RF cavity

The MA-loaded cavity has a length of 1.8 m. It contains six water tanks, each of which consists of three MA cores of size Φ 850 mm × 316 mm × 25 mm, forming three acceleration gaps. All the gaps are connected in parallel through the busbars. A schematic of the cavity is shown at the top of Fig. 1; the materials, manufacturing process, and performance tests of the cavity are described in detail in [18, 19]. The cavity is partitioned into upstream and downstream sections. The gap voltage is the vector sum voltage of the upstream and downstream sections, which represents the voltage difference between them. This serves as the accelerating voltage experienced by the beam as it passes through the acceleration gap.

Fig. 1Full-screen View

(Color online) Schematic of the MA-loaded cavity

The design of the Q-value and resonant frequency of the RF cavity follows the method outlined in [20]. This design approach aims to minimize the amplifier current, resulting in a Q-value of approximately 1.3 and resonant frequency of 2.1 MHz. Fig. 2 shows the cavity impedance, which clearly demonstrates the wideband characteristics of the cavity. The maximum cavity impedance is approximately 200 Ω.

Fig. 2Full-screen View

(Color online) Measured impedance of the MA-loaded cavity

RF amplifier

The RF signal-amplifier chain consists of two solid-state amplifiers (SSA) and a final-stage amplifier equipped with two TH558 tetrode tubes. The low-level RF (LLRF) control system generates an RF-driven signal that is directed to the SSA. The SSA output drives the control grid of the tetrode tubes. The tubes adopt a grounded-cathode structure and are powered by an anode power supply. Each tube is connected to the gaps using busbars. The two tubes form a push-pull configuration. A simplified schematic of the RF power system is shown at the bottom of Fig. 1.

The tubes are operated in Class AB with a conduction angle greater than 180°. Consequently, the tube current is cut off during a portion of each cycle. Because of this distortion and the wideband property of the cavity, the cavity voltage driven by the tube current exhibits several harmonics. This will be discussed in the next section. A multiharmonic feedback control is employed in LLRF to address these higher harmonics.

Independent RF voltage control

Numerous reports have confirmed that one of the greatest advantages of MA-loaded cavities is their high-gradient capability, which is effectively demonstrated through the push-pull operating mode [21, 22]. However, as discussed in previous sections, their wideband characteristics can lead to the excitation of a multitude of higher-order harmonic voltages, thereby making the suppression of unwanted harmonics a primary task for the LLRF control system. Traditional control algorithms that utilize signals synthesized from the vector sum of the voltages across the gap can effectively suppress the beam-induced harmonics and odd harmonics generated by the tube [17]. However, they fall short of addressing the even-order harmonics produced by the tube. These unsuppressed harmonics can lead to increased cavity-power dissipation, thereby limiting the further enhancement of the gradient of the MA-loaded cavity. The following provides a detailed explanation for this phenomenon.

