the anode, forming a virtual cathode on the inside after interacting with the anode, and make the transmission electrons move in the opposite direction, interact with the anode again. This process continues until the electron energy is completely absorbed, resulting in multiple reflections. The electrons absorbed by the anode during this process form the current in the circuit. The electron is subjected to the combined action of the electric field and the self-generated magnetic field during the reflection process, the E×B drift occurs, and the electron moves in the direction of the radiation window (the right side of figure 1(a)) so that the X-ray is strengthened in one direction. The design of CVCRT leads to multiple reflections of electrons, enhancing X-ray production in a specific direction with high energy conversion efficiency. Compared to CRTs, the CVCRT removes the inner solid cathode, features a simplified and compact structure, facilitating modular design and efficient series-parallel operation.

    Figure  1.  (a) The working principle and (b) structure diagram of CVCRT.

    The working process of CVCRT includes electron emission, the interaction between electrons and the target, and the transport process in a complex electromagnetic environment, and its beam density of the cathode can be expressed as [1, 17, 18]

    {J=fA(μ,U)gβ2Eβ2CJ0g=(Ra/RaRcRc)1/πβE=1+dπWβC=RcRaRcln(Rc/RcRaRa)J0=49ε02emeU3/2d2. (1)

    Where g represents the structural correction factor of the coaxial cylinder, βE denotes the electric field enhancement factor of the ring electrode surface, and βC signifies the electric field enhancement factor of the coaxial cylinder electrode surface; J0 stands for the beam density defined by the one-dimensional Child-Langmuir law; Ra corresponds to the outer radius of the anode foil, Rc denotes the inner radius of the cathode through hole, W represents the cathode thickness, U signifies the diode voltage, d stands for the AK gap [1, 18]. The target absorption factor fA of the reflex triode describes the total effect of processes such as electron-atom collisions, electron scattering, electron absorption, and radiation generation on the beam intensity. The electrons produced by the cathode are almost completely absorbed after acting on the target, and the reflected and transmitted electrons are few when the anode has a strong absorption effect on electrons. At this point, the target absorption factor fA approaches 1, and the CVCRT is equivalent to a CRT, its electron beam density is close to the maximum. The target absorption factor fA approaches 0 when the anode has a weak absorption effect on electrons. At this point, the current in the circuit is difficult to form, and the beam density is also close to 0. The absorption factor fA is difficult to accurately describe by analytical expression because of a large number of influencing factors and complex processes, so it is usually calibrated by numerical simulation. The absorption factor is determined by the anode thickness μ and gap voltage U when the anode material is determined [1]. The beam intensity of the CVCRT can be expressed as

    I=JS=fAgβ2Eβ2CJ02πRcW. (2)

    Where J represents the beam density of the cathode, S represents the emission area of the cathode.

    In accordance with equation (2), the beam intensity of the CVCRT is contingent upon both the emission area of the cathode and the electric field intensity of the cathode surface at a specific voltage. As the cathode ring width diminishes with a constant AK gap, the electric field enhancement factor of the cathode surface exhibits a nonlinear increase, leading to a substantial enhancement in beam density. Nonetheless, a reduction in the cathode emission area accompanies the decrease in ring width. The beam density J (1+d/(πw))2J0, wherein the cathode’s emission area varies linearly with the ring width, suggests that within a specific range of ring widths, the rate of increase in beam density surpasses the rate of decrease in emission area corresponding to the cathode ring width. As illustrated in figure 2, dividing a cathode with a width of w0 into n cathodes with widths of w0/n can elevate the electric field enhancement factor on the cathode surface under a consistent emission area. Consequently, this division boosts the current density of cathode emission and reduces the impedance of the CVCRT under the same AK gap.

    Figure  2.  The structure diagram of the multi-ring cathode.

