Magnet field error contributed by current error

The relationship between the magnetic field and the excitation current can be clearly expressed using the transfer function, which is the ratio of the integral magnetic field strength to the excitation current ($BL/I$) [27, 28]. The corresponding curves are shown in Fig. 6. The transfer function provides a visual representation of the effect of magnetic hysteresis on the working magnetic field of the magnet [29-32]. The upward and downward paths of the excitation current follow different curves, and the flat region of the curve during ideal magnet operation can be clearly identified. Beyond this range, the magnet becomes saturated as the excitation current continues to increase, resulting in a decrease in the transfer function.

Fig. 6Full-screen View

Transfer function of a dipole magnet

Considering that the peak work excitation current of the dipole magnet in RCS can reach approximately 2100 A, as shown in Fig. 7, $B/I-I$ reaches the edge region of the curve under practical operating conditions, causing a change in the excitation relationship between high and low fields. Taking these factors into account, the relationship between $\frac{\text{Δ}I}{I}$ and $\frac{\text{Δ}B}{B}$ can only be linearly correlated rather than strictly linear. The relative magnetic field error is a key parameter for simulating deviations in the horizontal orbit. To further analyze the impact of the current error on the magnetic field under actual operating conditions, it is necessary to utilize offline magnetic measurement curves and employ fitting methods to quantitatively determine the relationship between the relative current error and the relative magnetic field error.

Fig. 7Full-screen View

Dipole pole magnet excitation current curve

First, the current error sources must be determined. As shown in Fig. 8, this paper leverages old control strategies and actual feedback data to identify the 50 Hz current as the main source of disturbances. This observation considers the fact that both DC and 25 Hz currents are regulated through closed-loop control, the trend of the 50 Hz error variation is consistent with the trend of the outdoor temperature, and the amplitudes of harmonics at 75 Hz and higher frequencies are relatively small. Therefore, it is assumed that the 50 Hz error is the sole source, excluding the computational errors caused by harmonic superposition.

Fig. 8Full-screen View

(Color online) Output current data of DC and the harmonics at 25Hz fundamental frequency from BPS (ΔT = 13.3 ℃)

Under the parameter conditions of a 140 kW beam output power, the reference output current from Fig. 7 and the real current error from the original data, which was found to be abnormal, as shown in Fig. 8(c), were used to calculate the relative errors of the current and magnetic field for the RCS dipole magnets (for convenience of calculation and comparison, the error was approximately equal to 280 mA). Using the fitted equation obtained from the actual offline B-I magnetic field measurement data, Fig. 9 presents the relative error curves [33, 34]. The calculations show that the 280 mA error introduced by the 50 Hz harmonic current has a maximum relative current error of $\frac{\text{Δ}I}{I}=8.04\times {10}^{-4}$ at the lowest current point (the denominator equals 348.21A), while the maximum relative magnetic field error is $\frac{\text{Δ}B}{B}=7.22\times {10}^{-4}$.

Fig. 9Full-screen View

Magnetic Field Uniformity of RCS dipole magnets (Proton beam accelerates only while the rising edge of the current)

Calculation on orbital deviation by current error

In the RCS, dipole magnets are used to deflect the proton beam and complete its closed orbit. Magnetic field errors in the dipole magnets manifest as errors in the beam deflection angle. A thin-lens model with an integral field strength of $\text{Δ}Bl$ was used as the perturbation model of the dipole magnet on the beam orbit, as shown in Eq. (1). In this model, the dipole magnet provides an angular kick (angular error) as particles pass through it [35]. $\theta =\frac{\text{Δ}Bl}{B\rho }$ (1) ΔB represents the uniform variation in the magnetic field over the length l. This is the equivalent introduction of the relative magnetic field errors.

