Longitudinal impedance measurements and simulations of a three-metal-strip kicker

Longitudinal impedance measurements and simulations of a three-metal-strip kicker 1 2 first-author https://orcid.org/0000-0001-8239-7280 1 corresp † liuyd@ihep.ac.cn † liuyd@ihep.ac.cn liuyd@ihep.ac.cn 1 https://orcid.org/0000-0001-7928-5114 1 1 https://orcid.org/0000-0003-3998-6439 1 https://orcid.org/0000-0002-2889-0143 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China A kicker is a critical component for beam injection and accumulation in circular particle accelerators. A ceramic slat kicker plated with a TiN conductive coating was applied in the Beijing Electron Positron Collider (BEPC2). However, the ceramic slat kicker has experienced several sudden malfunctions during the operation of the BEPC2 in the past. With a reliable kicker structure, a three-metal-strip kicker can substitute the original ceramic slat kicker to maintain the operational stability of the BEPC2. A comparison of the numerical simulation was conducted for three kicker models, demonstrating the comprehensive advantage of the three-metal-strip kicker. Furthermore, impedance bench measurements were conducted on a prototype of a three-metal-strip kicker. The longitudinal beam-coupling impedance was measured using a vector network analyzer (VNA) via the coaxial wire method. A satisfactory agreement was obtained between the numerical simulations and measurements. Based on the numerical simulation data, the loss factor was 0.01721 V/pC, and the effective impedance was 3.59 mΩ ( σ =10 mm). The simulation of the heat deposition on each part of the kicker demonstrated that 84.4% of the parasitic loss of the beam was deposited on the metal strips, and the total heat deposition power on the kicker was between 113.3 W and 131.5 W. The obtained heat deposition powers can be considered as a reference for establishing the cooling system. Ceramic slat kicker Three-metal-strip kicker Impedance bench measurement Coaxial wire method Heat deposition power Introduction 1 A ceramic slat injection kicker was proposed and applied to the BEPC2, which meets the requirements of the BEPC2 storage ring injection system for a sufficient wide field region, high field uniformity, and low beam-coupling impedance. The detailed requirements for the injection kicker according to the BEPC2 design report are listed in Table 1 Table 1 . At the top and bottom sides of the beam pipe, there are two long ceramic slats plated with a TiN conductive coating to provide a continuous image current. Its longitudinal length is approximately 1.7 m, and it is composed of two current buses and two ceramic slats, which can fix the current buses and support the TiN coating. The aperture of the kicker magnet is 90 mm×44 mm. A TiN coating with a width of 34 mm was plated on the inner surface of the ceramic slats. The ceramic slat kicker sufficiently solved the contradiction between the beam impedance and field uniformity [ 1 1 ]. 1 1 Parameter 1 1 Value 1 1 Available length (m) 1 1 1.7 1 1 Beam stay clear (mm× mm) ( H × W ) 1 1 38× 82 1 1 Injection energy (GeV) 1 1 1.89 1 1 Deflection (mrad) 1 1 3 1 1 Integrated field strength (T ⋅ m) 1 1 0.02 1 1 Bottom width of field waveform (ns) 1 1 600 1 1 Good field region (mm) 1 1 X = ±20 1 1 Field uniformity ( Y =0 mm plane) 1 1 ±1% 1 1 Field uniformity ( Y =5 mm plane) 1 1 ±2% 1 1 Field uniformity ( Y =10 mm plane) 1 1 ±5% However, according to the BEPC2 2019 annual meeting proceedings, the ceramic slat injection kicker malfunctioned in April 2010 (e+k2), April 2011 (e-k2), December 2014 (e+k1), and August 2016 (e+k1). A possible reason for this failure is that the beam or synchrotron radiation directly hit the TiN coating, causing the coating to detach from the ceramic slat. Subsequently, the image current was blocked. A series of higher-order modes (HOM) was induced, resulting in an ignition inside the kicker. Consequently, the vacuum worsened, and the beam was lost. To prevent coating detachment, three metal strips were chosen to replace the TiN-plated ceramic slat as the image current path. To prevent the eddy current shielding effect and ensure magnetic field uniformity, two metal strips were disconnected from the middle with a slit of 1 mm. Two important issues for the design of a kicker are the beam-coupling impedance and heat deposition on the kicker. The beam-coupling impedance can cause collective beam instabilities. To ensure a good beam quality and stable operation, the impedance of the components should be