1. Introduction
In recent years, wireless power transfer (WPT) has become an attractive research topic which shows widespread application potential in smartphones, medical implant devices as well as electric vehicles (EVs). WPT offers more and more possibilities for EV charging by improving the power transmission distance and creating an electrically safe environment for the consumer market [29], [38]. Recently, high temperature superconducting (HTS) coils have been introduced to the WPT system to replace conventional copper coils. In the superconducting state, HTS coils demonstrate nearly zero DC resistivity and higher quality factors compared to copper coils [40]. WPT systems with HTS coils are expected to achieve similar efficiency as the conventional WPT system but with a more compact system size while allowing larger power delivery distance [7], [47]. Different HTS-WPT systems have been investigated by various research groups to realise higher power and efficiencies for EV charging over kilohertz level [9], [11], [18].
However, in the kHz frequency band, especially for the EV-WPT frequency range (around 85 kHz [43]), the AC losses from HTS coils increase $d_r$amatically, which increases the cooling burden and eventually reduces the system efficiency [53], [33], [52], [51]. Therefore, loss characteristics and potential loss reduction methods for HTS coils are essential research topics to enhance the practicability of the HTS-WPT systems.
AC losses within the HTS coils are closely related to the coil structures, which can be surprisingly high owing to the strong turn-to-turn electromagnetic interactions and the superposition of the magnetic field contributions from individual tapes [8], [46]. For the majority of HTS coil employed power applications, turns are closely packed to achieve high current density and better magnetic field profile [31], [12], [42], [15]. However, in the HTS-WPT applications, the overall system efficiency is the key factor to be considered. Therefore, the geometry of the HTS coils can be altered for WPT applications to improve AC power loss characteristics and power transmission performance. The losses from the HTS coil can be investigated through different aspects. By increasing the inter-turn gap, the total AC losses in the HTS stacks can be reduced significantly at low frequency levels [23], [24]. By changing the width of the HTS tape, the power losses from the HTS coils can vary significantly depending on the current ratio $i_r$ and its critical current [2], [36], [48]. Here, $i_r$ is the ratio between the applied transport current and the critical current of the HTS tape. HTS coils with different coil layouts have been proposed for HTS-WPT applications as resonator coils in the literature [22], [27], [41]. However, most previous works mainly focused on the specific coil design with fixed parameters, but rarely considered how coil geometry design can improve the performance of such WPT applications. Hence, studies regarding the AC loss characteristics within HTS coils at different inter-turn gaps and tape widths have been performed in this paper based on a 2D axisymmetric multi-layer model.
The work in this paper can be divided into three parts. The first part is to construct a 2D axisymmetric multi-layer numerical model that can derive the AC losses from the HTS coils through finite-element modelling. The second step then compares the AC loss characteristic, loss distribution, and magnetic field profiles for three coil geometries at different inter-turn gaps and tape widths. Finally, the optimal coil structure design and the potential benefit of implementing the structure modified HTS coils in the WPT system are also illustrated and discussed.
2. Numerical modelling
2.1. 2D a xisymmetric multi-layer model
Three commonly used HTS coil structures have been considered applicable for the HTS-WPT applications: the spiral coils, the solenoid coils, and the double pancake (DP) coils. The spiral-shaped coil has a relatively small vertical coil volume and thus can minimise the size of the cooling system. It is one of the most commonly used coils in the WPT system [12], [18], [46]. The solenoidal-shaped coil has a $d_r$amatically extended shape along the longitudinal direction and is suitable to work as the intermediate coil or the transmitting coil on the sending side to extend the power transfer distance for the HTS-WPT systems [30], [9], [10], [39]. Finally, the DP coil combines features from both spiral and solenoid coils and is widely used in high power applications such as WPT systems and HTS transformers [19], [20], [34].