According to [23], the anode currents in the two tubes in a push-pull configuration have the following cosine pulse forms: $\begin{array}{l}{y}^A=\{\begin{array}{l}A(\cos \varphi -\cos \alpha )\mathrm{,}-\alpha <\varphi <\alpha \hfill \\ \mathrm{0,}-\pi <\varphi <-\alpha \mathrm{,}\alpha <\varphi <\pi \hfill \end{array}\\ y^B=\{\begin{array}{l}A(\cos (\varphi +\pi )-\cos \alpha )\mathrm{,}-\alpha -\pi <\varphi <\alpha -\pi \hfill \\ \mathrm{0,}-2\pi <\varphi <-\alpha -\pi \mathrm{,}\alpha -\pi <\varphi <0.\hfill \end{array}\end{array}$ (1) Here, A is the amplitude and α is the conduction angle. We can perform a Fourier-series expansion of Eq. (1) to calculate the harmonic components. The equations for the harmonic components are as follows: $\begin{array}{l}{I}_n^A=\frac{1}{\pi }{\displaystyle {\int }_{-\alpha }^{\alpha }}A(\cos \varphi -\cos \alpha )\cos (n\varphi )\text{d}\varphi \\ Q_n^A=\frac{1}{\pi }{\displaystyle {\int }_{-\alpha }^{\alpha }}A(\cos \varphi -\cos \alpha )\sin (n\varphi )\text{d}\varphi \end{array}$ (2) $\begin{array}{l}{I}_n^B=\frac{1}{\pi }{\displaystyle {\int }_{-\alpha }^{\alpha }}A(\cos \theta -\cos \alpha )\cos (n(\theta -\pi ))\text{d}\theta \\ Q_n^B=\frac{1}{\pi }{\displaystyle {\int }_{-\alpha }^{\alpha }}A(\cos \theta -\cos \alpha )\sin (n(\theta -\pi ))\text{d}\theta \mathrm{,}\end{array}$ (3) where In and Qn are the cosine and sine coefficients of the n^th harmonic, respectively. By comparing Eq. (2) and Eq. (3), we observe that $I_n^A=\{\begin{array}{l}-I_n^B\mathrm{,}n=\text{odd}\hfill \\ I_n^B\mathrm{,}n=\text{even}\hfill \end{array}{Q}_n^A=\{\begin{array}{l}-Q_n^B\mathrm{,}n=\text{odd}\hfill \\ Q_n^B\mathrm{,}n=\text{even}.\hfill \end{array}$ (4) This suggests that the odd-order harmonics of the anode current between the two tubes are antiphase, whereas the even-order harmonics are in phase. Consequently, owing to the push-pull configuration, the even harmonics in both the upstream and downstream signals are inherently cancelled out in the vector-sum signal.

Figure 3 shows the measured voltage waveforms of the upstream and downstream sections, and their vector sum. The driving signal is a pure sinusoidal wave. The voltages in the upstream and downstream sections are distorted because of the presence of several harmonics. According to the FFT results, the fundamental, second and third harmonics in both sections are approximately 9.8 kV, 2.2 kV, and 0.25 kV, respectively. The other harmonics are negligible. In contrast, the vector-sum voltage exhibits few harmonic components. The fundamental, second, and third harmonics are 19.6 kV, 0.211 kV, and 0.408 kV, respectively.

Fig. 3Full-screen View

(Color online) Cavity-voltage waveform of upstream (blue), downstream (orange), and the vector-sum voltage (green)

Traditional vector-control algorithms face challenges in suppressing even-order harmonics. The existence of even harmonics in both the upstream and downstream sections does not contribute to beam acceleration because the beam only experiences the vector-sum voltage. Instead, their practical existence leads to an increase in power dissipation within the cavity, causing the temperature of the magnetic cores to further increase. This is particularly problematic when the cavity operates at a high duty cycle, which can hinder further improvements in the cavity gradient owing to the limitations imposed by the maximum temperature of the MA cores [19, 24].

Based on this, we propose an algorithm for independent control that refers to the separate regulation of voltage in the upstream and downstream gaps. In this approach, the upstream and downstream RF voltages are detected by sampling capacitors and then input into the feedback module for individual regulation. The output of the feedback module drives two amplifiers that supply currents to the upstream and downstream sections. The independent control method can effectively suppress the higher harmonics in both the upstream and downstream sections, leading to reduced power dissipation in the cavity. In addition, this enables the attainment of higher field gradients.

In the following section, we evaluate the effects of these two control algorithms on the power dissipation of the cavity using the data presented in Fig. 2 and 3.

The fundamental frequencies shown in Fig. 3 is approximately 2.75 MHz. In Fig. 2, the impedances at the fundamental and second-harmonic frequencies are approximately 175 Ω and 60 Ω, respectively. The power dissipated by the second harmonic accounts for approximately 14.6% of the total power. Compared with vector control, independent control can compensate for the second harmonic in both the upstream and downstream sections; therefore, an increase of approximately 1.6 kV in the fundamental harmonic of each gap (4.8 kV for a cavity because of three gaps) can be achieved with the assumption of the same total power dissipation for both control algorithms. This approach is meaningful for endeavors aimed at achieving high-gradient operations.

