1. Introduction
AC loss will be generated, when HTS REBCO coated conductors carry AC transport currents. AC loss is one of main obstacles for HTS applications such as HTS motors/generators, transformers, and superconducting magnetic energy storage systems (SMES) [1], [2], [3], [4]. While there have been many experimental and numerical studies on AC loss, most of them have focused on AC loss behaviours under sinusoidal transport currents [5], [6], [7], [8], [9], [10]. However, REBCO coated conductors in some applications carry non-sinusoidal current [11], [12], [13], [14].
Several studies have been conducted on AC loss measurement under non-sinusoidal transport currents [15], [16], [17], [18]. However, these studies have been limited to distorted currents due to harmonics [15], [16] and under non-sinusoidal waveforms [17], [18]. Furthermore, an experimental and numerical study has been conducted on AC loss in HTS coils under triangular and trapezoidal currents [19]. Nevertheless, these studies lack systematic numerical analysis on AC loss under various non-sinusoidal transport currents, such as square and different types of trapezoidal waveforms. To address the research gap and underpin practical applications, it is essential to numerically study AC loss in REBCO conductors carrying various non-sinusoidal currents.
This work presents AC loss simulation results in a 4 mm-wide REBCO SuperPower tape under sinusoidal and non-sinusoidal current waveforms at 77 K, using a finite element method (FEM) based on the T-A formulation [20] implemented in COMSOL. Non-sinusoidal transport current waveforms studied include square, five types of trapezoidal, and triangular waveforms. For convenience, transport current is normalized by the self-field critical current of the tape, termed as reduced current, i, which ranges from 0.1 to 0.9 at 8.8 Hz and 26.62 Hz. Transport AC loss dependence on the current waveform, frequency, and n-value is discussed. Furthermore, the simulated electromagnetic variables are analysed under each waveform to examine and reveal the loss behaviours.
2. Numerical method
A 2D FEM based on T-A formulation was implemented in COMSOL Multiphysics 6.0 to simulate transport loss in a SuperPower REBCO-coated conductor. In this method, the superconducting layer is considered as a sheet without thickness.
In the T-A formulation, the current density J and flux density B are defined as the curl of the current vector potential T and magnetic vector potential A, respectively:
$\nabla \times \boldsymbol{T}=\boldsymbol{J}$
$\nabla \times \boldsymbol{A}=\boldsymbol{B} $
Faraday’s Law and Ampere’s Law are applied to get the T - formulation and A - formulation, respectively:
$\nabla \times \rho(\nabla \times \boldsymbol{T})=-\frac{\partial \boldsymbol{B}}{\partial t}$
$\nabla \times(\nabla \times \boldsymbol{A})=\mu_{0} \boldsymbol{J}$
where $\mu_{0}$ is the permeability of free space, $4 \pi \times 10^{-7}$H/m.
The resistivity of the superconductor is derived from the E-J power law:
$\rho=\frac{E_{c}}{J_{c}(B)}\left|\frac{J}{J_{c}(B)}\right|^{n-1}$
$J_{c}(\boldsymbol{B})=J_{c 0}\left(1+\frac{B^{2}{ }_{\perp}+k^{2} B^{2}{ }_{\|}}{B^{2}{ }_{0}}\right)^{-\alpha}$
The $J_{c}(\boldsymbol{B})$ curves with three fitting parameters, k, α, and $B_{0}$ were fitted from the SuperCurrent V-I database [23]. $B_{\perp}$ and $B_{\|}$ are the perpendicular and parallel magnetic field components to the conductor wide-face. The $J_{c 0}$ is a constant current density which was obtained by dividing the cross-section of the superconductor from the self-field critical current. The fitting parameters and specifications of the conductor are listed in Table 1 [24].
Table 1. Fitting parameters in simulation [24] and sample specifications. |
| Parameters | 77 K |
|---|---|
| $I_{c 0}$(self-field) (A) | 108.0 |
| n-value | 23 |
| α | 0.19 |
| K | 0.82 |
| $B_{0}$(mT) | 25 |
| Width (mm) | 4.0 |
| Thickness of REBCO ($\mu \mathrm{m}$) | 1.0 |
| Substrate thickness ($\mu \mathrm{m}$) | 50.0 |
| Cu coper stabilizer each side ($\mu \mathrm{m}$) | 20.0 |
| Ag coper stabilizer each side ($\mu \mathrm{m}$) | 2.0 |
The transport AC loss Q (J/m/cycle) can be expressed as:
$Q=\int_{0}^{T} \int \boldsymbol{E} \bullet \boldsymbol{J} d S d t$
where T is the period of one cycle of the transport current and S is the cross-section of the tape. Fig. 1 shows different waveforms of the AC transport current: the square, trapezoidal, sinusoidal, and triangular waveforms. The amplitude, It, of each waveform is the same.
