Feedback-controlled flux modulation for high-temperature superconducting magnets in persistent current mode

High-temperature superconducting (HTS) magnets have found wide applications in high-field settings owing to their high current capabilities. Typically, these magnets are powered by high-current power supplies via current leads, which can complicate insulation between cryogenic and room temperature environments. However, new developments in flux pumps for HTS magnets have enabled charging of kA levels of current without power supplies. By combining flux pumps with HTS persistent current operation, it is possible to achieve accurate flux modulation and eliminate the need for power supplies and current leads. In this study, we report on a novel feedback-controlled flux modulation for HTS magnets in persistent current operations. This flux modulation is based on a flux pump mechanism that generates a DC voltage across the charging superconductor by applying a current higher than its critical current. With closed-loop feedback control, our flux modulation can achieve precise injection and reduction of HTS magnet current in increments of 0.5 A. This technology can lead to stable magnetic fields in HTS magnet designs. We anticipate that this work will enable future magnets to operate in a stable persistent current mode within a closed cryogenic chamber, significantly reducing the footprint and power demand of HTS magnets and opening up new opportunities for their applications.


Introduction
Superconducting technology has evolved over time, resulting in a wide range of industrial applications and commercial products. For instance, magnetic resonance imaging and * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. nuclear magnetic resonance (NMR) machines use superconducting magnets to generate the necessary magnetic fields [1][2][3][4]. These machines are essential for modern medical diagnosis [3,5]. Advancements in the manufacturing of hightemperature superconducting (HTS) coated conductors (CCs) [6][7][8] have made it possible to produce long, robust HTS-CCs with bending tolerance suitable for winding superconducting coils. As a result, HTS coils have become a competitive option for high magnetic field applications [3,4].
HTS-CC magnets are unmatched in their ability to generate strong, uniform, and ultra-stable magnetic fields. These advantageous properties result from confining supercurrents to high inductance coils with extremely low operational resistance. Studies have shown almost negligible attenuation, with the magnetic field produced sustained to better than 1 ppbh −1 in some cases, and extremely large decay times [9]. These so-called persistent mode-driven electromagnets transform into the world's highest field quasi-permanent magnets. Persistent current operation in an HTS coil becomes possible by including an HTS persistent joint in the circuit. Ohki et al presented an intermediate grown superconducting (iGS) joint between rare-earth barium copper oxide (REBCO) CCs and achieved a joint resistance of < 5 × 10 −13 − 3 × 10 −12 Ω [10]. It has also been observed that in the iGS joints between REBCO CCs, the critical current degrades to 45% of the CC critical current at 77 K in a self-field with a criteria of 1 µV. Recently, Yanagisawa et al used an iGS superconducting joint between CCs in their persistent-mode NMR magnet, achieving temporal stability of ∼1 ppm [11]. This allowed for highresolution NMR spectra to be obtained using a dc source and thermal switches (heaters) for charging the coil. In addition to a superconducting joint, Lee et al presented a wind-and-flip method to achieve a jointless HTS coil and demonstrated persistent current operation with zero field cooling [12].
A natural question that arises here is how HTS coils are energized. There are two possible options for energizing an HTS coil: direct current injection and indirect current induction. In direct current injection, the coils are connected to a power supply and energized through current leads [13]. This approach can be problematic, especially for high-current applications, because the current leads needed to transmit very high currents are extremely bulky, such as in the W7-X project, where 17.6 kA is required [14,15]. More importantly, the current leads create a path between the cryostat (cold) and the room temperature environment, imposing a heavy thermal load on the cooling systems [16][17][18] and resulting in significant additional capital and operating costs [19]. An alternative solution for indirect current induction is to incorporate a flux pump, which can constantly drive magnetic flux into a closed superconducting circuit without physical connections [20][21][22][23][24][25][26][27][28][29][30][31][32]. The flux pumping effect was originally discovered for low-temperature superconductors [33][34][35][36][37][38][39][40][41][42]. It was later discovered that eliminating superconductivity is not necessary for HTSs, leading to the new era of HTS flux pumps. Various developments in HTS flux pumps have been reported recently. These flux pumps include traveling magnetic field-induced flux pumps (dynamo, linear flux pumps) [22,24,[43][44][45] and HTS rectifier flux pumps (thermally switching, dynamic resistance switching, JcB switching, and selfswitching) [26][27][28][46][47][48]. HTS flux pumps can charge the magnet and compensate for any current decay, enabling the quasi-persistent operation of HTS-CC magnets. The contact voltage drop of soldering joints is considerably less than the voltage determined from the E-J relationship and is therefore considered negligible. The resistance of soldering joints still does not diminish to zero, so the current flowing through HTS magnets will still decay over time. In [49], a ring-shaped HTS magnet with a trapped field of 4.6 T after field cooling magnetization at 25 K is reported. This magnet can be operated in persistent current mode, but it will be difficult to compensate for the current decay when exposed to an external AC magnetic field without continuous charging using a flux pump. To date, it is still challenging to keep an HTS magnet working in a persistent current mode without soldering joints, and it requires continuous current injection with a flux pump.
This paper proposes a novel closed-loop feedback control for charging HTS magnets in persistent current operations. Closed-loop feedback control is widely used in many engineering applications, including motor drives [50], the motion of atoms in optical cavities [51], and stabilizing Rabi oscillation in superconducting qubits [52]. In an HTS CC, flux flow resistivity appears when there is flux motion. This phenomenon occurs when a superconductor is subjected to an AC field, transporting an AC current or a DC current above its critical current value. In [43], the authors proposed a self-switched HTS flux pump that works by driving the HTS-CC into the flux flow region on the E-J curve by injecting current above its critical current value. Building on this initial work, this paper presents a closed-loop feedback control system to induce a flux flow voltage and achieve bidirectional current control for an HTS coil in persistent current operation. The feedback control allows for effective flux modulation by accurately injecting and removing current in an HTS magnet, enabling precise field control of the magnet.