The 2D axisymmetric models implementing the H - formulation is constructed in this paper using COMSOL Multiphysics to simulate different HTS coils subjected to alternating input currents and varying external magnetic fields. Comparison with experimental measurements of the AC losses has shown that the 2D H - formulation method is sufficiently accurate in predicting AC losses for HTS stacks and HTS coils [16], [35], [49], [50], [17]. Meanwhile, 2D axisymmetric models can also effectively decrease the number of degrees of freedom and thus save computational time compared to 3D modelling. Recently, the 2D H - formulation method has also been extended to calculate the AC losses at high frequencies for WPT applications [33], [53]. Based on Faraday’s law (1), Ampère’s circuital law (2), Ohm’s law (3), and the E-J power law (4), the governing equations in the FEM model can be described as:
$\frac{\partial\left(\mu_{0} \mu_{\mathrm{x}} \boldsymbol{H}\right)}{\partial t}+\nabla \times \boldsymbol{E}=0$
$\boldsymbol{J}=\nabla \times \boldsymbol{H}$
$\boldsymbol{E}=\rho \cdot \boldsymbol{J}$
$E=E_{\mathrm{c}}\left(\frac{J}{J_{c}}\right)^{n}$
where H is the magnetic field density, E is the electric field, J is the current density, ρ is the material electrical resistivity, $μ_0$ is the vacuum permeability, μr is the relative permeability, $E_c$ = 10-4 V/m is the electric criterion for the critical current, $J_c$ is the critical current density and n is the power-law exponent.
An extended anisotropic Kim-like model was used to describe the magnetic field dependency of the critical current density $J_c(B)$ [44]. In the extended Kim-like model, the perpendicular and parallel magnetic field components are subjected to the superconductor:
$J_{\mathrm{c}}(B)=\frac{J_{\infty 0}}{\left(1+\frac{\sqrt{k_{\mathrm{s}}^{2} \Xi_{\|}^{2}+B_{\perp}^{2}}}{B_{0}}\right)^{\theta}}$
where $B_‖$ and $B_⊥$ are the parallel and perpendicular components of the magnetic flux density subjected to the wide surface of the superconductor tape; $B_0$ = 42.65 mT, which is the magnetic field constant; $k_a$ = 0.29515, and β = 0.7, both of which are the material-related parameter [53]. In this study, the E-J power law is assumed to be valid in the frequency range up to 85 kHz [52], [53].
From the literature, HTS coils with a turn number ranging from 4 to 13 were designed and tested for HTS-WPT applications according to specific system designs [26], [56], [9], [46]. In this paper, three coil layouts are investigated. For better comparison, each coil configuration consists of the same number of turns (8 turns). The side views of the three HTS coils are presented in Fig. 1. The spiral coil model includes eight-turn HTS tapes stacked along the r direction, and the solenoid coil model has eight-turn HTS tapes stacked in the z direction. The double pancake coil model contains two sets of four-turn spiral coils parallel in the z direction. Fig. 2 presents a cross-section view of the multi-layer structure for the HTS coils. It should be noted that the multi-layer structure on the figure has been modified from the real geometry for a better view. The simulation is performed with the specification of the 2G YBCO coated conductors (CCs) manufactured by SuperPower [45].The detailed specifications of the HTS tape used in the simulation are listed in Table 1.
Fig. 1. Side views of three HTS coil models: (a) spiral coil, (b) solenoid coil, and (c) DP coil. |
Fig. 2. Multi-coil model in 2D axisymmetric axis for three HTS coils, the inter-turn gaps between turns in r and z direction are parameterised by $d_r$ and $d_z$, respectively. The cross-section of an HTS tape with multiple layers is also presented on the top right corner. |
| Parameters | Value |
|---|---|
| Specification | SCS4050 |
| Total number of turns | 8 |
| Width of the HTS tape ($w_{\text {tape }}$) | 4 mm |
| Thickness of the HTS tape | 0.1 mm |
| Coil inner radius | 0.1 m |
| Thickness of Copper layer | 20 μm |
| Thickness of HTS layer | 1 μm |
| Thickness of Substrate layer | 50 μm |
| Thickness of Sliver layer | 2 μm |
| Vacuum permeability | 4π×10-7 H/m |
| Copper resistivity at 77 K | 1.97 × 10-9 Ω/m |
| Silver resistivity at 77 K | 2.7 × 10-9 Ω/m |
| Substrate resistivity at 77 K | 1.25 × 10-6 Ω/m |
| n-value | 30 |
| Critical current (77 K, self-field) | 135 A |
In this paper, the 2D axisymmetric multi-layer model is built in the time domain using COMSOL Multiphysics. Fig. 3 shows the mesh geometries for the HTS coil model. As shown, a free triangular mesh with a ’fine’ size is chosen for the air domain. Considering the precision of the simulation, the mesh number along the height of the tape can be vastly distinct for different layers. Two copper stabilisers have 4 vertical elements each. The silver layers and the HTS layer have 1 vertical element each. The substrate layer has 8 vertical elements. The horizontal mesh number along the width of the tape is chosen at 80. The mesh number along the tape width and the vertical element for each layer are reasonable to provide precise AC power loss results [1], [25].