Assuming that there is a disturbance $\left(\begin{array}{c}{u}_0\\ u_0{}^{\text{'}}\end{array}\right)$ at the exit of the error source, where u₀ is the position and $u_0{}^{\text{'}}$ is the angular. The corresponding closed orbit disturbance at the entrance of the error source is $\left(\begin{array}{c}{u}_0\\ u_0{}^{\text{'}}-\theta \end{array}\right)$, then using the single-turn transmission matrix $M\left(s_0+L|s_0\right)$ to solve the following Eq. (2): $\left(\begin{array}{c}{u}_0\\ u_0{}^{\text{'}}-\theta \end{array}\right)=M\left(s_0+L|s_0\right)\cdot \left(\begin{array}{c}{u}_0\\ u_0{}^{\text{'}}\end{array}\right)$ (2) It should be noted that on the investigated hypothetical condition assumes a transient single-point error source. The resulting closed orbit exhibits a sharp point at the error location, where the beam trajectory experiences an angular kick of θ. If errors occur throughout the entire ring and all 24 dipole magnets in the RCS are excited by the BPS alone, the presence of a 280 mA error will cause changes in the beam angle θ, leading to beam trajectory displacement. This can destabilize the accelerator operating point and, in severe cases, cause beam loss for RCS, especially given the limitations of the correction magnets and the sensitivity of the accelerator’s operating point to perturbations.

To quantify the impact of the 280 mA current error on the beam trajectory, this study employed MadX simulation software, developed by CERN for charged particle optics design and research on alternating gradient accelerators and beamlines [36].

Based on the lattice structural parameters of the RCS, the maximum relative current error data in Fig. 9 was used to introduce an equivalent error source for all the dipole magnets in the ring [37]. Assuming the 50 Hz current error is the sole contributor to magnetic field uniformity fluctuations, the simulated beam trajectory results, shown in Fig. 10, indicate that the maximum horizontal offset of the beam in the four arc sections of the ring due to the 280 mA current error is 2.81 mm.

Fig. 10Full-screen View

The results of the orbit offset simulation due to dipole magnet errors in CSNSRCS

This simulation quantitatively verifies the orbit deviation effect caused by the resonant power supply in the RCS. It demonstrates that the performance of the BPS significantly influences the horizontal orbit stability.

Load characteristics of resonant power supply

The BPS topology includes a three-phase rectifier bridge, BOOST circuit, inverter circuit, and output filtering circuit, as shown in Fig. 11. It consists of 10 power units connected in series with the load. The load of the BPS comprises 24 dipole magnet loads distributed throughout the RCS ring and 12 matching resonant units connected in series, forming the white-circuit structure.

Fig. 11Full-screen View

The topology structure of the main magnet power supplies in RCS

The series-connected white circuit structure is shown in Fig. 2. L_m represents the inductance of the magnet; L_ch is the inductance of the resonant choke; C_m and C_ch are the resonant capacitors; and R_m and R_ch are the equivalent resistances of the magnet and choke, respectively. The main components include a magnet, resonant capacitor, and resonant choke, which together form resonant units [37]. The number of resonant units varies with the power supply. The corresponding component parameters of each unit were essentially identical, ensuring that their resonant frequencies were consistently 25 Hz, as shown in Eq. 3. In fact, a load consisting of a matched resonant network exhibits nonlinear dynamic inductive characteristics under temperature variations and magnetic saturations [10, 38]. $\frac{1}{2\pi \sqrt{L_{\text{m}}{C}_{\text{m}}}}=\frac{1}{2\pi \sqrt{L_{\text{ch}}{C}_{\text{ch}}}}=25\text{ Hz}$ (3) In this structure, the power supply is directly connected in series with the magnetic branch of the resonant network, providing an AC current with a DC bias. It periodically replenishes the losses in each resonant unit and, in conjunction with the control system, synchronizes tracking and excitation across various types of magnets. Due to the resonant characteristics of the load [26], this circuit favors currents closely aligned with the resonant frequency ω₀ and effectively suppresses input currents at non-free oscillation frequencies. Under identical-intensity pulse current input conditions, the steady-state magnet current intensity exhibited variations exceeding a factor of 10. In engineering, BW denotes the range of frequencies through which a signal can pass effectively, as shown in Eq. 4. $BW={\omega }_2-{\omega }_1=\frac{{\omega }_0}{Q}$ (4) Frequencies ω₁ and ω₂ represent the boundary frequencies of the bandwidth, corresponding to a maximum value of 70.7% on the resonance curve. The quality factor $Q=\frac{{\omega }_{0}L}{R}$ indicates the circuit’s selective responsiveness. For a given natural oscillation frequency, circuits with higher Q values have narrower bandwidths.