Closed-loop feedback control
Closed-loop control systems are desirable in systems that are more sensitive to disturbances because of their ability to modify the natural dynamics of a system and stabilize it, which is a key feature of any closed-loop system. In contrast, open loop systems, such as previously reported flux pumps, do not have control over the current state of the system. In the closed-loop control system described here, the set point for the required magnetic field is assigned, and the error between the magnet current and the set point is obtained. Based on the magnitude of the error, a control block calculates the required voltage and corresponding current needed to charge the magnet. This current value is then fed to a limit block that is in place to protect the HTS-CC from quenching; in this case, the limit is set to be 140% of the critical current of the HTS bridge. The signal generation block generates an analogue signal based on the current value, which is fed into an amplifier. The amplifier is connected to the primary of the transformer, inducing the charging current on the secondary coil. When the charging current exceeds the critical current of the bridge, a voltage appears across the bridge, causing flux pumping into the HTS magnet.
The block diagram in figure 1 illustrates the proposed closed-loop feedback control HTS flux pump. The control algorithm, implemented using LabVIEW, estimates the required voltage for injecting flux into the HTS magnet and generates an input signal for the primary of the transformer to achieve flux pumping. In self-switching HTS flux pumps, the flux injection is achieved by driving the superconductor into the mixed state, but the non-linear V-I characteristics at this state can present a difficulty in accurately estimating and controlling the flux injected into the HTS magnet. Our results show that this difficulty can be overcome through the use of the proposed control system.

Voltage prediction in high temperature CCs
To predict the voltage across an HTS CC in the mixed state, the structure of the conductor must be taken into account. Second generation (2G) HTS conductors are composed of multiple layers, including a HTS layer, a substrate (Hastelloy) layer, a buffer stack layer, two silver overlays, and two electroplated copper layers [53]. The resistivity of the metal layers is wellknown, but the resistivity of the HTS layer can be characterized using the E-J Power Law, where, ρ hts is resistivity of HTS layer, V c is voltage criterion 10 −4 V, I c is critical current of HTS tape, I T is the transport current, and n is the exponent of power law. As the transport current exceeds the critical current value of the superconducting layer (Yttrium barium copper oxide (YBCO)), it enters the mixed state (flux flow regime) and redistributes itself to other metal layers [54]. The net voltage due to flux flow resistivity can be calculated using the following formula: where R ff is flux flow resistance, ρ hts is resistivity of HTS layer, S hts is cross-sectional area of HTS layer, ρ layer is resistivity of metal layers, S layer is cross-sectional area of metal layers and l is the length of the HTS tape. In equation (2) the resistivity of the metal layers is known [55,56] except the YBCO layer.
Assuming the current density and electric field to be uniform across the length (l) of HTS tape, the electric potential and current density of an HTS tape can be derived as; where V(t) is measured voltage across the HTS tape. The current in each layer can be obtained by integrating the current density over the cross section of each layer. If I total is the total measured current, the current in HTS layer I hts can be obtained as; Knowing the cross section area of the HTS layer, the resistivity of HTS layer ρ hts can be obtained as; The flux flow resistance value can be accurately calculated by substituting equation (5) into equation (2). To verify the validity of this system of equations, a sample of HTS tape was prepared and the critical current values were measured at 77 K under self-field conditions using a 1 µV criterion. The resulting value was 145 A. The sample was then subjected to a finite element simulation, in which a current pulse with an amplitude of 1.4% of the critical current was applied and the voltage across the layers of the tape was measured. These results were compared to those obtained from a laboratory experiment, as shown in figure 2. When the transport current exceeds the critical current value, it is distributed among the other metal layers, and the induced voltages across the sample are proportional to the resistivity of these layers. Higher currents flowing through the metal layers result in higher voltages, which can facilitate faster charging of the HTS magnet using a self-rectifying flux pump. However, it is important to note that excessive currents can cause quenching of the HTS tape, resulting in heat generation and potentially causing permanent damage to the HTS-CCs. The equations described in this paragraph allow the control block in figure 1 to estimate the required voltage for injecting flux into the HTS magnet.