Fig. 1. AC transport current waveform (a) square, (b) trapezoidal, (c) sinusoidal, and (d) triangular. |
To further study the influence of trapezoidal current on AC transport loss, five types of the trapezoidal waveform are used in this work by defining different constant periods (tw) of the trapezoidal currents, which are labelled as:
Trap_1: tw = 9 τ, Trap_2: tw = 7 τ, Trap_3: tw = 5 τ,
Trap_4: tw = 3 τ, Trap_5: tw = τ, where τ is defined as
$\tau=\frac{1}{22} \frac{1}{f}$
where f is the frequency of the transport current.
For each waveform, the coloured solid and dash lines mark the moments when the transport current reaches the positive and negative peaks. These marked moments will be used for discussions in Section 3.1 (see Fig. 6).
3. Results and discussion
3.1. Transport AC loss
$Q_{\mathrm{N}-\mathrm{E}}=\frac{I_{c}{ }^{2} \mu_{0}}{\pi}\left\{(1-i) \ln (1-i)+(2-i)\left(\frac{i}{2}\right)\right\}$
$Q_{N-S}=\frac{I_{c}^{2} \mu_{0}}{\pi}\left\{(1-i) \ln (1-i)+(1+i) \ln (1+i)-i^{2}\right\}$
where $Q_{\mathrm{N}-\mathrm{E}}$ and $Q_{\mathrm{N}-\mathrm{S}}$ are the Norris ellipse and strip models, respectively.
Fig. 2. Simulated results of transport AC loss (a) 8.8 Hz, (b) 26.62 Hz. |
As shown in the figure, the simulated AC loss values are different for the different current waveforms and lie between Norris strip and ellipse models. It is obvious that AC loss is the largest for the square current waveform and the smallest for the triangular current waveform.
$\text { ratio }=\frac{Q}{Q_{\text {tri }}}$
where Q is the AC loss for different waveforms, and $Q_{\text {tri }}$ is AC loss for the triangular waveform.
Fig. 3. Transport AC losses normalized by the AC loss of triangular current waveform at 8.8 Hz and 26.62 Hz. |
As shown in Fig. 3(a) and (b), the ratio varies between 1.2 and 2, and AC loss for the square current waveform is the greatest, followed by Trap_1, Trap_2, Trap_3, Trap_4, sinusoidal, Trap_5 and then the triangular current waveform. Except the case for the sinusoidal current waveform where the current ramp rate (dI/dt of each current waveform when the current rises from zero to the peak) varies all the time, the sequence of the AC loss coincide with the magnitude of dI/dt of each current waveform - the higher the dI/dt value, the greater AC loss generated in the conductor. The ratio of the Trap_3, Trap_4, sinusoidal, and Trap_5 waveform almost keeps constant when i ≤ 0.5, and then slowly increases with increasing i. While the ratio of the square, Trap_1, and Trap_2 shows monotonous increase over i. The results in Fig. 3 show that the current waveform has substantial influence on AC loss, especially for high-i situations. Furthermore, the ratios in Fig. 3(a) and (b) are slightly different. Taking the square waveform as an example, the ratio at 8.8 Hz (see Fig. 3(a)) ranges from 1.7 to 2.0, while 1.65 to 1.95 at 26.62 Hz (see Fig. 3(b)). This implies that the frequency of the transport currents also has influence on the AC loss values. Frequency dependence will be shown in Fig. 8.
To understand the loss behaviours, Fig. 4 and Fig. 5 show the normalized current density $\left(J / J_{c}\right)$, perpendicular magnetic field ($B_y$), electric field ($E_z$), and instantaneous power density (P) along the tape width for different current waveforms at i = 0.5 and f = 8.8 Hz, for Trap_3, square, sinusoidal, and triangular waveforms. To see the evolution of the electromagnetic variables within one applied current cycle, T, eight moments are chosen and labelled as $t_1$ to $t_8$ in Fig. 4((a)-(a′)) and Fig. 5((a)-(a′)).
