Flux pumps are superconducting DC current supplies [1], [2], [3], [4] that can energize superconducting magnets to large currents from within the cryogenic environment itself [5]. Flux pumps using 2nd-generation (2G) HTS are rapidly progressing in development [3], [6], [7], [8], [9], [10], [11], [12], but have yet to reach the high currents (>10 kA) of earlier low-temperature superconducting (LTS) flux pumps [13], [14]. By using flux pumps, high-field magnet systems may benefit from forgoing resistive power supplies at room-temperature and the associated current leads [15]. This can lead to system-level efficiency improvements [16], [17] that are attractive in such areas as magnetically-confined fusion energy [18], [19], high-torque direct-drive motor/generators [20], [21], [22], [23], and high-field analytic magnets [24].

Of note is the transformer-rectifier flux pump topology [3]. These devices use transformer action to introduce large AC currents in a superconducting circuit which is then rectified into a DC current output in a superconducting magnet. This magnetically couples the superconducting secondary circuit to the resistive primary. Switches rectify the AC secondary current into a DC voltage that is then applied to the load magnet. The DC voltage causes the current in the load magnet to increase, thus charging it.

The circuit can be arranged in a half-wave or full-wave circuit topology [2]. The ideal full-wave circuit diagram is shown in Fig. 1. While both circuits have been studied widely in LTS devices, recent studies on HTS devices have focused heavily on the half-wave topology, with two notable exceptions: Oomen et al. demonstrated an IGBT-switched flux pump using an HTS full-wave topology [25], and Ishmael et al. proposed a full-wave device using superconducting switches [26] but no report on a physical device was ever published. The full-wave circuit is advantageous due to a lower current ripple in the output and lower loss, leading to higher power efficiency [11], [17], [25].

Here, we demonstrate a full-wave, centre-tapped transformer-rectifier using only HTS conductor. Rectification is produced using a mixture of dynamic resistance [27] and non-linear resistivity [11] switching by applying magnetic field to the superconductor. Using applied magnetic field instead of semiconductor- or thermal-switches allows for the entire circuit to remain fully- or partially-superconducting at all times during operation. The finite ‘closed’-state resistance of a solid-state IGBT switch is eliminated, lowering the internal resistance of the flux pump to only that of solder joints. This lowers the thermal loss of the flux pump during operation.

Dynamic resistance relates the electrical loss of a superconductor under time-variant magnetic fields to an effective circuit resistance, $R_{\mathrm{dyn}, \perp}$ [27], [28].

with the width of the wire, 2a, the magnitude of applied perpendicular magnetic field, $B_{\mathrm{app}, \perp}$, the length of the applied magnetic field region, l, the frequency of the applied magnetic field, $f_a$, and the zero-field critical current, $I_{c0}$. The dynamic resistance is linearly proportional to the transport current and applied magnetic field. This formulation uses the Kim model [29] which defines the constant $B_0$ as

resulting in a threshold applied field, $B_{\mathrm{th}, \perp}$, which must be exceeded before dynamic resistance occurs.

Superconductors carrying close to their critical current, $I_c$, exhibit non-linear resistivity which can be used to generate useful voltage [11].

where $n\left(B_{\mathrm{app}, \perp}\right)$ is an experimental constant that characterizes the transition out of the superconducting state, l is the length of the switch, and $E_0$ is the critical current criteria, 1.0 μV/cm. Both $I_c$ and n can be readily reduced by the application of perpendicular, DC magnetic field, $B_{\mathrm{app}, \perp}$. Due to the non-linear relationship, relatively large voltages can be generated at relatively small currents.

Both linear ($B_{\mathrm{app}, \perp}$) and non-linear (E-J) forms of switching will be utilized in the present device. It will be seen in the experimental results that a negative DC offset in secondary current develops, limiting the voltage and current output of the device A feedback control system is proposed to account for this and increase load current output. This feedback control system is then demonstrated in both experiment and simulation to increase load current output.

Fig. 2 shows the experimental setup constructed for this study. A resistive primary (Ø 0.5 mm copper, $N_1$ = 400, $L_p$ = 45 mH) is connected to a superconducting secondary by a transformer. The superconducting circuit uses wire manufactured by SuNAM Ltd. Co. (SCN12700), with a nominal self-field critical current of 798 A (n-value ≈ 45) at 77 K. A centre-tapped secondary ($N_{21}=N_{22}=2$) is connected to a pair of superconducting switches and the load magnet. The loose-wound load coil (N = 20, i.d. = 25 mm, 12 mm SCN12700 wire) has an $I_c$ of 315 A at 77 K (n-value ≈ 30). Its self-inductance is 300 μH up to 30 A and decreases to 50 μH above 90 A due to the saturation of the ferrite core. Applied magnetic field is provided to the HTS switches by iron-core electromagnets with a saturated air–gap field of 0.7 T. Switch elements are soldered using 143 ^°C (Ag3/In97) face-to-face to provide an anti-parallel configuration in the air–gap to avoid unwanted suppression of self-field $I_c$ [30]. Current is provided to the primary and switch electromagnets using a trio of Takasago BWS40-15 bipolar power supplies controlled using LabView software. Analog voltage signals are handled by National Instruments compact DAq cards NI-9205 and NI-9264 for voltage measurements and controls, respectively. The type of 1 Hz waveform used is seen in Fig. 3a), and further details on it are found in Supplementary Materials. Currents are measured using open-loop current sensors of the type described in [11]. Cooling was provided by submerging the entire experiment in liquid nitrogen (77 K).

