Beam Coupling Impedance

Assuming that a bunched beam with total charge q₁ traveling through a structure with offset parallel to the z -axis with speed , the longitudinal and transverse wake potentials are defined as follows [36] (see Eq. (1), and Eq. (2)).(1)(2)where t=(s+z)/c and s is the distance between the test charge q₂ and the head of the excitation bunch in the direction opposite to the speed , as shown in Fig. 3. The longitudinal and transverse impedance and in the CST software can be calculated from the wake potential and using a Fourier transformation as follows [37] (Eq. (3), and Eq. (4)).(3)(4)where λ(s) is the longitudinal distribution of the bunch q₁.

Fig. 3Full-screen View

(Color online) Device of an arbitrary cross section along an accelerator

Simulation Model and Results

The collimator is installed in a vacuum chamber equipped with ion, molecular, and Ti pumps. A simplified model of the mechanical structure was used to accelerate the simulation process. The bellows and servomotor of the collimator were deleted because they did not significantly affect the beams. The ion pump, molecular pump, and titanium pump were not considered to improve the simulation efficiency because the internal structure of the vacuum pumps was highly intricate. A simplified structure of the collimator installed in a chamber with only three pump ports is shown in Fig. 4.

Fig. 4Full-screen View

(Color online) CST simulation model of movable collimator, please note that D is the distance of collimator block from beam center to the block edge

To study the beam-coupling impedance behavior, we performed a detailed simulation of the collimator, as shown in Fig. 4 using the wakefield method. The wakefield solver of Particle Studio solves Maxwell’s equations in the time domain using a particle bunch to excite the electromagnetic fields. A test charge was used to sample the fields and compute the wake fields and impedances. The wakefields were tracked over 50 m behind the bunch. We also used the Transient Solver of CST Microwave Studio to simulate the wire measurements. The main output of the simulation, which is a physically measurable quantity, is the scattering parameter S_21,DUT. Hence, we used the scattering parameter S₂₁ for both the Device Under Test (DUT) and reference (the collimator block outside the chamber). In the wire simulation, the collimators were perfectly matched at both ends. The longitudinal impedance is calculated from S_21,DUT and S_21,ref using the standard log formula [37] (Eq. (5)):(5)where S_21,ref is the reference defined as the collimator block outside the chamber and is the characteristic impedance of the coaxial line formed by the wire and the DUT wall. is the simulation output; Then, we can calculate and crosscheck the beam-coupling impedance. The main parameters of the CST Particle Studio and CST Microwave Studio simulations are summarized in Table 1.

Simultaneously, all 24 collimator chambers were installed near the quadrupole magnet and connected to racetrack-type quadrupole magnet vacuum chambers, as shown in Fig. 5a. The first mode of the rectangular waveguide is TE₁₀ mode, and the cutoff frequency is as follows (Eq. (6)):(6)where a is the length of the rectangular waveguide cross-section and is approximately 230 mm, c is the speed of light. To obtain an accurate cutoff frequency value, we conducted another simulation using CST Microwave Studio. As shown in Fig. 5a, one incident wave was set up to pass through a PEC tube with the same cross-section as the quadrupole magnet vacuum chamber. The frequency of the incident wave was changed from DC to 1 GHz, and the dependence of the S-parameter on the frequency was calculated, as shown in Fig. 5b. The S₂₁ curve simulated by CST illustrates that the cutoff frequency is 0.6549 GHz. The approximate analytical solution and the simulated results were consistent with each other. Because the cutoff frequency of the quadrupole-magnet vacuum chamber, which is connected to the collimator chamber, is approximately 0.65 GHz, the real and imaginary parts of the longitudinal and transverse impedances are displayed up to 0.65 GHz.

Fig. 5Full-screen View

(Color online) Cut-off frequency simulation using CST microwave studio:(a) Simulation model of quadrupole magnet vacuum chamber and (b) obtained dependence of S₂₁ on the frequency

Figure 6 shows a comparison of the longitudinal beam coupling impedance of CST-Wake-field solver simulations and the wire simulation. Evidently, it is mainly a narrowband impedance, and the longitudinal impedances at the first and second frequencies are approximately 360 MHz and 528 MHz, respectively. The longitudinal beam-coupling impedances of CST-Wake-field solver simulations and numerical measurement simulations were in good agreement. Because all collimators were installed in the horizontal plane, we mainly performed and focused on the horizontal simulation of the transverse impedance. In addition, the CST Particle Studio wakefield solver allowed us to define the positions of the source beam and test particle (as the observation or computation point). Therefore, we can separately calculate the transverse dipolar and quadrupolar terms [38]. Transverse dipolar kicks were obtained by displacing the beam’s transverse location to (x, y) = (10 mm, 0) for a horizontal wake. Figure 7 shows a comparison of the transverse dipolar beam coupling impedance in the horizontal plane of CST-Wake-field solver simulations and the wire simulation. Very good agreement was observed for the broadband behavior, as shown in Fig. 7. Comparable to the longitudinal, the horizontal dipolar impedance at the first and second frequencies was also approximately 360 MHz and 528 MHz, respectively.