strictly limited. The heat load can cause localized heating, which may burn the components or destroy the vacuum in severe cases. Therefore, it is necessary to perform detailed impedance simulations of the three-metal-strip kicker. It is also useful to conduct bench measurements of impedance to verify the impedance simulations. The remainder of this paper is organized as follows. The mechanical structure of the three-metal-strip kicker is described in Sect. 2. Section 3.1 presents a simplified CST model of kicker, an explanation of how it is simplified from the mechanical structure, and the difference between the simplified kicker model and another two CST models. Section 3.2 presents several formulae to describe the relationship between the wake function, wake potential, and impedance; the impedance simulation results of the three models are compared, and the possible source of narrow-band resonances is explained herein. Section 4.1 presents the coaxial wire measurement setup, measurement of the characteristic impedance with VNA-achieved TDR, conduction of another CST simulation to obtain the cutoff frequency of the kicker, comparison of the impedance between the simulations and measurements, explanation for the significantly large narrow-band resonance amplitudes of the measurements, and the calculated longitudinal loss factor and effective impedance. Section 5 presents the application of a module called time frequency power loss monitor in CST STUDIO SUITE to obtain the heat deposition power, and the simulation results are displayed. The conclusions are presented in Sect. 6. MECHANICAL STRUCTURE OF THE THREE-METAL-STRIP KICKER 1 With a longitudinal port-to-port length of 1706 mm and transverse tank width of 305 mm, the three-metal-strip kicker is quasi-symmetric, as shown in Figs. 1(a) and (b) Figs. 1(a) and (b) . The two kicker octagon ports have the same dimensions, and the outer diameter (52 mm) is slightly larger than the inner diameter (44 mm), forming a longitudinal taper, as shown in Figs. 1(c) and (d) Figs. 1(c) and (d) . The feedthrough was connected to the current buses through two power plates. Two long-current buses were located on both sides of the kicker. Two groups of three metal strips were located above and under the beam axis, functioning as the image current path. Among the three metal strips, two side strips have a slit of 1 mm in the middle, which ensures the maximum passage of the magnetic field, as shown in Fig. 1(f) Fig. 1(f) . The gap between the strips is 8 mm wide and that between strip and current buses is 10 mm wide. Owing to HOM heating, the metals of the kicker would be lengthened to a certain extent during operation with the beams. The metal strips should be carefully considered owing to the 1 mm slit. The fractional change in length is proportional to the change in temperature as follows: <math display="block" id="M1"><mrow><mfrac><mrow><mtext>Δ</mtext><mi>L</mi></mrow><mi>L</mi></mfrac><mo>=</mo><mi>α</mi><mo>⋅</mo><mtext>Δ</mtext><mi>T</mi><mn>,</mn></mrow></math> where Δ L/L is the fractional change in length, Δ T is the change in temperature, and α is the linear thermal-expansion coefficient [ 2 2 , 3 3 ]. Because the length change of the strips should be less than 1 mm, we obtain the following: <math display="block" id="M2"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><msub><mi>L</mi><mn>1</mn></msub><mo>⋅</mo><mi>α</mi><mo>⋅</mo><mtext>Δ</mtext><mi>T</mi><mo>+</mo><msub><mi>L</mi><mn>2</mn></msub><mo>⋅</mo><mi>α</mi><mo>⋅</mo><mtext>Δ</mtext><mi>T</mi></mrow><mo>)</mo></mrow><mo><</mo><mn>1</mn><mtext> mm</mtext><mn>,</mn></mrow></math> where L 1 and L 2 are the strip lengths, as shown in Fig. 2 Fig. 2 . The linear thermal expansion coefficient of copper at a temperature of 20 ℃ was 17e-06/℃. Therefore, we found that the temperature of the metal strips should not exceed 99 ℃; otherwise, the strips would touch each other during the operation with the beams. The temperature of the strips was determined based on the cooling system efficiency. Therefore, the heat deposition power on the strips is an important reference for establishing a cooling system. LONGITUDINAL IMPEDANCE SIMULATIONS AND COMPARISON 1 To acquire the complete information regarding the beam-coupling impedance, numerical simulations were performed using CST STUDIO SUITE [ 4 4 ]. The simulation results were also compared with those of two other models, illustrating that the three-metal-strip kicker is currently the most reliable model after considering the