# A Superconducting Half-Wave Rectifier: Principle and Experiment

Tianyong Gong, Loïc Quéval

► To cite this version:

Tianyong Gong, Loïc Quéval. A Superconducting Half-Wave Rectifier: Principle and Experiment. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0233 . hal-05095731

HAL Id: hal-05095731

https://utc.hal.science/hal-05095731v1

Submitted on 3 Jun 2025

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# A Superconducting Half-Wave Rectifier: Principle and Experiment

Tianyong Gong¹,² and Loïc Quéval¹

GeePs-CNRS Laboratory, CentraleSupélec, University of Paris-Saclay,
91192 Gif-sur-Yvette, France

School of Electrical Engineering, Southwest Jiaotong University,
Chengdu, Sichuan 610031, China

Tel.: +33 - 0786921913.

URL: https://orcid.org/0000-0003-2481-0350

E-Mail: tianyong.gong@centralesupelec.fr

## Acknowledgements

This work was supported by the SuperRail project funded by the French Government as part of the Plan de Relance and the Programme d'investissements d'avenir.

## Keywords

«AC-DC», «Circuits», «Experimental testing», «Cryogenic», «Superconductors».

## Abstract

The superconducting rectifier is a low-power consumption, high-current rectifying device with significant potential for application in AC-DC conversion systems. In this study, we have clarified its operating principle from a circuit perspective. It is demonstrated that the rectification function in a superconducting rectifier is attributed to its time-varying resistance, which is controlled by external sources. An AC field-controlled superconducting half-wave rectifier was developed. By constructing an experimental platform, we measured the rectified voltage and current of the rectifier under various operating conditions. The rectifier successfully converted an AC current into a 34 A DC current, powering a superconducting coil load.

## Introduction

Superconducting rectifiers, utilizing the zero-resistance property of superconductors, efficiently rectify high currents with minimal power loss. They are ideal for high-current, high-efficiency applications like superconducting motors, power transmission, and energy storage, with potential for further advancements as superconductor technology progresses [1], [2].

Superconducting rectifiers, also known as superconducting switches, have been applied in superconducting flux pump systems [3], [4], [5], [6]. Superconducting switches or rectifiers need to be driven by external energy sources, such as mechanical, electrical, magnetic, and thermal sources. Depending on the driving method, they can be classified into rotary type [7], self-regulating type [8], field-controlled type [9], heating-switched type [10], and others [11], [12].

There are theories explaining the operation of superconducting switches: (1) dynamic resistance theory [13], where variations in the resistivity of superconductors lead to the generation of rectifier voltage; (2) macroscopic magnetic coupling effect [14], where local magnetic field inhomogeneities induce a coupling effect between vortex clusters and applied magnetic fields; and (3) non-equilibrium superconducting effects [15], where non-linear resistivity and asymmetric eddy currents produce the DC output in a superconducting dynamo. Each theory offers a unique perspective on the mechanisms driving superconducting switch operation.

In this paper, we explain the working principles of superconducting rectifiers from a circuit perspective, focusing on the dynamic resistance theory. We then fabricate and experimentally validate the performance of an AC field-controlled superconducting half-wave rectifier.

## Working Principle

Based on the dynamic resistance theory, Fig. 1 illustrates the dynamic resistance generation mechanism in a superconducting rectifier, using a superconducting loop subjected to an alternating external field as an example.
<figure><img src="image_2.png" /></figure>

Fig. 1: Schematic of the dynamic resistance generation mechanism in a rectifier, consisting of a superconducting loop placed in an alternating magnetic field, (a) the superconducting loop subjected to the AC magnetic field $B_e$, where 1 and 2 represents branch 1 and branch 2, and $u_{AB}$ represents the voltage between points A and B, (b) the equivalent circuit of the rectifier: $R_1$, $L_1$ and $\varepsilon_1$ represent the resistance, inductance, and induced electromotive force (EMF) of branch 1, while $R_2$, $L_2$ and $\varepsilon_2$ represent branch 2. M is the mutual inductance, and i is the induced current in the loop, (c) the dynamic equivalent circuits of the rectifier, where the top subfigure illustrates the rectifier operating during the negative half-cycle, while the bottom subfigure illustrates its operation during the positive half-cycle. Switches S1 and S2 represent branches 1 and 2, respectively. $\Phi$ indicates the net magnetic flux during one AC cycle of $B_e$.