Fig. 6Full-screen View

(Color online) Longitudinal impedance using by wake field and single wire simulation results when D = 40 mm: (a) real part of longitudinal impedance, (b) imaginary part of longitudinal impedance

Fig. 7Full-screen View

(Color online) Transverse dipolar impedance in horizontal plane using by wake field and double wire simulation results when D = 40 mm: (a) real part of horizontal impedance, (b) imaginary part of horizontal impedance

Longitudinal impedance measurement

A measurement campaign on one of the movable collimators was launched in 2023 at the IMP. Longitudinal measurements are straightforward. A single wire was inserted into the Device Under Test (DUT) with matching resistors at both ends of the collimator chamber, and the signal transmission (S₂₁) was measured using the VNA, from which the longitudinal impedance was calculated according to Eq. (5): The VNA and connecting cables had a characteristic impedance (Z_ch) of 50 Ω. A TEM line composed of a single wire and a DUT generally has a higher line impedance Z_L. To perform the measurements, a coaxial line composed of the wire and beam pipe must be adapted to the 50 Ω impedance of the VNA cables. This can be achieved using a matching network. As a simple and practical solution, we consider a single resistor Z_m = Z_L – 50 Ω before and after the DUT, as shown in Fig. 8. To match the 50 Ω cables, we set Z_m = Z_L – 50 Ω. For a circular beam pipe and wire, the line impedance is given by the well-known expression (Eq. (7)).(7)where D is the beam pipe diameter and d is the wire diameter. We chose the wire diameter d = 0.62 mm and the diameter of the collimator chamber D = 300 mm. According to Eq. (7), we calculate Z_L = 370 Ω. Finally, the series-matching resistor Z_m = Z_L – 50 Ω=320 Ω was calculated. We chose a radio frequency (RF) carbon resistor for our matching network, which has low inductance, making it ideal for high-frequency applications. We chose A = 6 dB attenuators before and after the DUT for the matching network to reduce the port reflection. The S-parameters were measured using the Rohde and Schwarz (R&S) ZVN-20 2-port VNA, which has a frequency range of 100 kHz to 20 GHz. The transmission S₂₁ was measured at a step size of 100 Hz with 5000 sweep points, and the input power was set to 0 mW. N-type female adaptors were selected for the input and output of the DUT and were connected to the sucobox from Huber+Suhner, which contained the matching resistor. Figure 8 shows the collimator installation and RF measurements for the longitudinal impedance schematic setup.

Fig. 8Full-screen View

(Color online) Longitudinal impedance schematic setup. A single wire is stretched along the device under test (DUT). The impedance matching is ensured by the network (block "Z_m") and the series attenuator (block " A ")

The simulation and measurement results are in good agreement (Fig. 9). In addition, the real and imaginary parts of the longitudinal impedance are extremely small below 300 MHz, which is the most interesting frequency for BRing because the bunch length in the BRing exceeds 10 m, as mentioned before. The longitudinal-impedance measurement results for the movable collimator are shown in Fig. 10 for D = 40 mm, 50 mm, and 60 mm. The measurement results show that the longitudinal impedance is inversely proportional to the distance D, which is defined as the distance from the beam center to the collimator block edge. This indicates that the longitudinal impedance does not affect beam stability in the BRing of HIAF.

Fig. 9Full-screen View

(Color online) Comparison of the longitudinal impedance with the CST wake field, wire simulation and measurement results for movable collimator when D = 40 mm: (a) real part of longitudinal impedance, (b) imaginary part of longitudinal impedance

Fig. 10Full-screen View

(Color online) Longitudinal impedance measurement results for the movable collimator when D = 40 mm, 50 mm and 60 mm, respectively: (a) real part of longitudinal impedance, (b) imaginary part of longitudinal impedance

Transverse impedance measurement

The experience obtained using wire transmission methods for longitudinal impedance measurements has motivated the search for a similar technique for transverse measurements. Hence, direct wire transmission measurements were also performed for transverse impedance. The transverse impedance measurement consists of two wires driven by opposite phases. A dipolar field was excited in the DUT and interacted only with the fringe field. In practice, the phase opposition between the two wires can be obtained by splitting the input signal into an 180° hybrid and recombining it in the same manner to obtain the DUT output signal. The 180° microwave hybrid H-183-4 was obtained from MACOM company, and the entire operating bandwidth ranged from 0.03 GHz to 3 GHz with seven octave frequency range. The 180° microwave hybrid could suppress the unwanted "0" mode very well owing to its excellent isolation, which is greater than 30 dB below 1 GHz. The maximum insertion loss was less than 0.4 dB, and the phase imbalance was better than ±7.5° for the entire operating frequency range. For resistive matching after a 180° -microwave hybrid, each 50 Ω hybrid output must be matched to half of the difference-mode line impedance . If matching is performed with single-series resistors R_m for each wire, the value of each resistor is set as R_m = Z_L/2 - 50 Ω. Figure 11 shows the collimator installation and RF measurements for the transverse impedance schematic setup.