magnetic field uniformity, impedance, and operation stability comprehensively. Simulation models 2 A simplified model of the mechanical structure was applied to accelerate the simulation process. The underprop that supports the kicker below was deleted because it could not be seen by the beams. Screws, small holes, and fixing devices were deleted because they were shielded by the current buses and metal plates and did not have a significant effect on the beams. The outer parts of the kicker octagon ports and current buses were reshaped to be significantly smoother because they could not be seen by the beams. The cabling was not considered, and the power plates that connected the feedthrough with the current buses remained in the CST modeling. The fixer between the current buses and front octagon port was also modeled. A simplified structure of the three-metal-strip model is shown in Fig.3 Fig.3 . The mesh sizes can be automatically adjusted by the CST software according to the specific dimension of a region. Adjustable mesh sizes can improve the calculation accuracy while saving the computing resources. In our calculations, a total of approximately 1e9 meshes were used for the three-metal-strip kicker. A Gaussian beam with an RMS bunch length of 10 mm was selected for the simulation. To improve the accuracy of the simulation results, the wakefields were tracked over 10 m behind the bunch. Excluding the fixer connecting the current buses with the front kicker octagon port that was set as being composed of alumina, all other parts were set as being composed of a perfect electroconductive material (PEC). Another model, called the fully unconnected model, is based on the three-metal-strip model, in which the only difference is that all its strips are unconnected with the same slit of 1 mm. The third model, the metal-slat model, is also based on the three-metal-strip model, in which the only difference is that the group of the three metal strips is combined as an entire metal slat. The structural differences between the three models are shown in Fig.4 Fig.4 . Beam-coupling impedance 2 The longitudinal wake function in the CST software is defined as follows: <math display="block" id="M3"><mrow><msup><mi>W</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>q</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></msubsup></mrow></mstyle><msub><mi>E</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>s</mi><mn>,</mn></mrow></math> where q is the excitation charge, E s is the electric field along the s direction seen by the test charge, z is the distance at which the test charge is behind the excitation charge, and W’ ( z ) is the longitudinal wake function. The longitudinal impedance <math id="M4"><mrow><msub><mi>Z</mi><mo>∥</mo></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow></math> can be calculated from the wake function W’ ( z ) using a Fourier transformation as follows: <math display="block" id="M5"><mrow><msub><mi>Z</mi><mo>∥</mo></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mn>1</mn><mi>c</mi></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>ω</mi><mi>z</mi><mo>/</mo><mi>c</mi></mrow></msup><msup><mi>W</mi><mo>'</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>z</mi><mn>.</mn></mrow></math> However, it is impossible to apply a point charge to perform the simulation. A bunch with a certain length and charge distribution was set up for the simulation. Therefore, the CST software provides data regarding the wake potential <math id="M6"><mrow><msub><mi>V</mi><mrow><mo>|</mo><mo>|</mo></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></math> rather than the wake function W’ ( z ). The relationship between the wake potential and wake function can be expressed as follows: <math display="block" id="M7"><mrow><msub><mi>V</mi><mo>∥</mo></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></msubsup></mrow></mstyle><mi>λ</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><msup><mi>W</mi><mo>'</mo></msup><mrow><mo>(</mo><mrow><mi>z</mi><mo>−</mo><mi>ξ</mi></mrow><mo>)</mo></mrow><mtext>d</mtext><mi>ξ</mi><mn>,</mn></mrow></math> where λ ( ξ ) denotes the normalized line density of the bunch. The following is obtained by combining Eqs. (3) and (4): <math display="block" id="M8"><mrow><msub><mi>Z</mi><mo>∥</mo></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></msubsup></mrow></mstyle><msub><mi>V</mi><mo>∥</mo></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>ω</mi><mi>z</mi></mrow></msup><mtext>d</mtext><mi>z</mi></mrow><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></msubsup></mrow></mstyle><mi>λ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>ω</mi><mi>z</mi></mrow></msup><mtext>d</mtext><mi>z</mi></mrow></mfrac><mn>.