Fig. 4. Simulated results of electromagnetic variables along tape width for Trap_3((a)-(e)) and Square((a′)-(e′)) current waveform at the current level, i = 0.5, and the frequency is 8.8 Hz. Electromagnetic variables include the normalized critical current, J/Jc ((b)-(b′)), the magnetic flux density, By ((c)-(c′), the electric field, E ((d)-(d′)), and the instantaneous power density, P((e)-(e′)). |
Fig. 5. Simulated results of electromagnetic variables along tape width for sinusoidal((a)-(e)) and triangular((a′)-(e′)) current waveform at the current level, i = 0.5, and the frequency is 8.8 Hz. Electromagnetic variables include the normalized critical current, J/Jc ((b)-(b′)), the magnetic flux density, By ((c)-(c′), the electric field, E ((d)-(d′)), and the instantaneous power density, P((e)-(e′)). |
The electromagnetic variables for the Trap_3 are shown in Fig. 4(b) - (e). When the transport current increases or decreases from zero to peak, $\left|J / J_{c}\right|$ at the tape edge is greater than 1, as shown in Fig. 4(b). Consequently, AC loss is generated at the tape edges from t1 to t3 and t5 to t7, which results from the non-zero E as can be seen from Fig. 4(d) and (e). When the current amplitude keeps constant, at t4 and t8, AC loss is still non-zero at the edges of the tape even when $\left|J / J_{c}\right|$ is below 1. The non-zero AC loss should be due to the finite n-value in the simulation model, leading to non-zero $E\left(<E_{c}\right)$ when $\left|J / J_{c}\right| \geq 0.8$ at n≈20. More details will be shown in Fig. 7. When the current ramps up or down, the penetration depth varies at the tape edges and becomes greatest at t4 and t8, as shown in Fig. 4(c).
Fig. 4(b′) - (e′) shows the electromagnetic variables for the square current waveform, where moments t1 to t3 and t5 to t7 are overlapped, respectively. Therefore, only t3 and t7 are used here. The distributions of the electromagnetic variables here are very similar to the case under the trapezoidal waveform, i.e., when the current reaches the peak at t3 and t7, $\left|J / J_{\mathrm{c}}\right|$ is bigger than 1 at the tape edges (see Fig. 4(b')), and dominant AC loss is generated at these moments due to the large E value (see Fig. 4(d′) and (e′)). Furthermore, the penetration depth is still greatest at t4 and t8, as shown in Fig. 4(c′). However, the magnitude of E and instantaneous P is remarkably larger than those of the Trap_3 waveform.
Fig. 5(a) - (e) illustrates the electromagnetic variables for the sinusoidal current waveform. The moments, t3 and t4, t7 and t8 are positive and negative peaks, respectively, and overlapped. Hence only t4 and t8 are used for simplicity. When the current increases to the peak value, the current density at the tape edges is greater than its critical current density, as shown in Fig. 5(b). As a result, E and instantaneous loss P are generated from t1 to t2 and t5 to t6 at the tape edges as can be seen Fig. 5(d) and (e). At t4 and t8, $\left|J / J_{\mathrm{c}}\right|$ is slightly less than 1, which is attributed to the zero dI/dt at the positive and negative peaks of the sinusoidal waveform. However, a small E and non-zero AC loss is generated because $\left|J / J_{\mathrm{c}}\right| \geq 0.8$ as shown in Fig. 5(d) and (e).
Fig. 5(a′) - (e′) shows the electromagnetic variables for the triangular current waveform, showing similar patterns to those of the sinusoidal waveform. For example, $\left|J / J_{\mathrm{c}}\right|$ is larger than 1 at the tape edges, and AC loss is generated at all moments; the penetration depth increases when the current increases from zero to the peak and has the maximum value at t4 and t8. Nevertheless, $\left|J / J_{\mathrm{c}}\right|$ is still bigger than 1 at t4 and t8, which is entirely different from the square, Trap_3, and the sinusoidal waveforms. We attribute this feature to the triangular waveform’s non-zero dI/dt for the entire period.
Fig. 6(a) compares the penetration depth along the tape width for different current waveforms of Fig. 1 at i = 0.5 and f = 8.8 Hz and Fig. 6(b) shows an enlarged figure in the range −1.75 mm ≤ × ≤ −1.575 mm. To clearly illustrate the penetration depth for different current waveforms, two moments are chosen, as labelled in Fig. 1((a)-(d)), at which the penetration depth is the greatest.
Fig. 6. (a)The perpendicular magnetic flux density, By, along the tape width for different current waveforms (b) enlarged figure of (a) in the range −1.75 mm ≤ × ≤ −1.575 mm. |
Fig. 7 compares the simulated voltage response within one current cycle, T, for two types of trapezoidal waveforms, (a) Trap_1 and (b) Trap_5, at i = 0.5 and f = 8.8 Hz. These two trapezoidal waveforms have different tw values. The voltage along the conductor increases when the current increases from zero to peak. However, the magnitude of the generated voltage of the conductor is different for Trap_1 and Trap_5. The peak voltage in the conductor for Trap_1 is approximately six times of that for Trap_5, due to much larger dI/dt. When the current amplitude keeps constant over tw, a relatively small voltage is still generated due to flux creep, although the magnetic flux within the tape does not change. Therefore, the current ramping process dominates the AC loss generation. In other words, the magnitude of dI/dt determines the AC loss in this case.