Fig. 3. Mesh geometry for the coil model and the air domain in 2D axisymmetric model. The Dirichlet boundary is used at the edge of the air domain. |
The resistivity ρ for the non-superconducting material is a constant value as listed in Table 1, and the resistivity for the superconducting material is given by:
$\rho_{\mathrm{HTS}}=\frac{E_{0}}{J_{c}}\left(\frac{J}{J_{c}(B)}\right)^{n-1}.$
Finally, by integrating the power density of each domain we can calculate the AC power dissipation generated in each layer per unit length for AC input signals, e.g. transport current or magnetic field, we have:
$Q_{\mathrm{n}}=2 f \int_{1 / f}^{1 / 2 f} \int_{S_{\mathrm{n}}} E \cdot J \mathrm{~d} S_{\mathrm{n}} \mathrm{d} t$
where n is the layer number of the HTS CC, with n = 1, 2, 3, 4, 5, and $S_n$ represents the cross-section area of the studied layer.
The full tape losses from all layers in the HTS CC are thus given by:
$Q_{\text {Total }}=\sum_{n=1}^{6} Q_{\mathrm{n}} \text {. }$
3. Results and discussions
3.1. Numerical model validation
In this work, a finite-element model based on the 2D axisymmetric H - formulation is constructed to calculate the AC loss profile. To verify this model, it was firstly employed to calculate the AC losses of a single HTS tape at both low and high frequencies, and the results are compared to the experimental measurement in [55]. As plotted in Fig. 4, the simulated results agree well with the measurement data in the case of HTS CC at both low and high frequencies, which substantiate the effectiveness of this model. It is to be noted that Tape B has thicker Copper stabilisers (50 μm) compared to Tape A, and thus brings about much higher loss increasing speed in the Cu layers as frequency increases.
Fig. 4. Experiment measurement [55] and simulated AC power losses from two HTS tapes at increasing frequency up to 15 kHz with input current at 45 A. Tape A: Superpower, reference SCS4050, Tape B: Samri, reference SCL-50-180. |
Furthermore, the simulation results obtained from the HTS coil model are compared to the measurement data from [42] as presented in Fig. 5. It can be seen that the simulation results from the 2D H - formulation multi-layer model agree well with the measurement data for HTS coils, which further verifies the model.
Fig. 5. AC power losses of a 36-turn HTS DP coil under transport current at different frequencies from simulation and experiment [42]. |
3.2. AC loss characteristics with different inter-turn gaps
The AC losses and potential weak turns within the HTS coils are strongly related to the coil configurations. Different coil stacking parameters, for example, the inter-turn gap, could affect the total power losses and loss distributions within the coils. In previous studies, different HTS coil structures were examined for WPT applications [33]. However, the AC loss characteristics and the electromagnetic behaviors in HTS coils with different inter-turn gaps are still unclear for such high frequency WPT applications. Therefore, the study regarding changing inter-turn gap is crucial for potential AC loss reduction in the HTS coils.
This section examines the impact of changing inter-turn gaps on the AC loss characteristics and WPT system performance for three HTS coils at f = 85 kHz. As plotted in Fig. 2, the inter-turn gap between the individual turns in the r and z direction are parameterised by $d_r$ and $d_z$, respectively. For the spiral coil, $d_r$ increases from 0.2 mm to 8 mm, indicating an r aspect ratio ($d_r/w_{tape}$) varying from 0.05 to 2. For the solenoid coil, $d_z$ increases from 0.2 mm to 8 mm, indicating a z aspect ratio (dz/$w_{tape}$) from 0.05 to 2. For the DP coil, two different cases are simulated. In the first case, $d_r$ increases from 0.2 mm to 8 mm, and $d_z$ is fixed at 4 mm. In the second case, $d_z$ increases from 0.2 mm to 8 mm, and $d_r$ is fixed at 4 mm. Each coil investigated is comprised of 8 turns with an inner radius of 100 mm.
Before simulating the HTS coils with different inter-turn gaps, the AC losses from each layer of three HTS coils with increasing frequency are presented in Fig. 6. The specifications of the studied HTS coils can be found in Table 1 with $i_r$ = 0.5. It can be seen that at low frequency, $Q_{HTS}$ is the highest among all other losses, followed by $Q_{Cu}$. As frequency increases, $Q_{Cu}$ gradually increases and exceeds $Q_{HTS}$ at a ’transition frequency’ due to the skin effect and shielding effect [52]. The AC power losses from the substrate and Ag layers can be ignored at the chosen frequency range due to their neglectable loss contribution.