</mn></mrow></math> The wake potentials of the three models are shown in Fig. 5 Fig. 5 . The wake of the fully-unconnected model does not sufficiently decay to zero after 10 m. It must be a long-range wake owing to the gaps and slits. It is unnecessary to make the wake of the fully unconnected model sufficiently decay to zero after 10 m because it is only shown to compare the narrowband resonances. The wakes of the other two models decay to zero after 10 m. The exact impedances can be obtained only if the wake potentials can be obtained for an infinite length or if they damp out completely over a finite length. Because the calculation of the wake potentials must be terminated at a certain length owing to the limitations of the computing resource, the impedance obtained from the simulations demonstrates oscillatory behavior around the actual resonance frequencies, and its real part may become negative at certain frequencies. To alleviate this problem, data windowing is applied, and a cos 2 filtering function is selected in the CST software [ 4 4 ]. The real and imaginary parts of the longitudinal beam-coupling impedance of up to 2 GHz are shown in Fig.6 Fig.6 . This comparison illustrates that the narrowband resonances are caused by the tank. The diameter of the tank is 305 mm. Its cutoff frequency is approximately 1 GHz. The transverse dimension is sufficiently large to generate these resonances at low frequencies. The dimensions of the gaps and slits determine the effect of the tank. A greater number of gaps and slits lead to significantly higher resonances. The resonances of the fully unconnected model are significantly higher than those of the three-metal-strip model because the slits in the side strips of the fully unconnected model magnify the effect of the tank. The resonance amplitude of the metal-slat model is the smallest among these models because the metal slat weakens the effect of the tank. Impedance simulations and measurements of the ceramic slat kicker were conducted by Demin Zhou [ 5 5 ], demonstrating that its impedance is nearly the same as that of the three-metal-strip kicker. Owing to the shielding effect produced by the eddy currents, the magnetic field cannot pass through the metal slat. Therefore, the metal-slat kicker, although demonstrating the lowest impedance, is not a suitable structure for the BEPC2. The fully unconnected model is also not a suitable structure owing to its large impedance. According to our measurements, the three-metal-strip kicker and ceramic slat kicker demonstrate similar performances with respect to the magnetic field. However, during the practical commissioning and operation of the BEPC2 in the past 10 years, the accelerator breakdown caused by its TiN coating detachment occurred every three or four years. Therefore, the optimized three-metal-strip kicker with a lower longitudinal impedance and higher reliability may be the most suitable structure for the BEPC2 owing to its apparent robustness against damage caused by heating. IMPEDANCE BENCH EXPERIMENT ON A THREE-METAL-STRIP KICKER PROTOTYPE 1 A prototype of the three-metal-strip kicker was manufactured to conduct an impedance bench experiment. The coaxial wire method was applied in the experiments, which has been widely used for the impedance bench measurements of the accelerator components [ 5 5 - 13 13 5-13 ]. In particular, Chinese investigators have conducted several pioneering studies regarding impedance measurements [ 5 5 - 9 9 5-9 ], which significantly benefited our experimental setup. Coaxial wire measurement setup 2 The facilities for coaxial wire measurements include the VNA, tapers, device under test (DUT), and a smooth reference pipe (REF). The layout of the experimental setup is illustrated in Fig.7 Fig.7 . As shown in Fig.8 Fig.8 , the ports of the REF are the same as those of the DUT. The experimental bench is illustrated in Fig.9 Fig.9 . The taper developed by Huang et al. [ 14 14 ] is shown in Fig.10 Fig.10 . The N-type connectors, connecting blocks, and Teflon spacers are shown in Fig.11 Fig.11 . A copper wire with a diameter of 2 mm was used as the inner conductor. A combination of the copper wire on the beam axis with the connecting blocks, tapers, or DUT forms a coaxial wire that can be processed as a two-port network. Because the