Therefore, independent voltage control is implemented to minimize the cavity power and achieve a higher gradient.

consideration for A/ϕ and I/Q control

In terms of voltage regulation, an RF signal can be represented by its amplitude and phase (A/ϕ) or by its in-phase and quadrature-phase (I/Q) components; thus, both the A/ϕ and I/Q control methods can be employed. Although both methods have proven effective, their advantages and disadvantages must be carefully examined to achieve optimal performance [25].

The stability and reliability of the $A/\varphi$ control in the LLRF system for the CSNS/RCS has been demonstrated [26]. However, compensating for the heavy beam loading in the MA-loaded cavity requires setting the setpoint to zero, resulting in a very small-amplitude signal output by the $A/\varphi$ control. This causes an unstable phase loop, rendering it unsuitable for MA-loaded cavity-voltage regulation. By contrast, the I/Q loop delivers superior performance in low-amplitude scenarios because it can be driven negatively [27]. Therefore, an I/Q feedback-control structure is under consideration.

Loop-phase calibration for I/Q control

In the context of I/Q control, loop-phase calibration is required because a substantial loop phase can introduce significant crosstalk between the I/Q channels, potentially leading to instability in I/Q control. Throughout this study, the loop phase refers to the phase response of the feedback loop from the I/Q modulator to the demodulator. The primary contributors to the loop phase are the loop delay and frequency response of the cavity and analog devices. The impact of the loop phase on the feedback is crucial for I/Q control.

To illuminate the impact of the loop phase on feedback, [28] provides an analysis of a scenario in which a loop phase of 180° results in feedback instability, as depicted in Fig. 4. Owing to the large loop phase, after feedback correction, the cavity field is modified but moves in the opposite direction to the set point.

Fig. 4Full-screen View

(Color online) Example of feedback instability of the I/Q control with a loop phase of 180°

While analyzing the I/Q control for a more general case with an arbitrary loop phase, we observed that instability may also occur. This can be explained through the following analysis.

The current cavity field, denoted as yn, can be expressed as follows: $\left[\begin{array}{c}{y}_{In}\\ y_{Qn}\end{array}\right]=R\left[\begin{array}{c}{u}_{In}\\ u_{Qn}\end{array}\right]=R\left[\begin{array}{c}{u}_{I\mathrm{,}n-1}\\ u_{Q\mathrm{,}n-1}\end{array}\right]+G\left[\begin{array}{c}{e}_{I\mathrm{,}n-1}\\ e_{Q\mathrm{,}n-1}\end{array}\right]\mathrm{.}$ (5) Here, G represents the loop gain and R is the rotation matrix that depends on the loop phase. The variables u and e represent the control signal and error, respectively. R is expressed as follows: $R=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\mathrm{.}$ (6) We can further rewrite (5) as follows: $\begin{array}{ll}\left[\begin{array}{c}{y}_{In}\\ y_{Qn}\end{array}\right]\hfill & =\left[\begin{array}{c}{y}_{I\mathrm{,}n-1}\\ y_{Q\mathrm{,}n-1}\end{array}\right]+RG\left[\begin{array}{c}se{t}_I-y_{I\mathrm{,}n-1}\\ se{t}_Q-y_{Q\mathrm{,}n-1}\end{array}\right]\hfill \\ \hfill & =RG\left[\begin{array}{c}se{t}_I\\ se{t}_Q\end{array}\right]+(I-RG)\left[\begin{array}{c}{y}_{I\mathrm{,}n-1}\\ y_{Q\mathrm{,}n-1}\end{array}\right]\mathrm{.}\hfill \end{array}$ (7) In general, the cavity field converges if (8) is satisfied. $\left|EIG(I-RG)\right|<1$ (8) Therefore, the cavity field can be regulated to the set point, as indicated by (9). $\left[\begin{array}{c}{y}_{In}\\ y_{Qn}\end{array}\right]={\left(RG\right)}^{-1}\cdot RG\left[\begin{array}{c}se{t}_I\\ se{t}_Q\end{array}\right]=\left[\begin{array}{c}se{t}_I\\ se{t}_Q\end{array}\right]$ (9) The maximum allowable loop phase in the feedback loop is determined by the loop gain. Higher loop gains necessitate smaller loop phases to preserve stability. Consequently, calibrating the loop phase to a minimum value is advantageous, guaranteeing feedback stability and enhancing the feedback gain. This, in turn, contributes to a reduction in set-point tracking errors.