Fig. 6. AC losses from each layer of the HTS CC for three HTS coils as a function of increasing frequency up to 100 kHz, with $i_r$ = 0.5. |
3.2.1. AC losses in HTS coils with different inter-turn gaps
The transport losses and total losses from three HTS coils at different inter-turn gaps are studied in this section. The operating frequency is set at 85 kHz according to the SAE J2954 [43], as it is the standard operating frequency for EVs wireless charging.
Fig. 7 illustrates the transport losses from three HTS coils at different inter-turn gaps over a range of $i_r$ from 0.1 to 0.9. The transport losses from a single HTS CC under its self-field condition is also plotted in the figure as a reference. The external magnetic field is not applied in this condition.
Fig. 7. The transport losses in three HTS coils at different inter-turn gaps with $i_r$ from 0.1–0.9, and f = 85 kHz. (a) Spiral coil at $d_r$ increases from 0.2 mm to 8 mm, (b) solenoid coil at $d_z$ increases from 0.2 mm to 8 mm, (c) DP coil at $d_r$ increases from 0.2 mm to 8 mm, (d) DP coil at $d_z$ increases from 0.2 mm to 8 mm. |
It can be seen that, the transport losses from the HTS coils increase with about the second power of$i_r$ and decrease with increasing inter-turn gaps. For the spiral coil, when $d_r$ increases to larger than the tape width (4 mm), the loss decreasing rate becomes smaller. For example when $i_r$ = 0.5, the transport losses in the spiral coil reduce from 72.1 W/m to 14.8 W/m (79% decreasing) as $d_r$ increases to 4 mm ($d_r/w_{tape}$ = 1). When $d_r$ increases further from 4 mm to 8 mm, the losses in the spiral coil only reduce by 24% from 14.8 W/m to 11.2 W/m. The reason for this phenomenon is that, as $d_r$ increases to larger values, the turn to turn electromagnetic induction reduces $d_r$amatically, and the loss value for each turn starts to converge towards the loss value of a single tape.
The solenoid coil has a geometry with vertical parallel HTS coils, which results in a less significant turn-to-turn electromagnetic interaction. Therefore, with increasing $d_z$, the loss reduction effect for the solenoid coil is weaker than that in the spiral coil with increasing $d_r$. As $d_z$ increases from 0.2 mm to 2 mm, the losses in the solenoid coil reduce by 50% from 23.3 W/m to 12.7 W/m at $i_r$ = 0.5. As $d_z$ increases further, the losses from the solenoid coils converge towards the loss value of a single tape under self-field, and the decreasing speed slows down as well.
Comparing the two cases for the DP coil, increasing $d_r$ shows a notable loss reduction effect. For instance, $i_r$ = 0.5, the transport losses in the DP coil reduce by 63% as $d_r$ increases from 0.2 mm to 8 mm. While in the second case, the losses only reduce by 29% as $d_z$ increases from 0.2 mm to 8 mm. Moreover, the loss decreasing rate for the DP coil is almost constant owing to its multi-dimension structure.
Fig. 8, Fig. 9, Fig. 10, Fig. 11 plot the magnetic flux densities at 0.2 mm and 8 mm inter-turn gaps when $i_r$=0.5 for the spiral, solenoid, and DP coil, respectively. As shown in Fig. 8, the spiral coil shows a higher magnetic flux density at $d_r$ = 0.2 mm. As $d_r$ increases to 8 mm, the maximum flux density reduces $d_r$amatically from 102 mT to 53.3 mT. Meanwhile, the high field region (the area within the blackline) around the coil also reduces with increasing $d_r$. As shown in Fig. 8 (a), the high field region around the spiral coil keeps a high magnetic flux density above 35 mT. While at $d_r$ = 8 mm, the high magnetic flux density region is much smaller and can only be observed at the edges of each turn. As a result, the total losses generated in the spiral coil decrease $d_r$amatically with increasing $d_r$.