From Fig. 1, the induced current i and voltage $u_{AB}$ can be derived as follows:

$$ 
\begin{aligned}
i &= \frac{\varepsilon}{R_1 + R_2} - \frac{1}{R_1 + R_2} \frac{d(Li)}{dt} \\
&= -\frac{1}{R_1 + R_2} \frac{d(\Phi_e + Li)}{dt} \\
&= -\frac{d\Phi_e / dt}{R_1 + R_2}, \quad (R_1 + R_2 \neq 0)
\end{aligned} 
\quad (1) 
$$

$$ u_{AB} = \varepsilon_1 - R_1 i - \frac{d(L_1 i + M i)}{dt} \quad (2) $$

in which,

$$ 
\begin{aligned}
\Phi &= \Phi_e + Li \\
\varepsilon &= \varepsilon_1 + \varepsilon_2 \equiv -\frac{d\Phi_e}{dt} \\
L &= L_1 + L_2 + 2M
\end{aligned} 
\quad (3) 
$$

Here, L is the self-inductance of the loop, and $\varepsilon$ is the induced electromotive force (EMF) of the loop. $\Phi_e$ and $\Phi$ represent the external and net magnetic flux through the loop, respectively.

Assuming the inductance of the loop remains constant over time, by integrating $u_{AB}$ over one period T of $B_e$, we obtain the DC voltage $U_{AB}$ as:

$$ 
\begin{aligned}
U_{AB} &= \frac{1}{T} \int_T \left( \varepsilon_1 - R_1 i - \frac{d(L_1 i + M i)}{dt} \right) dt \\
&= \frac{1}{T} \left( \int_T \frac{d\Phi_e}{dt} dt + (L_1 i + M i) \Big|_{t+T}^{t} \right) \quad (4) \\
&= \frac{1}{T} \int_T \frac{d\Phi_e}{dt} dt, \quad (R_1 + R_2 \neq 0)
\end{aligned} 
$$

It can be inferred from (4) that when $d\Phi_e/dt \neq 0$ and $R_1/R_2 \neq$ constant, a non-zero $U_{AB}$ may be generated. The first condition $d\Phi_e/dt \neq 0$ is achieved by applying the AC field $B_e$, while the second condition $R_1/R_2 \neq$ constant, can be realized by adjusting the resistances of switches S1 and S2 shown in Fig. 1.

Besides, according to,

$$ 
\begin{aligned}
\frac{d\Phi}{dt} &= \frac{d\Phi_e}{dt} + \frac{d(Li)}{dt} \\
&= -(\varepsilon_1 + \varepsilon_2) + \frac{d(Li)}{dt}
\end{aligned} 
\quad (5) 
$$

we can transform (4) into,

$$ 
\begin{aligned}
U_{AB} &= \frac{1}{T} \int_T \left( \varepsilon_1 - R_1 \frac{\varepsilon - u}{R_1 + R_2} - u_1 \right) dt \\
&= \frac{1}{T} \int_T \frac{(\varepsilon_1 - u_1)R_2 - (\varepsilon_2 - u_2)R_1}{R_1 + R_2} dt \\
&= \frac{1}{T} \int_T \frac{\varepsilon_1 - u_1 - (\varepsilon_2 - u_2)\alpha}{1 + \alpha} dt, \quad (R_1 + R_2 \neq 0)
\end{aligned} 
\quad (6) 
$$