The loop phase is measured and calibrated using an open loop, as shown in Fig. 5. ϕ_A represents the initial phase of the RF signal prepared by the modulator, and ϕ_B in the demodulator includes the phase response of the feedback loop. The loop phase (denoted by ϕ_offset) is obtained as ϕ_B-ϕ_A. After calibration, a new RF signal is generated as follows: $rf=I\cdot \sin \left(2\pi f_{n}t+{\varphi }_{\text{A}}-{\varphi }_{\text{offset}}\right)\mathrm{.}$ (10)

Fig. 5Full-screen View

(Color online) Loop-phase measurement method

If ϕ_offset is measured accurately, the result of the loop phase is zero.

Separate calibrations were performed for each harmonic. Calibration of the loop phase was performed up to (h=8); however, only the loop phases for (h=2) and (h=4) are shown in Fig. 6 for clarity. ϕ_offset varies continuously with frequency. As indicated in the lower part of Fig. 6, a linear relationship exists between ϕ_offset and the frequency, indicating that the primary cause of the loop phase is delay.

Fig. 6Full-screen View

(Color online) (a) Loop phase and frequency of (h=2) versus time. (b) Loop phase and frequency of (h=4) versus time. (c) Loop phase versus frequency (h=2). (d) Loop phase versus frequency (h=4)

The ϕ_offset value for each harmonic is stored in the memory of a field-programmable gate array (FPGA) as a phase pattern. After calibration, the ϕ_offset values for all harmonics are close to zero, as shown in Fig. 7. At this stage, the feedback loops can be closed owing to successful calibration.

Fig. 7Full-screen View

(Color online) All values of ϕ_offset for harmonics after calibration

Multiharmonic feedback control

Figure 8 shows a simplified block diagram of the multiharmonic feedback-control system. To enable feedback control and monitoring, sampling capacitors are strategically placed in the upstream and downstream sections. The pickup signal is then converted into a digital signal using a high-speed analog-to-digital converter operating at a clock frequency of 120 MHz. The digital signal is subsequently directed to the feedback block of each harmonic.

Fig. 8Full-screen View

(Color online) Block diagram of the multiharmonic feedback control

A direct digital synthesizer (DDS) is used to generate sine and cosine signals of unity amplitude with a highly accurate frequency. The frequency signal (h=n) fed to the DDS is obtained by multiplying the revolution frequency (h=1) by the selected harmonic number. For demodulation, the phase signal (h=n) fed into the DDS is zero, whereas for modulation, it is derived from the phase pattern. The phase pattern stores the phase offset (ϕ_offset) obtained from the loop-phase calibration in Sect. 3.3. The feedforward (FF) module is used for open-loop measurements.

The low-pass filter, which facilitates the acquisition of the I/Q values of the selected harmonic from the mixed signals, plays a critical role in the feedback control. The filter must have sufficient bandwidth to enable a fast feedback response while also being narrow enough to reject other harmonics. Additionally, the delay and FPGA-logic requirements should be considered. In our implementation, we utilized a second-order infinite impulse-response low-pass filter to circumvent the significant delay associated with high-order finite impulse-response filters. The filter has a bandwidth of approximately 60 kHz and provides rejection at the 80 dB level at frequencies above 500 kHz.