characteristic impedance of the coaxial cables and VNA, 50 Ω, is significantly different from that of the coaxial wire formed by the copper wire and DUT, this large impedance mismatch can cause a severe reflection, inducing ripples above the real impedance. To solve the mismatch problem, tapers are commonly used in the measurements. Because the entire length of the structure, including the tapers, reaches approximately 3 m, and the end of the tapers has a diameter of only 6 mm, stretching the copper wire along the beam axis [ 15 15 ] is significantly difficult. As a result, the measurement accuracy significantly decreases. In our measurement, we soldered the copper wire to an N-type connector with tin and used connecting blocks to cover the copper wire. Half-round Teflon spacers, as shown in Fig. 11 Fig. 11 , were placed inside the connecting blocks to fix the copper wire. The characteristic impedance, which changes with the distance along the coaxial wire, is shown in Fig. 13 Fig. 13 . The flatness in the middle of the curve indicates that the copper wire was sufficiently stretched to a certain extent. To calculate the longitudinal beam-coupling impedance, the scattering coefficients S 21 of the DUT and REF were measured using an Agilent E5071C ENA VNA. Subsequently, the impedance of the DUT can be calculated based on the scattering coefficients using the Hahn-Pedersen formula [ 16 16 ]: <math display="block" id="M9"><mrow><msub><mi>Z</mi><mo>∥</mo></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msub><mi>Z</mi><mtext>c</mtext></msub><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>S</mi><mrow><mn>21,</mn><mtext>REF</mtext></mrow></msub></mrow><mrow><msub><mi>S</mi><mrow><mn>21,</mn><mtext>DUT</mtext></mrow></msub></mrow></mfrac><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> or the Palumbo-Vaccaro formula [ 17 17 ]: <math display="block" id="M10"><mrow><msub><mi>Z</mi><mo>∥</mo></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msub><mi>Z</mi><mtext>c</mtext></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><msub><mi>S</mi><mrow><mn>21,</mn><mtext>DUT</mtext></mrow></msub></mrow><mrow><msub><mi>S</mi><mrow><mn>21,</mn><mtext>REF</mtext></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> or using the Walling formula for a long DUT [ 18 18 ]: <math display="block" id="M11"><mrow><msub><mi>Z</mi><mo>∥</mo></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msub><mi>Z</mi><mtext>c</mtext></msub><mi>ln</mi><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>S</mi><mrow><mn>21,</mn><mtext>REF</mtext></mrow></msub></mrow><mrow><msub><mi>S</mi><mrow><mn>21,</mn><mtext>DUT</mtext></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where Z c is the characteristic impedance of the coaxial wire formed by the copper wire and DUT. The Walling formula was adopted for the experiment in this study. The VNA-achieved TDR was applied to measure the characteristic impedance [ 19 19 ]. Because the travel speed of the incident signal is nearly equal to the speed of light, the location z can be expressed by z = ct /2. As shown in Fig.12 Fig.12 , the horizontal axis of the characteristic impedance curve is defined along the kicker center, and the zero point is set at the connector. The measured characteristic impedances of the DUT and REF are shown in Fig. 13 Fig. 13 . In addition to the measurement, the characteristic impedance of the coaxial wire can also be estimated from the structural information using the following equation: <math display="block" id="M12"><mrow><msub><mi>Z</mi><mi>c</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><msqrt><mrow><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow><mrow><msub><mi>ϵ</mi><mn>0</mn></msub></mrow></mfrac></mrow></msqrt><mi>ln</mi><mfrac><mi>b</mi><mi>a</mi></mfrac><mn>,</mn></mrow></math> where b is the outer conductor radius, a is the inner conductor radius, <math id="M13"><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo>=</mo><mn>4</mn><mi>π</mi><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>7</mn></mrow></msup><mtext> T</mtext><mo>⋅</mo><mtext>m/A</mtext></mrow></math> , and <math id="M14"><mrow><msub><mi>ϵ</mi><mn>0</mn></msub><mo>=</mo><mn>8.85</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>12</mn></mrow></msup><mtext> F/m</mtext></mrow></math> . Because the outer dimension of the kicker ports is slightly larger than the inner dimension, as shown in Figs.1(c) and (d) Figs.1(c) and (d) , the TDR curve bulges at the location of the two kicker ports. Meanwhile, because