Fig. 8. Magnetic flux density around the spiral coil at the current phase of 3π/2, with$i_r$ = 0.5, and f = 85 kHz. |
Fig. 9. Magnetic flux density around the solenoid coil at the current phase of 3π/2, with $i_r$ = 0.5, and f = 85 kHz. |
Fig. 10. Magnetic flux density around the DP coil at the current phase of 3π/2, with $i_r$ = 0.5, and f = 85 kHz. |
Fig. 11. Magnetic flux density around the DP coil at the current phase of 3π/2, with $i_r$ = 0.5, and f = 85 kHz. |
As shown in Fig. 9, for the solenoid coil, the maximum magnetic flux density reduces slightly from 77.6 mT to 61.8 mT as $d_z$ increases from 0.2 mm to 8 mm. At $d_z$ = 0.2 mm, the high magnetic flux density can only be observed at the end turns of the solenoid coils, and its high field region is also much smaller due to its vertical parallel coil geometry. Meanwhile, the magnetic field cancelling effect can also be observed as the middle turns present a much lower magnetic flux density than the upper and lower turns. In fact, the perpendicular magnetic fields generated by the top and bottom turns cancel each other, and thus effectively shield the middle turns. As $d_z$ increases to 8 mm, the magnetic flux density around the solenoid coil also reduces and therefore the losses. Meanwhile, a less fluctuating magnetic flux density for each turn within the solenoid coil can be observed, which results in a more even loss distribution.
As depicted in Fig. 10 (a) and (b), for case one the highest magnetic flux density around the DP coil reduces from 89.5 mT to 54.7 mT as $d_r$ increases from 0.2 mm to 8 mm. Meanwhile, the high field region around the DP coil also decreases $d_r$amatically as $d_r$ increases to 8 mm. This variation trend is similar to that in the spiral coil. In the second case with increasing $d_z$, the highest magnetic flux density reduces slightly from 69.4 mT to 61.3 mT as $d_z$ increases from 0.2 mm to 8 mm. By comparing two cases, it can be seen that changing $d_r$ has a greater impact on the magnetic field than changing $d_z$ for the DP coil, and therefore the losses.
3.2.2. AC losses in HTS coils with different inter-turn gaps under external magnetic field
In practical WPT applications, the electromagnetic environment around the resonant coils can vary with the input and operating conditions. The secondary/inserted coils can experience an external magnetic field under HTS-WPT conditions [5], [33], [6], [28]. For the simulation in this section, an AC external magnetic field $B_e$ is applied to the HTS coils. Fig. 13 presents the total losses in three HTS coils with AC transport current at $i_r$ = 0.5 while subjected to an alternative $B_e$ up to 100 mT, at f = 85 kHz.
Fig. 12 shows the magnetic field around the HTS receiving coil generated by the HTS transmitting coil in the cross-section view. The red arrows indicate the magnetic field vector around the HTS coils. It is to be noted that, the magnetisation losses of the HTS coils are dominated by the magnetic field components perpendicular to the wide surfaces of the HTS tapes and thus $B_z$, the parallel field component, has been neglected. $d_{gap}$ represents the power transmission distance between the two resonant coils.
Fig. 12. Magnetic field distributions $B_e$ around HTS coil. $B_r$ represents the magnetic field perpendicular to the wide surfaces of the HTS tape. |
From Fig. 13 (a), it can be seen that for the spiral coil with different $d_r$, different loss variation trends can be observed under low and high $B_e$. The effective loss reduction with increasing $d_r$ can only be obtained if $B_e$ is smaller than the threshold field level ($B_{thre}$). Here, Bthre indicates the external magnetic field level where total losses from the HTS coil with smaller inter-turn gaps exceed the total losses from the HTS coil with larger inter-turn gaps. As presented in Fig. 13 (a) and (c), $B_{thre}$ for spiral coil and DP coil in case 1 is 30 mT and 18.5 mT, respectively. Under small external fields ($B_e$ $B_{thre}$), the total losses decrease with $d_r$, because the applied transport current dominates the total losses. For example, when $B_e$ = 10 mT, the total losses decrease from 74 W/m to 21 W/m as $d_r$ increases from 0.2 mm to 8 mm. However, this trend reverses at higher $B_e$ ($B_e$ >$B_{thre}$). For instance, when $B_e$ = 40 mT, the total losses increase from 112 W/m to 182 W/m as $d_r$ increases from 0.2 mm to 4 mm. This reverse trend is due to the stronger shielding effect in the spiral coil with smaller $d_r$. In the spiral coil, the inner turns can be effectively shielded from the external magnetic field and therefore experience lower losses than the outer turns. Meanwhile, this shielding effect is also strengthened with the reduced inter-turn gap. At larger $d_r$, this shielding effect gets weaker, and the spiral coil becomes delicate to the variation of $B_e$. Therefore, the spiral coils with larger $d_r$ experience higher total power losses when subjected to higher $B_e$. It is to be noted that, for practical WPT applications, the external magnetic field $B_e$ over the coils reduces rapidly with increasing gap distance. In this paper, $B_e$ is much smaller than the threshold field $B_{thre}$.