The I/Q errors are continuously updated by comparing the I/Q values with the designated I/Q set points and are subsequently minimized by the feedback controller. Although various new feedback controllers have been reported, such as the linear quadratic regulator [29, 30] and model predictive control [31], we chose to use a proportional and integral (PI) controller. This decision was based on the challenges associated with obtaining the model of an RF system operating in sweeping mode, which is crucial for controllers relying on the system model, as well as the advantages of simplicity, robustness, and easy parameter tuning associated with the PI controller. The PI values were carefully selected to achieve satisfactory tracking performance in the presence of disturbances, while simultaneously maintaining low levels of high-frequency noise.

In addition, to provide flexibility, the harmonic feedback loop includes a switch that allows the closing or opening of the feedback loop.

The RF-driven signal is formed by summing all feedback output signals, which are then prepared by the I/Q modulator. Finally, the driven signal is converted back to an analog signal using a digital-to-analog converter and transmitted to the amplifier chain.

Beam-off experiments

Prior to the official implementation of the multiharmonic feedback control, an assessment was conducted to reduce the voltage-waveform distortion caused by the nonlinearity of the RF amplifiers.

The selected driven harmonic is (h=2), with the programmed voltage and frequency patterns matching those used during operation. Fig. 9(a) presents the harmonic components of the upstream cavity voltage when only the feedback for (h=2) is closed. The voltage amplitude for (h=2) is effectively regulated at 5 kV. The primary observed harmonic component is the 2^nd harmonic, which reaches a maximum amplitude of approximately 2.2 kV. Additionally, a small amount of the 3^rd and 4^th harmonic components exist, both measuring less than 0.3 kV. Owing to the tube operation (class AB1) and cavity frequency response, the other higher-order harmonics exhibit minimal components.

Fig. 9Full-screen View

(Color online) Harmonic components of upstream voltage in the case of (a) the feedback closed for (h=2) only and (b) the feedback loops closed for (h=2, 4, 6, 8)

Further closure of the feedback loops for (h=4, 6, 8), with (I_set, Q_set) set to (0, 0) as the set points, effectively suppresses the higher harmonics, as shown in Fig. 9(b). The fundamental harmonic remains the same, whereas the 2^nd harmonic is reduced from 2.2 kV to 0.15 kV, and the other harmonics are below 0.1 kV. After compensating for the higher harmonics, the cavity power is reduced from 40 kW to 36 kW.

The experimental results confirmed the functionality of the multiharmonic I/Q feedback control. The even harmonics in both streams were successfully reduced by independent control, resulting in a significant decrease in cavity-power dissipation.

Beam-loading compensation

With the beam commissioning for dual-harmonic acceleration, the MA-loaded cavity was successfully operated, providing both fundamental and second-harmonic voltages. The following experimental results further validate the effectiveness of multiharmonic feedback control under the beam-loading effect.

Initially, the beam loading in the MA-loaded cavity is measured at a beam intensity of 1.95×10¹³ protons per pulse (ppp) for two bunches, corresponding to a beam power of 125 kW. The wake voltages are shown in the upper graphs in Fig. 10(a). During this phase, no feedback control is applied to any of the harmonics. The MA-loaded cavity does not supply RF voltage, and the accelerating voltage is provided by the ferrite-loaded cavities. The injection-pulse width is 514 μs.