in which,

$$ 
\begin{aligned}
u &= u_1 + u_2 = \frac{d(Li)}{dt} \\
u_1 &= \frac{d(L_1 i + M i)}{dt}, \quad u_2 = \frac{d(L_2 i + M i)}{dt} \quad (7) \\
\alpha &= \frac{R_1}{R_2}
\end{aligned} 
$$
When $R_1 = 0$ and $R_2 = 0$, the induced current $i$ and the voltage $u_{AB}$ can be derived as:

$$i = \frac{1}{L} \int \varepsilon dt = - \frac{1}{L} \int \frac{d\Phi_e}{dt} dt = - \frac{\Delta\Phi_e}{L} + C \quad (8)$$

$$u_{AB} = \varepsilon_1 - \frac{d(L_1 i + M i)}{dt} \quad (9)$$

where $C$ is a constant.

Then, the DC voltage, $U_{AB}$, is as follows:

$$\begin{aligned} U_{AB} &= \frac{1}{T} \int_T \left( \varepsilon_1 - \frac{d(L_1 i + M i)}{dt} \right) dt \\ &= -\frac{1}{T} \int_T \frac{d(L_1 i + M i)}{dt} dt \quad (10) \\ &= \frac{1}{T} \left[ (L_1 i + M i)|_T - (L_1 i + M i)|_{T+T} \right] \end{aligned}$$

By combining (6) and (10), we can calculate the average voltage of the rectifier over a complete cycle. In Fig. 1(c), when the rectifier operates in the negative half-wave mode, with switch S1 off and switch S2 on (i.e., $R_1 \neq 0$ and $R_2 = 0$), $\alpha \to \infty$ in (6). Conversely, when the rectifier operates in the positive half-wave mode, with switch S1 on and switch S2 off (i.e., $R_1 = 0$ and $R_2 \neq 0$), $\alpha \to 0$ in (6). In both half-wave modes, the rectified voltage for the remaining half-cycle is calculated using (10).

## Experimental setup

We fabricated a field-controlled superconducting half-wave rectifier, as shown in Fig. 2, which consists of a 6 mm wide YBCO high-temperature superconducting (HTS) tape placed in the air gap of an electromagnet (EM). In contrast to Fig. 1, the tape here serves as the superconducting loop, and the electromagnet provides the alternating magnetic field $B_e$. More details about this rectifier can be found in [16].

<figure><img src="image_3.png" /><figcaption>Fig. 2: Photograph of the superconducting rectifier, which consists of a 6 mm wide YBCO high-temperature superconducting (HTS) tape placed in the air gap of an electromagnet (EM).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3: (a) Testing circuit of the superconducting rectifier, where $i_1$ denotes the secondary current of the transformer, $i_2$ is the electromagnet current in the rectifier, and $i_L$ is the rectified load current, (b) the waveform of $i_1$ and $i_2$, with $n$ denoting the number of periods of $i_2$ during one cycle of $i_1$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4: (a) Photographs of testing platform for the superconducting rectifier and (b) RT-LAB interface connected to the Opal-RT.</figcaption></figure>
To test the rectifier, we constructed a testing platform, and its circuit is shown in Fig. 3. The testing system primary consists of a transformer, the rectifier and a superconducting coil load. The transformer has a copper primary winding with turns $N_1$ and a superconducting secondary winding with turns $N_2$. The load is a single-turn superconducting coil, connected to both ends of the HTS tape inside the rectifier using the copper block compression method. The joint resistance is denoted by $R_j$. The key parameters of the testing system are listed in Table I.

To generate the specific current waveform $i_2$ required for the rectifier, as depicted in Fig. 3(b), we used the RT-LAB to control the AC sources. Fig. 4 shows photographs of the testing platform. The entire testing system is immersed in a liquid nitrogen (LN₂) environment.