Fig. 13. The total losses in three HTS coils at different inter-turn gaps at $i_r$ = 0.5 and a range of $B_e$ up to 100 mT at f = 85 kHz. (a) Spiral coil at $d_r$ from 0.2 mm to 8 mm, (b) solenoid coil at $d_z$ from 0.2 mm to 8 mm, (c) DP coil at $d_r$ from 0.2 mm to 8 mm, (d) DP coil at $d_z$ from 0.2 mm to 8 mm. |
As presented in Fig. 13 (b), for the solenoid coil, the AC loss decreasing rate with increasing $d_z$does not vary with increasing external magnetic fields. Each turn within the solenoid coil is fully subjected to the external magnetic field without any shielding effect. Therefore, the loss variation at different $d_z$ follows the same trend with increasing $B_e$.
As shown in Fig. 13 (c), with increasing $d_r$, the loss variation in the DP coil is comparable to that in the spiral coil. As $B_e$ increases, the DP coils with smaller $d_r$ experience lower power losses due to the shielding effect from the outer turns. Compared to the spiral coil, Bthre for the DP coil is slightly lower due to the weaker shielding effect. As shown in Fig. 13 (d), with increasing $d_z$, the DP coil shows a similar loss variation trend as the solenoid coil. However, its total losses at higher $B_e$ are much smaller than that in the solenoid coil due to the shielding effect.
3.2.3. Coil loss distributions with different inter-turn gaps
To evaluate the potential weak turns and better describe the shielding effect in the outer turns when subjected to an external magnetic field, Fig. 14, Fig. 17 present the loss distributions from three HTS coils with different inter-turn gaps at Be = 0 mT and $B_e$ = 60 mT. The input current $i_r$ = 0.5 is applied to three HTS coils. Both $i_r$ and $B_e$ are set at f = 85 kHz. The normal average loss contribution (12.5%) is also illustrated in the figure as a reference.
Fig. 14. Loss distributions from the spiral coil at different $d_r$ with (a) $B_e$ = 0 mT and (b) $B_e$ = 60 mT. |
As shown in Fig. 14, when $B_e$ = 0 mT, two end turns in the spiral coil experience higher losses at $d_r$ = 0.2 mm due to the strong electromagnetic interaction within the closely stacked HTS coils. For $d_r$ >0.2 mm however, the loss distributions are almost constant. At this point, the loss deviation between turns is smaller because of the weaker field interaction between the adjacent turns. At $B_e$ = 60 mT, the loss differences between the inner and outer turns gradually decrease with $d_r$. When $d_r$ = 0.2 mm, the two end turns produce 43% of the losses due to the shielding effect under a high external magnetic field. While at $d_r$ = 8 mm, the loss share at two end turns reduce to 27% because the shielding effect is reduced at larger $d_r$. It should be pointed out that the loss distributions are concentrated on the innermost turns due to the uneven loss contribution from the upper and lower copper stabilisers [32].
In Fig. 15, due to the field cancelling effect, the two end turns of the solenoid coil generate perpendicular magnetic fields that cancel each other, thus shielding the middle turns. Meanwhile, those middle turns generate perpendicular magnetic fields superposed on both end tapes, increasing the magnetic field strength and the power losses. It can be seen that when $B_e = 0$ mT the loss distribution in the solenoid coil becomes more balanced as $d_z$ increases to 8 mm, which indicates that the field cancelling effect becomes less significant at larger inter-turn gaps. Nevertheless, at $B_e = 60$ mT, the high external magnetic field determines the power losses within the solenoid coil, and the field cancelling effect is not significant. From Fig. 15 (b), it is obvious that the loss distributions are skewed to the upper turns under the superimposed effect of both the external magnetic field and the self-field, and the uneven loss distribution also reduces with increasing inter-turn gaps due to the weaker self-field effect generated by the adjacent turn.