A frequency-response analysis was conducted on the actual resonant load of QPS03 (with characteristics similar to those of other loads), as shown in Fig. 12. This analysis demonstrates a strong ability to suppress higher-order harmonics of the response current, which significantly influences the design of subsequent harmonic wave compensation controls. However, it is important to note that while this load effectively suppresses frequencies outside the resonance point, the harmonic disturbances caused by current distortion in the actual load output are already attenuated, and the remaining interference is considerable.

Fig. 12Full-screen View

Bode scheme of resonant load of QPS03

Control characteristics with magnets saturation situation

Ideally, in the absence of magnet saturation, the magnetic field gradient should follow the variations of a 25 Hz sinusoidal current gradient. However, in practice, magnet saturation causes the load inductance to vary nonlinearly, as shown in Fig. 13. This nonlinearity leads to changes in the resonant circuit’s load characteristics and distortion of the output current waveform. To ensure the magnetic field gradient meets the required specifications and eliminate the higher-frequency components of the magnetic field, the power supply system must adopt appropriately modified current reference curves based on the actual nonlinear B-I relationship, as shown in Fig. 6. This process involves superimposing additional anti harmonic components on the resonant power supplies’ output and correcting the reference current [39, 40]. The modified current reference should include a 50 Hz component with an amplitude of 4.59 A and a phase shift of 10.3°, along with higher-order harmonics determined through offline magnetic field measurements and FFT (Fast Fourier Transform) calculations. For resonant power supplies, precise control of the frequency, amplitude, and phase of the output current is critical to achieving optimal tracking of the magnetic current across multiple resonant networks.

Fig. 13Full-screen View

The inductance measurement of a dipole magnet

Design of the high-order harmonic current control

This design emphasizes the accuracy and stability of the output current in RCS resonant magnet power supplies. More precise control of the 50 Hz harmonic current’s amplitude and phase is essential. Based on given the control issues and characteristics described above, further optimization of algorithms and code structure is necessary to achieve precise control of the 50 Hz harmonic current while operating within the constraints of limited chip resources. As shown in Fig. 14, an optimized control scheme based on the DPSCM controller is proposed. This scheme incorporates PI control for the amplitude and phase of the 50 Hz harmonic into the multi-harmonic reference loop. The resulting output is then used as the reference input for rear synthetic current-loop control after the Ref-Waves combination. This configuration establishes a sophisticated dual closed-loop control system.

Fig. 14Full-screen View

The control algorithm block diagram for the CSNSRCS main magnet power supplies

Considering the nonlinearity of the load inductance, disturbances from the power grid, and the characteristics of the resonant circuit, background distortion currents are generated at multiple frequencies. Under actual operating conditions, an approximately 6.7 A 50 Hz harmonic current (referred to as the background current) with a phase shift, as well as higher-order harmonic components (not detailed in this paper), are observed. The background current needs to be controlled to a 50 Hz reference value of 4.59 A. Although the fluctuation of the load current is corrected through the closed-loop feedback of the regulation circuit, closed-loop control inherently introduces a response delay, resulting in a lagging reaction. PI closed-loop control alone cannot achieve the desired reference value for the 50 Hz harmonic current, which prevents the attainment of the expected magnetic field strength. To address this limitation, this study incorporates extra harmonic compensation through an open-loop adjustment to counteract the influence of the background current.