<table><caption>**Table I: Specification of the superconducting rectifier testing system**</caption><thead><tr><th>Items</th><th>Specification</th></tr></thead><tbody><tr><td colspan="2">Superconducting rectifier</td></tr><tr><td>Tape</td><td>Material: Superconductor<br>Tape number: 1</td></tr><tr><td>Electromagnet</td><td>Material: Copper<br>Number of turns: 251</td></tr><tr><td colspan="2">The transformer</td></tr><tr><td>The primary</td><td>Material: Copper<br>Number of turns $N_1$ = 1000</td></tr><tr><td>The secondary</td><td>Material: Superconductor<br>Number of turns $N_2$ = 1</td></tr><tr><td colspan="2">Superconducting load</td></tr><tr><td>Coil</td><td>Material: Superconductor<br>Number of turns: 1<br>Joint resistance $R_j$: ~ 10 μΩ<br>Inductance: 458 nH (simulated)</td></tr></tbody></table>

<table><caption>**Table II: Abbreviations used in the following**</caption><thead><tr><th>Symbol</th><th>Description</th></tr></thead><tbody><tr><td>n</td><td>Period numbers of the electromagnet current $i_2$ in the rectifier</td></tr><tr><td>$i_1, I_1$</td><td>Instantaneous and RMS values of the transformer secondary current</td></tr><tr><td>$i_2, I_2$</td><td>Instantaneous and RMS values of the electromagnet current</td></tr><tr><td>$i_L, I_L$</td><td>Instantaneous and average values of the load current</td></tr><tr><td>$f_1$</td><td>Transformer frequency</td></tr><tr><td>$f_2$</td><td>Electromagnet frequency</td></tr><tr><td>$I_{dc}$</td><td>DC current in the tape of the rectifier</td></tr><tr><td>$V_{dc}$</td><td>The rectified voltage of the rectifier</td></tr></tbody></table>

## Results

We firstly measure the rectified voltage of the rectifier. Then, the superconducting coil load is charged by the rectifier system under various operating conditions. For clarity, some abbreviations which will be used in the following sections are summarized in Table II.

### Rectified voltage of the rectifier

Fig. 5 shows the rectified voltage $V_{dc}$ produced by the rectifier under varied $I_{dc}$ and $I_2$. Fig. 5(a) illustrates that $V_{dc}$ monotonically increases with $I_{dc}$. However, this increase is nonlinear, which can be attributed to two factors: first, the resistivity of superconducting rectifier is inherently nonlinear; second, as $I_{dc}$ increases, the temperature inside the rectifier rises more significantly, enhancing the nonlinearity of the resistance. In Fig. 5(b), $V_{dc}$ increases linearly with $I_2$. However, when $I_2$ is below 0.3 A, the voltage remains near zero because the rectifier is not activated until the magnetic field applied to the tape in the rectifier exceeds a certain threshold [17].

<figure><img src="image_6.png" /><figcaption>Fig. 5: Rectified voltage $V_{dc}$, (a) as a function of $I_{dc}$, and (b) as a function of $I_2$.</figcaption></figure>
## Load current under varied *n*

Fig. 6 shows the load current over a range of *n* values from 4 to 52, with increments of 4. In this measurement, *f*₁ = 10 Hz and *f*₂ = 500 Hz, giving a frequency ratio of 50. Therefore, when *n* is greater than 50, the rectifier stays in the off-state for the entire rectification period.

<figure><img src="image_7.png" /><figcaption>Fig. 6: Load current under varied *n*, (a) the instantaneous load current *i*<sub>L</sub>, smoothed using MATLAB's *smoothdata* function to eliminate high-frequency noise and current fluctuations, (b) *I*<sub>L</sub> as a function of *n*.</figcaption></figure>

In Fig. 6(a), the load current *i*<sub>L</sub> remains negligible before the rectifier is activated. Once activated, the current increases rapidly and then saturates within approximately 1 s, regardless of *n*. When *n* = 52, *i*<sub>L</sub> = 0, as the switch stays in the off-state for the entire rectification cycle. This results in *R*₁/*R*₂ being constant, which violates the necessary condition for generating rectified voltage as outlined in (4).