Fig. 15. Loss distributions from the solenoid coil at different $d_z$ with (a) $B_e = 0 mT and (b) $B_e = 60 mT. |
The DP coil consists of two four-turn spiral coils. Numbers 1–4 represent the upper turns, and numbers 5–8 represent the lower turns. From Fig. 16 and Fig. 17 it can be seen that, in the first case with increasing $d_r$, each 4-turn spiral coil within the DP coil presents a loss distribution comparable to the spiral coil at both $B_e$ = 0 mT and $B_e$ = 60 mT. While in the second case with increasing $d_z$, the loss distributions are almost constant. This is because the loss distributions within the DP coils are mainly determined by the turns in the r direction. Moreover, one of the pancake from the DP coil can present a higher loss share than the other pancake due to the superimposed effect of the external magnetic field and the self-field. For example in the Fig. 16 and Fig. 17, the upper pancake presents a higher loss share at the first half current cycle because Be and the self field from the lower pancake superimposed at the upper pancake. At the second half current cycle, $B_e$ and the self field from the upper pancake superimposed at the lower pancake and the situation reversed.
Fig. 16. Loss distributions from the DP coil at different $d_r$ with (a) $B_e$ = 0 mT and (b) $B_e$ = 60 mT. |
Fig. 17. Loss distributions from the DP coil at different $d_z$ with (a) $B_e$ = 0 mT and (b) $B_e$ = 60 mT. |
3.3. AC loss characteristics with varying tape widths
This section studies the impact of tape width variation on the AC losses and WPT performance in terms of the studied three HTS coil topologies. In the previous section, the width of the HTS tape was set at a fixed value of 4 mm. Other than the 4 mm-wide YBCO tape, tapes with 2 mm, 3 mm, 6 mm, and 12 mm widths are also available from SuperPower. Since tapes at different widths have the same layer structures (material, thickness, etc.), the material-related parameters for samples with different tape widths are assumed to be identical in the simulation. Meanwhile, the critical currents for the HTS tape are essentially proportional to the tape width, with a typical $I_c$ for the chosen tape sample at 135 A/4 mm [45]. Five typical tape widths are selected at 2 mm, 3 mm, 4 mm, 6 mm, and 12 mm. The inter-turn gaps $d_r$ and $d_z$ are set at 4 mm for three HTS coils.
3.3.1. AC losses in HTS coils with varying tape widths
Fig. 18 (a), (b), and (c) present the transport losses from three HTS coils at different tape widths with It ranging from 20 A to 120 A. Frequencies for both It and Be are set at 85 kHz. The applied transport current for cases with $w_{tape}$ = 2 mm and $w_{tape}$ = 3 mm are smaller than 120 A to enforce $I_t$ to be lower than the critical current. Fig. 18 (d) compares the transport losses from three HTS coils at different tape widths with $I_t$ =50 A and $B_e$ = 0 mT.
Fig. 18. Transport losses from three HTS coils at different tape widths with $I_t$ = 20–120 A, $B_e$ = 0 mT and f = 85 kHz. (a) Spiral coil at tape widths from 2 mm to 12 mm, (b) solenoid coil at tape widths from 2 mm to 12 mm, (c) DP coil at tape widths from 2 mm to 12 mm, (d) comparison of the transport losses from three HTS coils at $I_t$ = 50 A as a function of the increasing tape widths. |
At zero $B_e$, the total AC losses are naturally dominated by the transport current as well as the self-field produced by the transport current. In this case, the losses are inversely proportional to the tape width at a fixed It, and the current ratio $i_r$ is smaller in the HTS tapes with wider tape widths as they have much higher critical currents. From Fig. 18 (d), it can be seen that, the loss decreasing rate for the spiral coil reduces at $w_{tape}$ >4 mm ($d_r$). While for the solenoid coil and the DP coil, the decreasing rates are almost constant. Among the three coils, the solenoid coil experiences the highest loss reduction effect with AC power losses reduced by 86.2% as $w_{tape}$ increases from 2 mm to 12 mm. In contrast, the DP coil experiences the lowest loss reduction at 71.4%. It should be pointed out that, the current ratio $i_r$ in the 12 mm tapes has reduced by 83.3 % compared to $i_r$ in the 2 mm tape. For the solenoid coil, the loss reduction rate is larger than the current reduction rate, while for the spiral and DP coil, the loss reduction rate is smaller than the current reduction rate.
3.3.2. AC losses in HTS coils with varying tape widths under external magnetic field
To study the AC loss variation within three HTS coils at different tape widths with respect to increasing $B_e$, Fig. 19 presents the total losses from three HTS coils at different tape widths with $B_e$ ranging from 0 mT to 100 mT. The input transport current $I_t$ = 50 A is applied to three HTS coils. For both input current and applied external magnetic field, f = 85 kHz.