Fig. 11Full-screen View

(Color online) Transverse impedance schematic setup. The two wires in the DUT are driven in phase opposition via two 180° microwave hybrids. Each hybrid output is then matched separately to the two-wire line

To precisely measure the transverse impedance using the direct wire transmission method, one of the key points is to calculate the precise matching resistor. In this study, we utilized CST Microwave Studio to calculate the line impedance of double wires. The diameter of each wire was 0.62 mm, and the center distance between the two wires was 35.75 mm. The CST simulation model is illustrated in Fig. 12. A waveguide port and differential mode excitation were adopted to simulate the line impedance of the double wires, as shown in Fig. 12. The simulation results indicate that the line impedance was 585 Ω, as shown in Fig. 13. Finally, R_m = Z_L/2 - 50 Ω = 242.5 Ω. In our measurements, we soldered an RF carbon-matching resistor between the copper wire and an N-type connector pin with tin, and used sucobox to cover the enamelled copper wire, as shown in Fig. 14.

Fig. 12Full-screen View

(Color online) CST simulation model for the calculation of the line impedance of the double wires. Please note that the wave port is set to be differential mode excitation (pull-push mode)

Fig. 13Full-screen View

Line impedance numerical calculation results of the double wires using CST software

Fig. 14Full-screen View

(Color online) Photograph of sucobox, RF carbon resistor, enamelled copper wire and N type adaptor for the transverse impedance bench measurement

The beam-coupling impedance was measured using a vector network analyzer via the double-wire transmission method and calculated as shown in the longitudinal case. Subsequently, the transverse impedance in the horizontal plane is determined using [38, 39] (Eq. (8)):(8)where c denotes the speed of light, Δ denotes the center distance between the two wires, ω = 2πf. According to Eq. (8), the transverse dipolar impedance is finally calculated.

Figure 15 shows a photograph of the double-wire transmission measurement setup of the movable collimator. Ports 1 and 2 are the VNA ports used for transmission measurements of DUT. Figure 16 compares the transverse dipolar impedance, including the real and imaginary parts, with CST wire simulation and the measurement results for the movable collimator when D = 40 mm. Good agreement is also observed between the simulated and measured results for the horizontal impedance of the movable collimator when D = 40 mm, as shown in Fig. 16. The transverse dipolar impedance measurement results of the movable collimator are shown in Fig. 16 when D = 50 mm and 60 mm, respectively. The measurement results illustrate that the transverse dipolar impedance is inversely proportional to distance D, which is similar to the longitudinal behavior. Evidently, the measured result has four narrow band transverse dipolar impedances including real and imaginary part below 300 MHz approximately at 69 MHz, 99 MHz, 124 MHz and 169 MHz, respectively. Nevertheless, narrowband impedances, including the real and imaginary parts below 300 MHz, are not observed in the CST numerical simulation possibly because the water-cooling pipe, signal wire, and screws on the collimator block are not considered; alternatively, this result was obtained possibly because the feed-through, ion pump, molecular pump, and titanium pump were not considered in the simplified simulation model.

Fig. 15Full-screen View

(Color online) The movable collimator used during the RF horizontal dipolar impedance measurements campaign. Ports 1 and 2 are the VNA ports used for the transmission S₂₁ measurements, please note that the impedance measured with the collimator mounted in the beam line, and the bellows at both ends were removed for the campaign

Fig. 16Full-screen View

(Color online) Comparison of transverse dipolar impedance with CST wire simulation and measurement results for the movable collimator, when D = 40 mm: (a) real part of horizontal impedance, (b) imaginary part of horizontal impedance. Note that the operating frequency bandwidth of the 180° hybrid is from 0.03 GHz to 3 GHz

The total real and imaginary parts of the horizontal dipolar impedance for the 24 movable collimators are approximately 120 kΩ m^-1 and 43 kΩ m^-1 at 124 MHz, respectively, which are significantly lower than the threshold impedance for the transverse mode coupling instability. The transverse beam-coupling impedance may stimulate transverse mode-coupling instability when it exceeds a specific threshold. DELPHI (discrete expansion over Laguerre polynomials and Headtail modes for instabilities) was used to calculate the beam stability for a typical ⁷⁸Kr¹⁹⁺ heavy-ion beam of the booster ring. In the DELPHI simulation, all structures, including the movable collimator for BRing, were included in the weighting of their local betatron functions. The simulation results showed that wideband impedances and other structures, including collimators, are not expected to have a significant impact on the beam stability in the BRing of HIAF [33, 40].