The results of this experiment are compared to an effective-circuit model formulated using the Simulink package in MatLab. This model is an adaption of the model by Mallett et al. [12]. Here, the most important features are described for convenience, as well as any changes and adaptations. The detailed magnetic and electrical circuit is shown in Fig. 4.

The primary circuit is characterized by primary current, $I_P$, series resistance, $R_P$, and primary inductance, $L_P$. The resistive primary is coupled magnetically to the superconducting circuit by a non-linear reluctance that represents the iron-core transformer. The primary current waveform is matched such that it produces the same secondary current as in the experiment. From the definition of magnetomotive force (MMF) as $ \mathscr{F}=\mathscr{R} \phi$, the non-linear reluctance $ \mathscr{R}_{\text {core }}(H)$ was modelled as

with B-H curve defined by

where $ \mu(H)$ is the permeability relationship of the iron, $ B_{\text {sat }}$ is the saturation magnetic field, and $ H_{\text {sat }}$ is the induction required to produce a saturated iron-core. The magnetic circuit is characterized by $ \mathscr{R}_{\text {core }}(H)$ and leakage flux paths, $ \mathscr{R}_{\mathrm{P}}, \mathscr{R}_{1}$ and $ \mathscr{R}_{2}$ for the primary, secondary 1 and secondary 2 windings, respectively. This produced almost perfect coupling provided the iron-core did not saturate.

The secondary rectifier circuit contains a pair of identical secondary windings, $L_s$, carrying currents $I_1$ and $I_2$, respectively. These are each connected in series to the switches, which are modelled as variable resistors $ R_{s \mathrm{sw} 1}\left(I_{1}\right)$ and $ R_{\text {sw } 2}\left(I_{2}\right). The applied field $B_{app,⊥}$ is simulated using a separate (not depicted) magnet circuit to emulate the air–gap field of the iron-core electromagnets.

$ R_{\mathrm{sw} 1}\left(I_{1}\right)$ and $R_{\text {sw } 2}\left(I_{2}\right)$ are implemented as non-linear resistances from Eq. (3) by the use of two-dimensional (2-D) look-up tables that reference both transport current $I_t$ and $B_{app,⊥}$ to resolve the resistivity of both switches independently. The look-up tables utilise experimentally derived critical current and n-values taken from the SuperCurrent Facility at Paihau-Robinson Research Institute [31].
Linear resistance from Eq. (1) was implemented by a similar look-up table method. This considered a discrete differential of $B_{app,⊥}$ to determine when a predetermined dynamic resistance should be applied. In this specific case, $B_{app,⊥}$ only changes when $I_t$ is at significant fractions of $I_c$, thus $ B_{\text {th }, \perp} \approx 0$. This instantaneous approximation of $ R_{\mathrm{dyn}}$ is then

When $ B_{\mathrm{app}, \perp}>B_{\mathrm{th}, \perp},|d B / d t|>0$ and $ I_{\mathrm{t}}<I_{\mathrm{c}}\left(B_{\mathrm{app}, \perp}\right)$; and 0 otherwise. In the trapezoidal waveform used, only 2% of the cycle (0.02 s) is used ramp from zero to peak $I_p$, which represents 1/4 of an ac cycle. To match this, $ f_{a}=4 \times f_{o}$ / 2% = 200 Hz is used, and $B_{app,⊥}$ is taken to be 0.7 T.

As seen from the comparison of switch voltages in Fig. 3d), this model closely replicates both linear and non-linear switching mechanisms.

The secondary loops are bridged in the centre by the load coil of inductance, $L_L$, carrying load current, $I_L$. Solder joints add a series resistance of $R_1,R_2$ and RL to the both secondary arms and the load coil, respectively.

The present model differs from previous implementations in its use of the full-wave circuit topology. Additionally, the effect of dynamic resistance was not modelled in previous versions. Later, the feedback control system was included to scale the primary waveform. Thermal effects are not modelled. A full list of parameters used in the model are found in the Appendix A.