The design principle for the current feedforward involves adjusting the modulation coefficient (duty cycle) m to compensate for the effect of the background current on the output. Assuming I_fb and I_out are the current feedback and output, respectively, and $I_{\text{fb,0}}$ is the constant average component of the feedback current, the following incremental expression can be derived by considering the introduction of error interference: $I_{\text{fb}}=I_{\text{fb,0}}+\text{Δ}{I}_{\text{fb}}$ (5) $m=m_0+\text{Δ}m$ (6) The output current is: $I_{\text{out}}=m{I}_{\text{fb}}$ (7) $I_{\text{out}}=m_0\cdot I_{\text{fb,0}}+m_0\cdot \text{Δ}{I}_{\text{fb}}+\text{Δ}m\cdot I_{\text{fb}}+\text{Δ}m\cdot \text{Δ}{I}_{\text{fb}}$ (8) Because the output current change caused by the feedback current is zero and ignoring the higher-order terms Δm·Δ_fb, it follows that $\text{Δ}{I}_{\text{out}}=m_0\cdot \text{Δ}{I}_{\text{fb}}+\text{Δ}m\cdot I_{\text{fb}}=0$ (9) Obtaining the change in m through the transformation: $\text{Δ}m=-\frac{m_0}{I_{\text{fb,0}}}\cdot \text{Δ}{I}_{\text{fb}}$ (10) This design is illustrated in Fig. 15. Referring to Eq. 10, the modulation coefficient adjustment requires obtaining the AC component of the harmonic output current, ΔI_fb. A digital high-pass filter is used to calculate the peak-to-peak values of each harmonic’s parameter, ΔI_fb, while a low-pass filter calculates $I_{\text{fb,0}}$, the average component of the feedback current.

Fig. 15Full-screen View

Block diagram of current feedforward control

To meet the requirements for high-precision output current and ensure future scalability and flexibility, FFT is employed to determine the amplitude and phase of each harmonic component up to the fifth-order harmonic. This method can provide current feedforward for each harmonic component.

The overall control scheme includes a 50 Hz harmonic current compensation section and a PI closed-loop section for the 50 Hz harmonic current within the multi-harmonic reference loop. As illustrated in Fig. 16, this approach generates antiphase harmonic currents and subtracts background currents.

Fig. 16Full-screen View

The algorithm design for the harmonic injection module

To implement harmonic compensation, the amplitude and phase information of the harmonic background current induced by the magnet are pre-obtained using the controller’s ADC board and FFT algorithm module. The open-loop response is then calculated. The Cordic algorithm is incorporated to accelerate the iterative computation of trigonometric functions with harmonic compensation phase parameters. A new "Deduct Background" submodule was developed to execute wave superposition processing with phase reversal. This module provides the reference input for the rear synthetic current loop.

In the controller, a time-multiplexing and pipeline scheme is employed for floating-point multiplication, addition modules, and the Cordic algorithm, among other IP cores, to minimize resource utilization, such as the controller’s BRAM and CLB.

Current error measurement and comparison

In this study, online experiments were conducted in the RCS to visually compare the improvements achieved by the new control scheme in the performance of the resonant power supplies and the horizontal orbit deviation of the proton beam. Data were extracted from the original accelerator operation database, known as the CS-Studio archive.

The relationship between the actual 50 Hz harmonic current output and outdoor temperature variations(the temperature sensor is far from the resonant units) was taken as a comparison in this experiment. To demonstrate the effectiveness of the proposed control scheme, two sets of data showing significant temperature fluctuations in 2022 and 2023 are presented. These datasets correspond to a beam power of 140 kW and employ various control strategies.

As shown in Fig. 17 and 18, the actual output current is significantly affected by the outdoor temperature. After implementing the proposed new control scheme, the experimental results show a significant reduction in both the amplitude and phase errors of the 50 Hz harmonic current, while effectively suppressing the current temperature drift. Within a temperature variation range of 13 ℃, the error of the 50 Hz harmonic current amplitude is reduced to 132 mA, and the phase error is reduced to 1.68.