    The electric field enhancement effect is the basis of the impedance control method. The electric field enhancement effect of the ring width w on the single-ring cathode can be expressed by the field enhancement factor βE = 1+d/(πw), which is in good agreement with the numerical results [18]. Compared with the single-ring cathode, the electric field distribution of the multi-ring cathode is more complex, and the average electric field intensity of the multi-ring cathode is weaker than that of the single-ring cathode due to the shielding effect between the rings. With the increase of the ring gap, the electric field shielding effect is weakened, and the field enhancement factor gradually increases, but it is always less than 1+d/(πw).

    In multi-ring cathodes, the electric field shielding between the rings reduces the emission beam density, with this shielding effect gradually diminishing as the ring gap u increases. However, a clear standard for selecting the ring gap is currently lacking. For a ring with width w, when its emission current matches the Child-Langmuir beam intensity emitted by a region of width w0 without a field-enhanced structure, it can be deduced that the cathode ring has no impact on electron emission beyond the w0 region. This implies that the cathode ring provides sufficient space for achieving maximum beam emission. According to equations (1) and (2), the following relationship is presented:

    Hw(1+dπw)2=Hw0. (3)

    Where H(Ra,Rc,μ,U)=2πRcfAgβ2CJ0. The minimum ring gap can be expressed as:

    umin (4)

    The current of the multi-ring cathode CVCRT reaches its peak when the ring gap is sufficiently large and interactions between cathode rings can be disregarded, as shown in equation (5):

    {I_{{\text{max}}}} = Hnw{\left( {1 + \frac{d}{{{\text{π}}w}}} \right)^2} . (5)

    The width of a single-ring cathode with an equivalent emission area is nw, and its current can be denoted as:

    {I_{{\text{sr}}}} = Hnw{\left( {1 + \frac{d}{{{\text{π}}nw}}} \right)^2} . (6)

    In the case of multi-ring cathodes, as the number of rings n increases, the ring width w decreases, Imax increases, while Isr remains constant. Figure 3 depicts the changes in Imax and Isr as n varies, with d = 5 mm and nw = 12 mm. It is clear that, in comparison to a single-ring cathode with a similar emission area, the beam emitted by the multi-ring cathode intensifies, with the current growth rising in tandem with the increase in the number of rings n.

    Figure  3.  The curve of the maximum current of the multi-ring cathode CVCRT Imax and the current of the single-ring cathode CVCRT Isr with an equivalent emission area with the number of rings n.

    The electric field distribution on the surface of the double-ring cathode is simulated and compared with that of the single-ring cathode of the same width. The results are shown in figure 4, the AK gap d = 3 mm, the ring width w = 4 mm, and the ring gap u = 4 mm. Due to the shielding effect, the surface electric field of the multi-ring cathode decreases significantly compared with that of the single-ring cathode, especially at the edge of the emission region near the other ring. The effect of ring gap u on the field enhancement factor of the double-ring cathode is further simulated. As shown in figure 5, the electric field enhancement factor βE increases with the increase of ring gap u and approaches the limit value 1+d/(πw). The simulation results confirm the electric field enhancement effect of the multi-ring cathode, and the variation law of its variation with the ring width and ring gap is given.

    Figure  4.  Comparison of electric field distribution on the surface of multi-ring and single-ring cathodes.
    Figure  5.  Simulation results of the electric field enhancement factor of the multi-ring cathode variation with the ring gap u.

    Electron reflection and drift within the CVCRT modify the spatial charge distribution, leading to interactions among the cathode rings. The relationship between beam intensity and the configuration of a multi-ring cathode is highly intricate. To further explore the impact of structural parameters on beam intensity, the multi-ring cathode, defined by a specific cathode radius Rc, is characterized by four parameters (n, w, u, h), where n represents the number of rings in the cathode, w denotes the ring width, u indicates the ring gap, and h signifies the groove depth between the rings. Its structure is illustrated in figure 6. The parameters w and d determine the electric field enhancement factor. At the same time, the ring gap u influences the shielding effect between the rings, and the groove depth h modifies the distribution of the gap electric field. These parameters collectively contribute to fluctuations in beam intensity.