The first cycle of the flux pump experiment is detailed in Fig. 3. A primary current waveform, a), is transformed into the secondary, b), with switching applied from the electromagnets when the current is positive, c). This produces two forms of voltage generation seen in d) and e); namely two spikes that bookend the switch actuation mostly caused by dynamic resistance, and a smaller, steady voltage caused by non-linear E-J loss. Half of the switch voltage is then applied to the load, seen in f); the polarity of which is positive in both halves of the cycle. This is seen in both experiment and simulation. The instantaneous circuit voltages are compared to the simulated values in Fig. 3d-f).

Fig. 5 shows the load current trace of the experiment and corresponding circuit simulation. Both are operating at a cycle-frequency of 1 Hz. The experiment charges to a maximum load current of 35 A in about 100 s. The current ripple is only 2%. Current ripple could be improved by using a faster operational frequency, a larger load magnet inductance, or by lowering the circuit resistance [9]. At t = 150 s, the load current is deemed saturated and $i_P$ is set to zero to observe the free decay of current. Current decays quickly due to the resistance of the solder joints in the secondary circuit (Rtot = 9.34 μΩ). The simulation closely replicates the charging curve of the experiment.

The slight difference between experiment and simulation can be explained by coupling of the non-linear behaviours at play in the system. Namely, the non-linear reluctance of the iron-core transformer and sensitivity of non-linear voltage. Slight inaccuracies in non-linear transformer reluctance can cause minor inconsistencies in secondary current. This slight mismatch in secondary current can cause reasonably large differences in switch voltage and charging rate. Therefore, the model was quite sensitive to non-linear transformer reluctance.

As shown in Fig. 5, the load current rapidly charges in the earlier stages before gradually reducing in ramping speed. This charging cycle closely matches the functional form of existing half-wave HTS flux pumps [11], despite the change in flux pump topology.

The dynamic nature of the circuit is highlighted in Fig. 6 with cycle-averages, defined as the integral of a value divided by the cycle period, 1/f. Fig. 6a) shows the cycle-average of current through switch 1 and switch 2 against $-1/2I_L$ in experiment, highlighting the mechanisms driving this functional form. The averages initially diverge, with one or other of the switch currents increasing and the other decreasing. In previous HTS flux pumps, this divergence is related to the generation of a DC offset from magnetization of the transformer core, as annotated in Fig. 6a). It is clear that in this experiment and simulation, the load current is ultimately split approximately evenly between circuit arms, in opposite polarities. Note that the secondary windings have inverse current polarities due to being co-wound with respect to each other.

Two factors drive the generation of a DC offset: 1) distribution of the load current and 2) unbalanced voltage generation during operation. Both experiment and simulation demonstrate that the load current is evenly distributed into both loops during operation leading to a $-1/2I_L$ DC offset. To the best of our knowledge, such an even split of current has not been seen previously in any flux pump device. The offset generated by the load current is therefore a natural and inevitable result in a full-wave HTS flux pump with superconducting switches.

However, as $I_{os}$ is distributed in opposite directions through the transformer windings its net induction on the transformer core is zero. It therefore does not lead to transformer magnetization or iron-core saturation. Instead, the offset acts to reduce the generated secondary current in their respective charging phases.

The second DC offset operates in addition to $I_{os}$ and is caused by imbalanced voltage generation in either half of the charging cycle. Switch voltage is applied to both the load coil and opposing secondary windings, which induces magnetization in the iron-core by Faraday’s law.

where t is the period of a single cycle, $\phi_{\text {core }}$ is the flux offset captured in the core. In half-wave flux pumps, the magnetization is removed using a series switch during the maintenance phase. Here, it is instead removed by the action of the opposing switch. However if the switch voltage magnitudes are unbalanced, the excess voltage manifests as $B_{core}$, which can be related to the induction of the secondary by DC currents, $H_S$.

$l_{core}$ is the length of the magnetic circuit in iron and $μ_r$ the relative permeability of the magnetic circuit. The associated DC current is then half of the difference between $I_1$ and $I_2$.

The total DC current in each secondary is therefore composed of $I_{os}$ and the induced current from core magnetization.

Despite the application of a symmetric waveform in the present study, Fig. 6a), both $I_{os}$ and also $I_{mag}$ are generated during charging.

Either form of DC current alters the voltage output of the switches by reducing the absolute current present in the switch during the charging phase. The effect of negative DC current is seen in the cycle-averaged switch voltages in Fig. 6b). The voltage output is seen to decrease in successive cycles from an initial output voltage to a final voltage in tandem with the load current increase. There is some additional separation between switch voltages caused by uneven circuit resistance (see Appendix A for measured resistance values). The experiment shows an initial voltage of 0.3 mV which decreases to only 0.15 mV. It is this reduction in voltage which causes the reduced charging rate with increasing load current observed in Fig. 5.