Fig. 6(b) shows *I*<sub>L</sub> as a function of *n*. Initially, *I*<sub>L</sub> rises as *n* increases, peaking at *n* = 28, and then decreases. For *n* values between 16 and 32, the variation in *I*<sub>L</sub> is not significant. Since a larger *n* prolongs the off-state duration, resulting in increased losses in the rectifier, it is preferable to choose a smaller *n* to reduce losses and enhance rectification efficiency. Based on this consideration, *n* = 18 was selected for the subsequent tests.

## Load current under varied *I*₁

Fig. 7 shows the load current under varied *I*₁. After activating the rectifier, *i*<sub>L</sub> increases quickly and stabilizes within about 1 s, irrespective of *I*₁. Fig. 7(b) presents the *I*<sub>L</sub>–*I*₁ curve, which exhibits nonlinear characteristics. Specifically, *I*<sub>L</sub> rises more rapidly with increasing *I*₁. This behavior is attributed to the nonlinear *V*<sub>dc</sub>–*I*₁ relationship of the rectifier, as demonstrated in Fig. 5(a).

<figure><img src="image_8.png" /><figcaption>Fig. 7: Load current under varied *I*₁, (a) the instantaneous load current *i*<sub>L</sub>, smoothed using the same method as in Fig. 6, and (b) *I*<sub>L</sub> as a function of *I*₁.</figcaption></figure>
## Load current under varied $I_2$

Fig. 8 presents the load current for varying $I_2$. As shown in Fig. 8(a), $i_L$ remains zero when $I_2$ is below 0.24 A. This is because, at low $I_2$, the rectifier is not activated and does not generate a rectified voltage, as demonstrated in Fig. 5(b). From Fig. 8 (b), it is evident that once $I_2$ exceeds the threshold, here 0.24 A, $I_L$ increases linearly with $I_2$ with a slope of 54. This linear trend is consistent with the rectified voltage behavior depicted in Fig. 5(b).

<figure><img src="image_9.png" /><figcaption>Fig. 8: Load current under varied $I_2$, (a) the instantaneous load current $i_L$, smoothed using the same method as in Fig. 6, and (b) $I_L$ versus $I_2$.</figcaption></figure>

## Discussion

In our experiment, the maximum rectified current, powering a superconducting coil load, reached only 35 A. This limitation is primarily due to the joint resistance, which is in the microohm level. By using a soldering method, this resistance could be reduced to the nano-ohm level [18], potentially allowing for a higher rectified current.

## Conclusion

This paper outlines the operating principles of superconducting rectifiers using a circuit model. It demonstrates that the rectified voltage in a superconducting rectifier arises from its time-varying resistance, which can be controlled by external factors such as magnetic fields. An AC field-controlled superconducting half-wave rectifier was tested under various operating conditions. While the rectifier successfully rectified a current of 34 A, further advancements, such as reducing joint resistance, are needed to improve its rectifying capacity.

## References

[1] J. H. P. Rice et al., "Design of a 60 kA flux pump for fusion toroidal field coils," *IEEE Trans. Appl. Supercond.*, vol. 32, no. 4, Jun. 2022, doi: 10.1109/TASC.2021.3137762.

[2] J. H. P. Rice et al., "Report on progress towards a 10 kA transformer-rectifier flux pump," *IEEE Trans. Appl. Supercond.*, vol. 34, no. 5, Aug. 2024, doi: 10.1109/TASC.2024.3369589.

[3] T. Coombs, J. Geng, L. Fu, and K. Matsuda, "An overview of flux pumps for HTS coils," *IEEE Trans. Appl. Supercond.*, vol. 27, no. 4, Jun. 2017, doi: 10.1109/TASC.2016.2645130.

[4] T. A. Coombs, "Superconducting flux pumps," *J. Appl. Phys.*, vol. 125, no. 23, p. 230902, Jun. 2019, doi: 10.1063/1.5098384.

[5] W. Wang, J. Wei, C. Yang, C. Wu, and H. Li, "Review of high temperature superconducting flux pumps," *Superconductivity*, vol. 3, p. 100022, Sep. 2022, doi: 10.1016/j.supcon.2022.100022.