    Figure  6.  Structure diagram of CVCRT with multi-ring cathode.

    To analyze and simulate the impedance characteristics of multi-ring CVCRT, a coefficient is introduced

    k = \frac{{{I_{{\text{mr}}}}}}{{{I_{{\text{sr}}}}}} , (7)

    where Imr is the beam intensity for a multi-ring cathode, and Isr is the beam intensity for a single-ring cathode with the same emission area. k can directly represent the extent to which the emission current is increased by the enhanced structure with the same emission area, it can also reflect the degree of electric field shielding between rings. When n = 1, k = 1, and if n > 1, k > 1.

    Initially, the impact of the groove depth h is examined through simulation. In the computational model, the anode radius of the CVCRT is set at 7.5 mm, the length of the AK gap is 5 mm, and the anode consists of tantalum foil with a length of 80 mm and a thickness of 20 μm. Maintaining consistency with other parameters, the variation in impedance Z of multiple CVCRTs employing multi-ring cathodes with different h values is simulated using the particle-in-cell (PIC) method. The outcomes are illustrated in figure 7. For h < 2 mm, the impedance of each triode experiences a notable decrease as h increases, weakening the shielding effect between the rings. Conversely, when h \geqslant 2 mm, the impedance of each triode stabilizes and remains relatively constant.

    Figure  7.  Simulation results of triode impedance variation with groove depth for multi-ring cathodes.

    The variation in the k value of a multi-ring cathode concerning the ring width w, ring gap u, and ring number n was computed using the PIC method, as depicted in figure 8. In the examined multi-ring cathodes, the beam intensity surpasses that of the corresponding single-ring cathodes with equivalent emission areas. The k value escalates with an increase in the ring gap u, signifying that as the ring gap widens, the shielding effect diminishes gradually. Consequently, the triode impedance decreases gradually, leading to a rise in the current of the multi-ring cathode, approaching its maximum value. These simulation outcomes corroborate the findings of the theoretical analysis.

    Figure  8.  Simulation results of k for different multi-ring cathodes. (a) Under n = 2, (b) under n = 3.

    The simulation outcomes reveal a notable disparity in beam intensity among the cathode rings within a multi-ring configuration. Specifically, the emission intensity from the central cathode rings is markedly lower than that of the end cathode rings, as illustrated in figure 9. The beam intensity ratio across the three cathode rings, from left to right, stands at 1.79:1:1.28. This discrepancy arises due to a significant pinching effect, attributed to the shielding of the middle cathode ring by its adjacent rings, while the end rings are predominantly shielded by a single adjacent ring, allowing more space for beam drift. Owing to the robust shielding effect exerted by the central cathode rings, their contribution to the total emission beam intensity is relatively minor. Moreover, a higher number of rings would escalate the complexity of cathode processing and assembly. Hence, it is advisable to limit the number of rings in a multi-ring cathode to prevent undue complications. Considering the simulation findings, a prudent choice would be to keep the number of rings at n \leqslant 4 for optimal performance.

    Figure  9.  Schematic diagram of cathode emission when n = 3.

    During the initial phase of electron emission, the unemitted segment on the cathode surface becomes shielded by the emitted area, thereby impeding emission from the unemitted region. For a single-ring cathode, electron emission commences at the electric field-enhanced zones located at the upper and lower edges of the cathode aperture. Subsequently, as the triode voltage escalates, emission gradually extends to the middle of the cathode surface, culminating in complete surface emission. Research indicates that the swift emission of the cathode surface promotes stable emission [19]. The duration from electron emission initiation to full cathode surface emission is termed the cathode emission stability time te. Simulation analysis of the variation in cathode emission stability time with diverse ring widths was conducted to assess its impact on emission stability. In figure 10, the simulation results illustrate the alteration in emission stability time for a single-ring cathode concerning cathode thickness. As the cathode thickness increases, te exhibits a gradual incline. This trend arises from the reduction in ring width, which diminishes the disparity in electric field intensity between the edge and central regions. Consequently, the electric field shielding effect of electrons initially emitted from the edge on the central region weakens relatively.