The behaviour of the E-J and dynamic loss components can be compared by isolating their relative contributions. The experiment reveals that the ratio between these components changes during the charging process. Initially, the output voltage is about 66% dynamic resistance, however this increases to 90% at the final load current. The dynamic resistance component remains relatively constant throughout the charging process. As $I_{os}$ increases, both types of switching are reduced, but the effect is more pronounced for the non-linear E-J switching than the linear dynamic resistance.

$I_{os}$ can be mitigated by dynamically altering the applied primary using feedback control. The initial secondary current, $I_{S,0}$, to produce a desired voltage, $v_{set}$, is determined from (3).

This corresponds to an initial primary current,

Additionally, a constant charging voltage can be produced by increasing $v_{set}$ along with $I_L$.

The E-J component of switch voltage can be maintained by increasing the initial primary current magnitude proportional to half the load current. This gives a consistent secondary current for the non-linear switch given by Eq. 3.

where k is experimental scaling of the current transfer through the transformer. The load current is monitored on a per-cycle basis and the waveform scaled by

where superscripts show the $j^{\text {th }}$ cycle after the first. Provided $I_L$ does not change much between cycles, such a control scheme can effectively compensate for $I_{os}$. A flow-chart for this control system is shown in Fig. 7.

Fig. 8 shows the results of experiment with the feedback control scheme in Eq. (16). The load current initially reaches 35 A, as was the case in Fig. 5. Feedback control is initiated at 100 s.

Fig. 8a) shows the primary current is then increased proportionally to $I_L$. This increases the secondary current, seen in Fig. 8b). The load current output then increases until it is turned off at 275 A (159 s) to avoid quenching the load coil ($I_c$=315 A). As $I_L$ increases, $I_{os}$ also increases in magnitude, resulting in only a small, unintended increase in peak secondary current. The current ripple at 275 A is 3%, similar to the case without feedback control.

These results are compared to the circuit model simulation utilizing the same feedback control scheme. Both show increases in current beyond that which was possible with a static waveform, with the experiment increasing more than expected. The difference between the experiment and simulation could be related to local heating of the switches causing a decrease in $I_c$ and therefore a power-law increase in switch voltage [32]. This has recently been shown to improve the accuracy of simulated output [33].

Fig. 9 shows the cycle-averaged voltages of the feedback-controlled experiment and simulation. Fig. 9a) shows the E-J voltage immediately returns to its initial value (t = 0 s). This increase is almost entirely due to E-J dissipation, with only a slight increase in dynamic resistance switching. The simulation (Fig. 9b)) shows an ideal implementation of the feedback control mechanism. Here, the voltage returns to the initial value and the load continues to charge. The simulation eventually approaches a maximum of 47 A.

Fig. 10 shows the simulated magnetization of the core. There is an oscillatory waveform as voltage is induced in opposite directions within the transformer during charging. This behaviour is similar to half-wave simulations [12] because the second half of the charging cycle performs the function of the series switch in a half-wave flux pump. The DC magnetization initially increases strongly, then decreases towards zero.

We have demonstrated a full-wave, centre-tapped full-wave transformer rectifier flux pump using HTS materials employing both AC loss and E-J dissipation to produce switching. The device initially reached a maximum load current of 35 A using 0.3 mV of voltage applied to the load. Voltage output was limited by a DC offset of $I_{\mathrm{os}}=-1 / 2 I_{\mathrm{L}}$ developing in the superconducting secondary during charging. A feedback control system is proposed and demonstrated to account for $I_{os}$. Similar to earlier devices’ use of a commutation step, this DC offset can be accounted for by increasing the primary current magnitude proportional to the load current and the ratio of transformer windings. Doing so allowed the flux pump to reach 275 A of load current output using an output of 0.6 mV. The load current output was limited only by the critical current of the load coil. This is a major step towards transformer-rectifier flux pumps of >10 kA current output.

This work was financially supported in part by the New Zealand Ministry of Business, Innovation and Employment (MBIE RTVU1916). It has also been part-funded by STEP, a UKAEA programme to design and build a prototype fusion energy plant and a path to commercial fusion.

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: [Rodney A. Badcock reports financial support was provided by New Zealand Ministry of Business Innovation and Employment. James H.P. Rice reports financial support was provided by UK Atomic Energy Authority.]

The authors would like to thank B. Leuw for helpful discussions. This work was supported in part by the New Zealand Ministry of Business, Innovation and Employment (MBIE RTVU1916), and has been part-funded by STEP, a UKAEA programme to design and build a prototype fusion energy plant and a path to commercial fusion.

All model parameters are listed in Table 1.

Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.supcon.2023.100064.