there is a gap between the front port and the current buses, as shown in Fig.14 Fig.14 , a slight asymmetry in the TDR curve is induced. The REF ports are similar to that of the DUT; however, the DUT has a significantly more complicated structure. Therefore, the characteristic impedances of the REF and DUT would be the same at the octagon ports but different at other locations. Benchmark between measurements and simulations 2 Because the cross-section of the kicker port is octagonal, it is impossible to analytically calculate the cutoff frequency. As shown in Figs.1(c) and (d) Figs.1(c) and (d) , the largest and smallest aperture dimensions are 108 mm and 44 mm, respectively. Therefore, we can obtain an approximate range for the cutoff frequency as follows: <math display="block" id="M15"><mrow><msub><mi>f</mi><mrow><mtext>min</mtext></mrow></msub><mo>=</mo><mfrac><mi>c</mi><mrow><mn>2</mn><mi>π</mi><msub><mi>b</mi><mrow><mtext>max</mtext></mrow></msub></mrow></mfrac><mo>=</mo><mn>0.88</mn><mtext> GHz</mtext><mn>,</mn></mrow></math> <math display="block" id="M16"><mrow><msub><mi>f</mi><mrow><mtext>max</mtext></mrow></msub><mo>=</mo><mfrac><mi>c</mi><mrow><mn>2</mn><mi>π</mi><msub><mi>b</mi><mrow><mtext>min</mtext></mrow></msub></mrow></mfrac><mo>=</mo><mn>2.17</mn><mtext> GHz</mtext><mn>,</mn></mrow></math> <math display="block" id="M17"><mrow><msub><mi>f</mi><mrow><mtext>avg</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>f</mi><mrow><mtext>min</mtext></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mtext>max</mtext></mrow></msub></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1.53</mn><mtext> GHz</mtext><mn>.</mn></mrow></math> To obtain an accurate cutoff frequency value, we conducted another simulation using the CST software. As shown in Fig.15(a) Fig.15(a) , one incident wave was set up to pass through a PEC tube that had the same cross-section as the kicker. The frequency of the incident wave changed from 0 GHz to 5 GHz, and the dependence of the S-parameter on the frequency was recorded, as shown in Fig.15(b) Fig.15(b) . The S 21 curve illustrates that the cut-off frequency is approximately 1.61 GHz. As the cutoff frequency of the kicker is approximately 1.61 GHz, the real and imaginary parts of the longitudinal impedance are displayed up to 2 GHz. The results are compared with those obtained from the CST simulations. The comparison in Fig.16 Fig.16 illustrates that the kicker longitudinal impedance mainly exists in the low-frequency range. The narrow-band resonances of the bench measurement and simulation were located at nearly the same frequencies. However, the resonance amplitude of the bench measurement was significantly larger. This may be caused by the multiple reflections owing to the impedance mismatch. No absorber was used in the measurements. Uncertain HOMs are induced owing to the multiple reflections. Their frequencies may be equal to or near the frequencies of the narrow-band resonances. Subsequently, the resonance amplitudes were magnified. To verify this hypothesis, we plan to deposit absorbers in the tapers and select a copper wire with a diameter of 3 mm by referring to Lambertson et al. [ 20 20 ]. Based on the impedance spectrum shown in Fig. 16 Fig. 16 , the loss factor was calculated within different frequency ranges ( σ =10 mm), as shown in Fig. 17 Fig. 17 . Based on the bench measurements, the loss factor up to 2 GHz was 0.01723 V/pC and the longitudinal effective impedance <math id="M18"><mrow><mfrac><mrow><msub><mi>Z</mi><mrow><mo>|</mo><mo>|</mo></mrow></msub></mrow><mi>n</mi></mfrac></mrow></math> up to 2 GHz was 5.45 mΩ ( σ =10 mm). Based on the simulations, the loss factor up to 2 GHz was 0.01721 V/pC, and the longitudinal effective impedance <math id="M19"><mrow><mfrac><mrow><msub><mi>Z</mi><mrow><mo>|</mo><mo>|</mo></mrow></msub></mrow><mi>n</mi></mfrac></mrow></math> up to 2 GHz was 3.59 mΩ ( σ =10 mm). These data demonstrate good consistency between the measurement and simulation results. Furthermore, the loss factor for the ceramic slat kicker was 0.017±0.002 V/pC based on the measurements in a previous study by Demin Zhou [ 5 5 ], which is significantly close to the value indicated above. HEAT DEPOSITION ON DIFFERENT PARTS OF THREE-METAL-STRIP KICKER 1 As a beam passes through the kicker, it loses a certain amount of energy. The energy loss is called the parasitic loss of the beam and can be calculated using the following equation: <math