    Figure  10.  Curve of emission stability time of single-ring cathode with the width of single-ring cathode.

    Compared the single-ring cathode and multi-ring cathode with the same emission area, the cathode emission stability time is shown in table 1. Under the condition of a voltage rise rate of 7 kV/ns, the time of complete emission of multi-ring cathode decreases compared with that of single-ring cathode. When the number of rings and the ring width of a multi-ring cathode increase, the stable emission time of the cathode also increases. Therefore, to improve the emission stability, the cathode ring width should be reduced as much as possible.

    Table  1.  Emission stabilization times for different cathodes at 7 kV/ns voltage rise rate.
    Multi-ring Single-ring
    n w (mm) u (mm) te (ns) wsr (mm) te (ns)
    2 2 4 5.55 4 6.59
    4 4 7.46 8 7.50
    3 2 4 6.89 6 7.35
    4 4 8.12 12 8.87
     | Show Table
    DownLoad: CSV

    The variation in ring width w and ring gap u alters the point of electron interaction with the anode, impacting X-ray absorption by the cathode and subsequently influencing X-ray output. Photons moving towards the radiation window are termed forward photons, with their total energy ratio to the total energy of the electric pulse loaded on the triode defined as the conversion efficiency ηp.

    Through simulation and analysis, the conversion efficiency of CVCRT with varying cathodes was investigated under identical electrical pulse energy conditions (i.e., consistent voltage and current). The structural parameters (n, w, u, h) of the multi-ring cathode are shown in figure 11, where the units of w, u, and h are millimeters. As depicted in figure 11, the conversion efficiency was primarily influenced by the overall cathode width wtotal, exhibiting a gradual decline as the total cathode thickness increased. When the total cathode thickness expanded from 8 mm to 20 mm, the conversion efficiency decreased from 0.7% to 0.68%. While the utilization of multi-ring cathodes can enhance beam intensity and bolster emission stability, its impact on X-ray conversion efficiency is relatively minimal.

    Figure  11.  Variation of X-ray conversion efficiency of different triodes with total cathode thickness.

    Addressing the challenge of reducing the impedance of high-intensity large-area pulsed hard X-ray loads, this paper introduces a method involving the utilization of a multi-ring cathode to regulate the impedance of the CVCRT. The impact of various parameters of the multi-ring cathode on the CVCRT’s impedance is meticulously simulated and analyzed, alongside an examination of changes in cathode emission stability time and its implications on overall stability. Furthermore, a comparative analysis of X-ray conversion efficiency among multi-ring cathodes with different total widths is conducted. In scenarios featuring consistent cathode gap and emission area, employing a multi-ring cathode leads to a significant amplification in beam intensity and a substantial reduction in diode impedance. Notably, due to the factors like electron reflection and drift affecting intermediate cathode ring emission, the beam enhancement in the cylindrical reflection triode falls short of theoretical expectations. Enhanced efficiency is observed particularly when the ring count remains below 4. Decreasing ring width results in a reduction in cathode emission stability time, thereby enhancing operational stability of the CVCRT. While an increase in the total width of the multi-ring cathode alters the electron target position and X-ray absorption by the cathode, the resultant effect remains modest, typically below 3%.

    The findings highlight that electric field enhancement and beam density augmentation facilitated by the multi-ring cathode can enhance impedance control and operational stability, potentially leading to a further reduction in CVCRT impedance. This advancement holds promise for resolving the technical challenges associated with generating high-intensity large-area pulsed hard X-rays. The subsequent phase involves experimental validation of these outcomes.

    This work was supported by National Natural Science Foundation of China (Nos. 12027811 and 12275222).

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