[6] Z. Wen, H. Zhang, and M. Mueller, "High temperature superconducting flux pumps for contactless energization," *Crystals*, vol. 12, no. 6, p. 766, May 2022, doi: 10.3390/cryst12060766.

[7] A. Morandi, G. Russo, M. Fabbri, and L. Soldati, "Energy balance, efficiency and operational limits of the dynamo type flux pump," *Supercond. Sci. Technol.*, vol. 35, no. 6, p. 065011, Jun. 2022, doi: 10.1088/1361-6668/ac662e.

[8] G. Li, C. Li, Y. Xin, and B. Li, "A superconducting switch using induced current in the closed loop circuit," *Supercond. Sci. Technol.*, vol. 37, no. 4, p. 045009, Apr. 2024, doi: 10.1088/1361-6668/ad2ed6.

[9] G. Li, C. Li, Y. Xin, and B. Li, "A full-wave high-temperature superconducting rectifier based on AC field-controlled switches," *IEEE Trans. Power Electron.*, pp. 1-10, 2024, doi: 10.1109/TPEL.2024.3452500.
[10] L. Zhu et al., "A flux pump driven non-soldering closed-loop HTS magnet and its electromagnetic-thermal semi-analytical modelling method," *Supercond. Sci. Technol.*, vol. 37, no. 1, p. 015014, Jan. 2024, doi: 10.1088/1361-6668/ad10b7.

[11] L. Zhu, Y. Wang, Y. Guo, W. Liu, and C. Hu, "Current decay and compensation of a closed-loop HTS magnet in non-uniform magnetic fields based on electro-magneto-thermal semi-analytical analysis," *Energy*, vol. 277, p. 127560, Aug. 2023, doi: 10.1016/j.energy.2023.127560.

[12] C. Wu, W. Wang, R. Long, H. Li, L. Zhou, and P. Liu, "Fast current regulation and persistent current maintenance of high-temperature superconducting magnets with contact power supply and flux pump," *IEEE Trans. Power Electron.*, pp. 1-10, 2023, doi: 10.1109/TPEL.2023.3342471.

[13] J. Geng et al., "Origin of dc voltage in type II superconducting flux pumps: field, field rate of change, and current density dependence of resistivity," *J. Phys. D: Appl. Phys.*, vol. 49, no. 11, p. 11LT01, Feb. 2016, doi: 10.1088/0022-3727/49/11/11LT01.

[14] W. Wang and T. Coombs, "Macroscopic magnetic coupling effect: The physical

origination of a high-temperature superconducting flux pump," *Phys. Rev. Appl.*, vol. 9, no. 4, p. 044022, Apr. 2018, doi: 10.1103/PhysRevApplied.9.044022.

[15] R. C. Mataira, M. D. Ainslie, R. A. Badcock, and C. W. Bumby, "Origin of the DC output voltage from a high-T$_{c}$ superconducting dynamo," *Appl. Phys. Lett.*, vol. 114, no. 16, p. 162601, Apr. 2019, doi: 10.1063/1.5085226.

[16] T. Gong and L. Quéval, "A field-controlled high-temperature superconducting switch: experiment and simulation," *IEEE Trans. Appl. Supercond.*, early access, 2025, doi: 10.1109/TASC.2025.3539269.

[17] M. P. Oomen, J. Rieger, M. Leghissa, B. ten Haken, and H. H. J. ten Kate, "Dynamic resistance in a slab-like superconductor with $J_c(B)$ dependence," *Supercond. Sci. Technol.*, vol. 12, no. 6, p. 382, Jun. 1999, doi: 10.1088/0953-2048/12/6/309.

[18] B. Shen, Y. Ozturk, L. Fu, J. Yang, and T. Coombs, "Low resistance soldering and installation for a kA level HTS flux pump," *IEEE Trans. Appl. Supercond.*, vol. 31, no. 8, Nov. 2021, doi: 10.1109/TASC.2021.3091100.