display="block" id="M20"><mrow><mtext>Δ</mtext><msub><mi>W</mi><mrow><mtext>beam</mtext></mrow></msub><mo>=</mo><msup><mi>q</mi><mn>2</mn></msup><msub><mi>k</mi><mo>∥</mo></msub><mn>,</mn></mrow></math> where q is the bunch charge and <math id="M21"><mrow><msub><mi>k</mi><mo>∥</mo></msub></mrow></math> is the longitudinal loss factor [ 21 21 ]. An energy loss occurred only during the passage, which was a relatively short time. The parasitic power loss over one turn of the storage ring can be calculated as follows: <math display="block" id="M22"><mrow><msub><mi>P</mi><mrow><mtext>beam</mtext></mrow></msub><mo>=</mo><msup><mi>q</mi><mn>2</mn></msup><msub><mi>k</mi><mo>∥</mo></msub><msub><mi>f</mi><mtext>r</mtext></msub><mn>,</mn></mrow></math> where f r denotes the revolution frequency. The variable P beam is a certain value determined by the Kicker impedance, which is independent of time. Note that <math id="M23"><mrow><msub><mi>k</mi><mo>∥</mo></msub></mrow></math> is calculated using frequencies up to the kicker cutoff frequency. Certain modes with frequencies higher than the kicker cutoff frequency can travel out of the kicker and eventually deposit their energy on lossy materials elsewhere. Only modes with frequencies lower than the kicker cutoff frequency trapped inside the kicker are considered herein [ 22 22 , 23 23 ]. However, once the beam travels into the kicker, the electromagnetic field begins to continually deposit energy on the kicker. After passage of the beam, the HOM trapped in the kicker continues to deposit energy on the kicker until all the parasitic energy lost by the beam is deposited. Therefore, the heat deposition power P beam is a continuous function of time and can be obtained through simulations. As the total energy is conserved, the following relationship between P beam and P heat exists: <math display="block" id="M24"><mrow><mtext>Δ</mtext><msub><mi>W</mi><mrow><mtext>beam</mtext></mrow></msub><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>T</mi><mtext>r</mtext></msub></mrow></msubsup></mrow></mstyle><msub><mi>P</mi><mrow><mtext>beam</mtext></mrow></msub><mtext>d</mtext><mi>t</mi><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>T</mi><mrow><mtext>end</mtext></mrow></msub></mrow></msubsup></mrow></mstyle><msub><mi>P</mi><mrow><mtext>heat</mtext></mrow></msub><mtext>d</mtext><mi>t</mi><mn>,</mn></mrow></math> where T r is the revolution period and T end is the heat deposition duration. To obtain the heat deposition power on each part of the kicker, a module called the time frequency power loss monitor in CST STUDIO SUITE was used [ 4 4 ]. In the simulation, the bunch charge, bunch RMS length, and wakefield tracking length were 1 nC, 10 mm, and 50 m, respectively. Because the heat deposition power is proportional to the square of the bunch charge, and the bunch charge in the BEPC2 is 7.78 nC (9.8 mA), the final simulation results were multiplied by the square of 7.78, as shown in Figs.18 Figs.18 and 19 19 . The practical heat deposition process prolongs for an extremely long duration, whereas the simulation process only lasts 175 ns owing to the limited time and computing resources. Data fitting was performed to obtain more information regarding the deposition. To improve the data fitting results, all the data of the power plates and 98.29–172.75 ns data of the other parts of the kicker were used. An exponential function was chosen as the fitting function to describe the decay process of the deposition. The heat deposition curves over time, including a part of the fitted data, are shown in Fig. 19 Fig. 19 . To obtain the deposited energy proportion for each part of the kicker, the heat deposition power should be integrated to infinity over time. The energy deposited on each part of the kicker is shown in Table 2 Table 2 , which demonstrates that approximately 84% of all the parasitic energy lost by the beam was deposited on the metal strips. 