Equivalent circuit model
To establish a relationship between the primary current (input) and the voltage across the bridge, a simplified magnetic circuit model was developed. When the transport current is below its critical value, the secondary coil is short-circuited by the bridge and the flux flow resistance does not appear. Thus, the load side of the circuit can be neglected under this condition. By applying Kirchhoff's voltage law, the following expressions can be obtained: where L p is the inductance of the primary coil, L s is the inductance of the secondary coil, M is mutual inductance, R 1 is ohmic losses in the primary winding and R s is the total resistance due to soldering joint in the secondary side. The mutual inductance of the transformer can be regarded as; The flux ϕ c through the transformer core can be calculated by solving the magnetomotive force due to the applied current over the reluctance of the transformer core. Substituting equation (8) into equations (6) and (7) yields an expression for the induced current i s with respect to the applied current i p . The voltage across the bridge is caused by the flux flow resistivity. By understanding the relationship between the applied and induced currents, we can represent the flux pump in a simplified circuit diagram by replacing the excitation circuit with a current-controlled voltage source, as shown in figure 3(b).
The following equation represents the relationship between the current flowing through the load magnet and the voltage across the bridge caused by flux flow resistivity:

Applied current
An asymmetric signal is applied to the primary of the transformer as input. The positive peak has a larger amplitude in order to drive the superconductor into the flux flow regime, while the negative peak has a much lower amplitude in order to maintain the superconducting state. The ratio of the positive period to the negative period is kept inversely proportional to the peak values in order to nullify the dc component. During the positive cycle, the current ramps up to the positive peak I p at a constant rate and then down to zero at the same rate. Similarly, during the negative cycle, the current ramps down to the negative peak I n and then up to zero at the same rate. I p is the amplitude of the positive peak, I n is the amplitude of the negative peak, i p is the primary current, and T is the period. These equations are used by the signal generation block in figure 1 to estimate the required input current for injecting flux into the HTS magnet,ˆT

Experiment
The experimental setup, shown in figure 4, consists of a transformer and a jointless HTS magnet. The HTS magnet is made of 12 mm-wide, 6 m-long YBCO 2G HTS tape with an average critical current of 495 A. It was cut longitudinally at its center and wound into a double pancake structure, with each pancake having a width of 6 mm and a former diameter of 70 mm. One pancake was flipped to align the orientation of the field in the same direction. One side of the jointless HTS coil was extended to form the 'bridge'. The secondary side of the 300:1 transformer was made of HTS tape with a critical current of 700 A, and the terminals were soldered to either side of the bridge, creating the return path for the induced current. The critical current of the bridge is ⩽248 A. Two pre-calibrated current transducers were used to measure the induced voltage on the secondary side and the current in the magnet and provide a feedback signal. A twisted lead voltage tap across the bridge was used to record the voltage. The feedback control was implemented in LabVIEW, with voltage and current readings measured by an NI-9238 DAQ card and the current signal generated by an NI-9263 DAQ card. The power amplifier AE-Techron 7766 was used to generate the primary current for the transformer.

Critical current of a jointless HTS magnet
The preparation of the jointless HTS magnet involves cutting the HTS tape down the center using mechanical roller blades, which can potentially cause degradation of the critical current due to the mechanical stress applied on the tape [57]. Measuring the critical current of a jointless coil using the four-probe method can be challenging [58], but one approach is to saturate the coil to the point where the phenomenon of flux creep occurs [59]. In HTS coils, there should be no DC losses in theory, but when the coil approaches the critical state, flux flow and flux creep-related losses may occur. The current plot of the jointless magnet used in this study is shown in figure 5, demonstrating that the magnet saturates at 110 A and decays to 85 A due to flux creep after the charging is switched off.