Fig. 10Full-screen View

(Color online) Harmonic components of the wake voltage of 125 kW beam at both ends of one gap without feedback (a). Harmonic components of the voltage at both ends of one gap with the acceleration of a 125 kW beam with feedback for (h=2, 4, 6) (b). Harmonic components of the voltage at both ends of one gap with the acceleration of a 140 kW beam with feedback for (h=2, 4) (c). Harmonic components of the voltage at both ends of one gap with the acceleration of a 160 kW beam with feedback for (h=2, 4) (d). The left and right plots show the RF voltages at the upstream and downstream ends of one gap, respectively

Only even harmonics are observed in the gap voltage because two bunches are accelerated. The maximum amplitudes of the 1^st (h=2), 2^nd, and 3^rd harmonics of the wake voltage in one gap between the injection and extraction are approximately 8 kV, 2 kV, and 1 kV, respectively. The slight discrepancies in wake voltages between the upstream and downstream sections can be attributed to minor manufacturing differences between the MA cores housed within the tanks.

Next, the LLRF system generates an RF signal for the MA-loaded cavity. In this scenario, the RF signal is composed of the driving signal for beam acceleration and the compensation signal for beam-loading compensation. The performance of the MA-loaded cavity supplying a maximum (h=4) second harmonic voltage of approximately 66 kV (11 kV for each end) under 125 kW beam loading is tested with feedback for (h=2, 4, 6). The beam test results are shown in Fig. 10(b). With feedback, the second-harmonic voltage is well regulated according to the voltage program from the beginning to 2.5 ms. After 2.5 ms, the second-harmonic voltage is effectively suppressed below 0.3 kV. The (h=2) beam-loading is effectively mitigated, decreasing from 4 kV to 0.4 kV, and the (h=6) harmonic is consistently suppressed below 0.4 kV throughout the cycle.

A beam power of 140 kW is achieved [32] with successful beam commissioning. The voltage program for the 140 kW beam acceleration and harmonic components of (h=2, 4, 6) in the upstream and downstream sections are shown in Fig. 10(c). Given the small size above the (h=6) harmonic during beam acceleration [33], feedback is applied only for (h=2, 4) in the beam commission. The (h=4) harmonic remains well regulated at 11 kV according to the voltage pattern. In addition to the (h=4) harmonic voltage, the MA-loaded cavity also features an (h=2) harmonic-voltage program from 2 ms to 20 ms with a maximum of 18 kV (3 kV for each end), aimed at reducing beam loss in the arc region and alleviating beam instability. The (h=2) harmonic is maintained well below 0.35 kV during the initial 2 ms to 20 ms. However, some oscillations occur in the (h=2) voltage range from 3 ms to 8 ms, which are likely caused by the longitudinal oscillation of the beam and are currently under investigation. However, this does not result in a significant beam-intensity loss.

After the installation of an additional MA-loaded cavity, the beam power has recently been increased to 160 kW. Each MA-loaded cavity provides both (h=2) and (h=4) harmonic voltages with a voltage program that differs from that used for the 140 kW beam acceleration. Each cavity delivers a maximum (h=4) harmonic voltage of 50 kV from the beginning to 4 ms and provides a (h=2) harmonic voltage of 25 kV from 4 ms to 20 ms, as shown in Fig. 10(d). The performance of the (h=4) harmonic-voltage regulation has not deteriorated despite the heavier beam loading, and is well regulated throughout the entire acceleration cycle, with a maximum deviation of 0.5 kV at approximately 6.5 ms. However, the oscillations in the (h=2) voltage have increased, with the amplitude reaching 1 kVpp. These oscillations also align with changes in the voltage program. They do not significantly impact the beam acceleration at the current stage; however, we plan to implement an adaptive feedforward control based on the approach described in [34] to enhance the tracking performance.

The disparity in the (h=6) harmonics between the upstream and downstream sections can also be observed in Fig. 10(c) and (d). This discrepancy occurs because the harmonics stem not only from the wake voltage, but also from the distorted tube-output current, as the two tube output-current waveforms exhibit asymmetry under beam-loading effects [35]. This characteristic is inherent to multiharmonic operations. Despite the different tube operating conditions, the feedback demonstrates effective functionality in all cases.