1 1 Part 1 1 Simu. energy (J) 1 1 Fit energy (J) 1 1 Total energy (J) 1 1 Proportion 1 1 Current buses 1 1 2.7103e-8 1 1 6.2350e-8 1 1 8.9455e-8 1 1 8.41% 1 1 Metal strips 1 1 1.7490e-7 1 1 7.2265e-7 1 1 8.9758e-7 1 1 84.36% 1 1 Power plates 1 1 2.2722e-9 1 1 1.7268e-8 1 1 1.9540e-8 1 1 1.84% 1 1 Back port 1 1 6.4081e-9 1 1 3.8535e-9 1 1 1.0262e-9 1 1 0.96% 1 1 Front port 1 1 6.8252e-9 1 1 9.6337e-9 1 1 1.6459e-8 1 1 1.55% 1 1 Sidewall 1 1 6.6581e-9 1 1 2.4038e-8 1 1 3.0696e-8 1 1 2.89% 1 1 Kicker 1 1 2.2416e-7 1 1 8.3977e-7 1 1 1.0640e-7 1 1 100% The aforementioned discussions are based only on a single bunch passing through the kicker, whereas a storage ring contains multiple bunches. There were 93 bunches in the BEPC2, the bunch current was 9.8 mA, and the revolution time was 791.77 ns. Therefore, the time interval between the two bunches is 791.77 ns divided by 93, that is, 8.51 ns. To obtain the final heat deposition power generated by multiple bunches, the heat deposition power of the bunch should be overlaid with those of the later bunches. The results after approximately 10 turns are shown in Fig. 20 Fig. 20 . Figure 20 Figure 20 illustrates that after the heat deposition power of one bunch is overlaid with others, the final deposition power on each part of the kicker oscillates within a range. The power range rises rapidly and reaches a stable region after approximately 10 turns. The lower limit, upper limit, and average powers of each part of the kicker are listed in Table 3 Table 3 , which demonstrates that among all the parts of the kicker, the metal strips have the largest power and require more consideration. 1 1 Part 1 1 Min. power (W) 1 1 Max. power (W) 1 1 Avg. (W) 1 1 Current buses 1 1 9.897 1 1 10.976 1 1 10.51 1 1 Metal strips 1 1 95.29 1 1 112.82 1 1 105.08 1 1 Power plates 1 1 2.222 1 1 2.366 1 1 2.29 1 1 Back port 1 1 0.57 1 1 13.72 1 1 1.21 1 1 Front port 1 1 1.28 1 1 13.20 1 1 1.93 1 1 Sidewall 1 1 3.582 1 1 3.645 1 1 3.61 1 1 Kicker 1 1 113.3 1 1 131.5 1 1 124.63 The heat deposition data can be further processed by obtaining the average. Considering 8.51 ns as the smallest time unit, the average deposition power can be calculated for each time unit and designated as the instantaneous deposition power in the middle of the time unit. The final averaged heat deposition powers over time for each part of the kicker are shown in Fig. 21 Fig. 21 . The final convergent power averages are presented in Table 3 Table 3 . The parasitic power loss of the beam over one turn can be calculated using <math id="M25"><mrow><msub><mi>P</mi><mrow><mtext>beam</mtext></mrow></msub><mo>=</mo><msub><mi>n</mi><mtext>b</mtext></msub><msup><mi>q</mi><mn>2</mn></msup><msub><mi>k</mi><mo>∥</mo></msub><msub><mi>f</mi><mtext>r</mtext></msub></mrow></math> . The measured longitudinal impedance provides the loss factors 0.01723 V/pC and P beam =122.50 W. The CST simulation yields the loss factors 0.01721 V/pC and P beam =122.35 W, respectively. The direct CST simulation of the heat deposition on the kicker provides the average heat deposition power of 124.63 W, which is consistent with the parasitic power loss of the beam. CONCLUSIONS 1 After comprehensively considering the beam coupling impedance, magnetic field uniformity, heat deposition on the kicker owing to the parasitic loss of the beam, and operational stability, the three-metal-strip kicker is determined to be the most suitable structure for the BEPC2. The impedance and loss factor of the three-metal-strip kicker are nearly the same as those of the ceramic-slat kicker. The impedance simulations and bench measurements of the kicker prototype using the coaxial wire method were in good agreement. The heat deposition power on the kicker obtained from the CST simulations sufficiently matches the beam parasitic loss power calculated from the loss factor. The final heat deposition power on each part of the kicker can be considered as a reference for designing the cooling system. References W. Kang , Y.D. Hao , Q. 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Material preparation, data collection and analysis were performed by Jin-Liu Su, Yu-Dong Liu, Sai-Ke Tian, Lei Wang, Na Wang, and Sen Yue. The first draft of the manuscript was written by Jin-Liu Su and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. We would like to thank the IHEP accelerator team for their support and discussions, particularly, the kicker team for providing an accurate mechanical drawing of the kicker and patient explanation. 10.1007/s41365-023-01212-4 This was supported by the National Natural Science Foundation of China (Nos. Y8113C005C and U1832132). 2022-05-20 2022-12-07 2022-12-18 2023-04-24 2023-06-07 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2023 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-023-01212-4 http://dx.doi.org/10.1007/s41365-023-01212-4 . Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. 2023 38408149 The data that support the findings of this study are openly available in Science Data Bank at https://www.doi.org/10.57760/sciencedb.07738 and http://resolve.pid21.cn/31253.11.sciencedb.07738. Data Availability Statement Nuclear Science and Techniques 4 34