Field modulation in HTS magnet using closed-loop control
The magnetic field of the jointless HTS magnet is modulated by injecting and removing current. The current plot of the HTS magnet is shown in figure 6. The set point is initially set to 80 A. The current in the magnet is zero, so the error is at its maximum. The required voltage for flux pumping is calculated based on the bridge voltage and a primary current signal is generated and sent to the amplifier. This process is repeated continuously until the set point is reached. Once the set point is reached, the required voltage is reduced to zero and the primary signal becomes zero, resulting in the HTS magnet operating in self-field persistent current mode (shown in figure 6  The relation between the charging current (secondary) i s , the bridge current i b and the load current ∆i can be given as;  With the increase in the load current, the DC current starts to run through the bridge and the bridge current now has both AC and DC components, due to this a DC biasing in the bridge voltage plot can be seen in figure 7. The charging current plot is shown in figure 8(a). During the first cycle, the charging current is equal to the bridge current since the load current (∆i) is zero. As the load current accumulates, the bridge current becomes biased in the opposite direction. The magnitude of the charging current is automatically adjusted to keep the peak of the bridge current constant in each cycle, as illustrated in figure 8(b). When the load current reaches its set point limit, the input signal returns to zero and the corresponding charging current is also reduced to zero. However, due to the jointless geometry of the HTS magnet, we observe a persistent current flowing in the bridge. Figures 8(c) and (d) show the charging and discharging cycle of the HTS magnet and its corresponding bridge and charging current in detail. The magnitude of the peaks in the bridge current is nearly constant, indicating the accurate performance of the voltage calculation block in the feedback control.
Closed-loop feedback control is widely used in a variety of applications due to its ability to enable real-time system modification. In this study, we demonstrate the ability to precisely modulate the current of the jointless HTS magnet by 0.5 A, as shown in figure 9(a). However, due to the relatively low inductance of the HTS magnet (∼0.5 µH), small changes in load current may not be accurately quantified due to significant ripple noise. Improved precision can be achieved for magnets with higher inductance values. Figure 9(b) illustrates the induced voltage across the bridge required to inject or remove current from the magnet, which is a step toward developing an ultra-stable DC source for HTS magnets.

Discussion and conclusion
Previous flux pumps have been effective at rapidly increasing currents in large magnets, but they may not be ideal for maintaining field stability. The sharp V-I curve of the bridge superconductor makes it difficult to control the bridge voltage effectively, as small variations in the bridge current can cause significant errors in the bridge voltage. This problem has been addressed in previous studies by adding an additional field source perpendicular to the bridge superconductor [24,25]. However, this solution increases the heat load on the cryogenic system and enlarges its footprint. Moreover, these flux pumps operate using an open loop control. In contrast, the proposed closed-loop control system can provide stable DC voltages on demand, enabling flux pumping at stable DC voltages and achieving persistent current operation in HTS magnets. In theory, a 55 mV voltage can instantly establish an 80 A current in the HTS magnet, but achieving this high voltage in the HTS bridge is only possible in the flux flow regime. In the flux flow regime, the current is shared by other metal layers, which can produce heat and potentially quench the superconductor. Therefore, the superconductor is briefly driven into the flux flow regime to inject current into the magnet. During each cycle, the feedback loop calculates the required voltage and continues injecting current until the set point is reached. The closed-loop feedback control can limit excessive current into the HTS bridge, protecting the system from quenching or overcurrent damage.
We are presenting a closed-loop feedback control system for modulating the magnetic field of HTS magnets in persistent current mode. The experimental results demonstrate that this proposed control technique can accurately induce a DC voltage across the HTS bridge, enabling effective flux modulation. In contrast to traditional flux pumps, which do not control the DC voltages across the bridge and rely on the frequency and amplitude of the applied signal for pumping speed, the precision of our flux modulation is demonstrated by the ability to remove a specific current of 0.5 A from the magnet on demand. The control block calculates the required voltage of 7 mV to generate a corresponding current signal that drives the bridge into the flux flow regime and achieves the desired voltage needed to inject 0.5 A. This feedback-controlled flux pump can be used for precise flux modulation of HTS magnets in persistent current operation, addressing the technical challenges associated with using high-current power supplies and current leads and reducing the cooling requirements and footprint of HTS magnets.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.