Fig. 7. Simulated voltage response (Red) for (a)Trap_1 and (b)Trap_5 at I = 0.5, and the frequency is 8.8 Hz. |
3.2. Frequency and n-value dependence
The Norris model was derived from the Bean model [26], where the n-value is infinite, when a superconducting ellipse or strip carries sinusoidal transport current. Later, an analytical equation was derived for finite n-value to describe the relationship between the frequency and the transport AC loss in HTS coated conductors shown in Eq. (12) [27]:
$\frac{Q}{Q_{0}}=\left(\frac{f}{f_{0}}\right)^{\frac{-2}{n}} \quad\left(I_{t} \ll I_{c}\right)$
where Q0 is the AC loss at the frequency, f0, and It is the amplitude of transport current. However, there has been no report on the frequency dependence of the transport AC loss in coated conductors carrying non-sinusoidal current.
Fig. 8 shows the simulated transport AC loss results when i = 0.5, and n = 23 for three types of waveforms: square, Trap_3, and triangular waveforms. The loss values from Eq. (12) are included in the figure. As shown in Fig. 8, the simulated AC loss values decrease exponentially with increasing frequency, have a good agreement with the values from Eq. (12) even when the transport current has non-sinusoidal waveforms, such as square, trapezoidal, and triangular current waveforms. The simulation result implies the applicability of Eq. (12) to non-sinusoidal current waveforms.
Fig. 8. Simulated AC loss for different waveforms, square, Trap_3, and triangular at the current level, i = 0.5, and n-value is 23. |
Fig. 9 plots the simulated AC loss values as a function of n-value at i = 0.5 and f = 8.8 Hz, for three types of current waveforms: square, Trap_3, and triangular waveforms. When n-value increases, the AC loss for the square and Trap_3 current waveform decreases exponentially, while the AC loss for the triangular current waveform hardly changes. As a result, the difference in the AC loss values becomes smaller with increasing n-value, although the difference between the AC loss values for square and triangular at n = 200 is greater than 10%. Finite n seems to be the main reason for the difference in AC loss values for different current waveforms. For infinitely large n value, all the AC loss values will be the same according to the Bean model.
Fig. 9. Simulated AC loss for different waveforms, square, Trap_3, and triangular at the current level, i = 0.5, and the frequency is 8.8 Hz. |
4. Conclusion
In this work, dependence of transport AC loss in a 4 mm - wide single REBCO coated conductor carrying sinusoidal and non-sinusoidal currents on waveforms, frequency and n-value has been numerically investigated. Non-sinusoidal transport current waveforms include square, five types of trapezoidal, and triangular waveforms.
Simulation results show that transport current waveforms have substantial influence on transport AC loss in the conductor: AC loss for the square current waveform is the greatest, followed by Trap_1, Trap_2, Trap_3, Trap_4, sinusoidal, Trap_5 and then the triangular current waveform. Except the case for the sinusoidal current waveform, the sequence of the AC loss is consistent with the magnitude of dI/dt of each current waveform - the higher the dI/dt value, the greater AC loss generated in the conductor. The sequence of AC loss in the above coincides with the magnitude of the penetration depth: i. e. penetration depth of Square > Trap_1 > Trap_2 > Trap_3 > Trap_4 > Sinusoidal > Trap_5 > Triangular. This implies that the penetration depth determines AC loss in the conductor.
Furthermore, the transport AC loss in the conductor was found to decrease with frequency as $f^{-2 / n}$ even for non-sinusoidal transport current.
AC loss values for different current waveforms are significantly influenced by n-values: when n-value increases, AC loss for the square and trapezoidal current waveforms decrease exponentially, while AC loss for the triangular current waveform hardly changes. As a result, the difference in the AC loss values becomes smaller with increasing n-value, although the difference between the AC loss values for the square and triangular waveforms at n = 200 is greater than 10%. Finite n seems to be main reason for the difference in AC loss values for different current waveforms. Infinite n will eliminate the difference in AC loss values according to Bean model.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This work was partially supported by the New Zealand Ministry of Business, Innovation and Employment under Catalyst Space and Fusion project “International Science Co-operation on Superconductor Technologies” contract number RTVU1916. This work was also partially supported by the New Zealand Ministry of Business, Innovation and Employment under the Advanced Energy Technology Platform program “High power electric motors for large scale transport” contract number RTVU2004.