Fig. 17Full-screen View

(Color online) The BPS output current with temperature under old control condition

Fig. 18Full-screen View

(Color online) The BPS output current with temperature under the new control condition

Improvement in horizontal orbital shifts of the beam

To show the main contribution of 50 Hz output current error to the beam trajectory horizontal displacement, another set of past representative operational data of 100 kW is listed, which has a large temperature difference, as shown in Fig. 19. The data were detected by a Beam Position Measurement (R4BPM05), which reflects the negative displacement in the horizontal direction caused by temperature variations. Under open-loop control with the main disturbing contribution, a 274 mA error in the 50 Hz harmonic current, the beam trajectory experienced a horizontal displacement change of approximately 4.78 mm. To ensure the generalizability of the experimental results, it was necessary to consider the injection, extraction, and high-frequency acceleration stages in the four arc segments. As shown in Fig. 20, the average horizontal trajectory displacement data from six representative Beam Position Measurements (BPMs) scattered throughout the RCS were collected during the first millisecond of each acceleration cycle for one minute. Also to ensure the consistency of the experiments as much as possible, the experimental data presented in the follow all come from after the RCS reached 140 kW, and the temperature difference, both before and after implementing the new control, exceeds 10 ℃. With the implementation of the new control scheme, the disturbance in the beam trajectory displacement is significantly reduced, as shown in Fig. 21(a). The data from these BPMs are summarized in Table. 1. Except for R1BPM01, which was located in the initial straight section and experienced anomalous data owing to other factors during the experiment, the results from the other BPMs indicated that the new control scheme designed in this study effectively suppressed the temperature drift in the horizontal trajectory of the RCS.

Fig. 19Full-screen View

(Color online) Impact on the 50 Hz current and horizontal orbit after the large temperature disturbance in 100 kW. (a) 50 Hz current parameter versus temperature; (b) horizontal orbit versus temperature

Fig. 20Full-screen View

Diagram of BPMs Location

Fig. 21Full-screen View

Comparison of beam trajectory horizontal offset under different control strategies (The data is extracted from the operational database of 2022 and 2023. The offset anomalies in the graph are due to machine faults occurring during accelerator operation. These anomalies can be ignored for this study.) (a) new control; (b) old control

Verification of the simulation model

In this subsection’s experiment, the new control scheme reduces the influence of relevant interference factors to a low level, which is a prerequisite for verifying the simulation results.

By manually superimposing an amplitude of 280 mA onto the 50 Hz reference value in the BPS, the quantitative impact of changes in the 50 Hz current on the beam’s horizontal orbital deviation can be intuitively observed. In the second section, it was assumed that the 50 Hz component was the sole contributor to orbital disturbance, and the simulation process adhered to this assumption. To verify the accuracy of the model in this experiment, the assumption was temporarily maintained. During the experiment, other frequency parameters of the reference waveform remained unchanged, with adjustments made only to the 50 Hz reference. Under operating conditions of 140 kW, the average horizontal trajectory displacement was calculated. Using experimental data from the RCS, where the 50 Hz current served as the independent variable and the horizontal orbit as the dependent variable, a linear fit was performed. This equation was then used to calculate orbit variations resulting from current errors and compared with the simulation results.

Measurements were taken at the 50 Hz reference harmonic current setpoint of 4.59 A, along with two additional experimental current points of 4.31 A and 4.87 A (4.59 A ± 280 mA). The results are presented in Table. 2, where x represents the 50 Hz harmonic current, and y represents the horizontal trajectory displacement in the RCS. A current error ($\text{Δ}x$) of 280 mA was substituted into the fitting equation to calculate the relative trajectory displacement ($\text{Δ}y$) using the expression $\text{Δ}y=k\cdot \text{Δ}x$.

Based on the “Relative Error” column in the table, which represents the relative error between the simulated horizontal trajectory displacement results from the CSNS Lattice model and the actual horizontal trajectory displacement in the RCS, some differences were observed. These differences are attributed to performance variations in the BPMs and other internal factors within the accelerator. However, the remaining results indicate that the model aligns well with experimental expectations, confirming that the 50 Hz harmonic is the primary contributor to interference. This model provides a reliable reference for follow-up studies.