Fig. 19. Total losses from three HTS coils at different tape widths over a range of $B_e$ with $I_t$ = 50 A, f = 85 kHz and $B_e$e are increased from 0 mT to 100 mT. (a) Spiral coil at tape widths from 2 mm to 12 mm, (b) solenoid coil at tape widths from 2 mm to 12 mm, (c) DP coil at tape widths from 2 mm to 12 mm, (d) comparison of the total losses from three HTS coils at $B_e$ = 60 mT as a function of the increasing tape widths. |
It can be seen that at low $B_e$, the total losses from three HTS coils decrease with tape width. However, at larger $B_e$, the total losses from the HTS coils with larger tape widths increase $d_r$amatically with $B_e$. This is because when the perpendicular magnetic field is large, it determines the total losses generated from the HTS coils. For example, for the spiral coil as shown in Fig. 19 (a), at $B_e$ >$B_{thre}$, the effect from the induced self-field can be ignored, and the total losses in the spiral coil increase with tape width. Here, $B_{thre}$ indicates the field level where total losses at $w_{tape}$ = 12 mm exceed the total losses at $w_{tape}$ = 4 mm. For the spiral, solenoid and DP coils, $B_{thre}$ is 13.7 mT, 3.7 mT and 10 mT, respectively. It should be noted that when subjected to a high external magnetic field, the total losses from the 12 mm $w_{tape}$ spiral coil are slightly smaller than that from the 6 mm $w_{tape}$ spiral coil. It is because the coil with 12- mm-wide tape experiences a stronger shielding effect with a small aspect ratio ($d_r/w_{tape}$) [4], [37]. The DP coil also presents a similar variation trend with less shielding effect at higher Be as shown in Fig. 19 (c). Meanwhile, the threshold field level for the DP coil is smaller than that of the spiral coil. For the solenoid coil, as shown in Fig. 19 (b), the threshold field can be much lower than that in the other two coils, indicating that the solenoid coil with larger tape width is very sensitive to the increment of $B_e$. Therefore, even with $B_e$ at a few mT, the total losses from solenoid coil with larger tape width can be much higher.
3.3.3. Coil loss distributions with varying tape widths
To evaluate the turns with higher energy dissipation, this subsection presents the loss distributions within three HTS coils with different tape widths at $I_t$ = 50 A. $B_e$ are set at 0 mT and 60 mT, indicating the WPT operating conditions at low and high external magnetic fields. Fig. 20, Fig. 22 illustrate the loss distributions in the spiral, the solenoid, and the DP coil, respectively.
Fig. 20. Loss distributions from the spiral coil at different tape widths with (a) $B_e$ = 0 mT and (b) $B_e$ = 60 mT. |
As shown in Fig. 20, at $B_e$ = 0 mT, the applied transport current dominates the AC losses within the spiral coils. Due to the unbalanced loss contribution from upper and lower stabilisers and the superimposed effect from the self-field and the externally applied field, the loss distributions within the spiral coil are concentrated on the innermost turns. Additionally, more losses will be generated in the innermost turns with increasing tape width due to its smaller aspect ratio ($d_r/w_{tape}$). As the tape width increases to 12 mm, the loss distribution from the innermost turn (turn 1) increases from 12% to 19%, while the contribution from the outermost turn (turn 8) decreases from 10% to 7%. At $B_e$ = 60 mT, the power losses are determined by $B_e$. At smaller tape widths, the aspect ratio ($d_r/w_{tape}$) for the spiral coil is large, which results in a weaker shielding effect from the end turns. For example, with 2 mm $w_{tape}$, the maximum loss deviation within the spiral coil is less than 2%. While at $w_{tape}$ = 12 mm, the shielding effect is stronger, and almost 40% of the total losses are generated from the two end turns.
For the solenoid coil as shown in Fig. 21, at $B_e$ = 0 mT, the loss deviation between the inner and outer turns increases with tape width due to the field cancelling effect. For instance, at $w_{tape}$ = 2 mm, the loss contribution is 32% at two end tapes, while at $w_{tape}$ = 12 mm the loss contribution from two end tapes increases to 45.7%, which is almost half of the overall power losses from the coil. This tendency is expected because wider HTS tape is more sensitive to the magnetic field variation from the adjacent turns. Therefore, the solenoid coil using wider HTS tapes experiences a stronger field cancelling effect and also higher loss deviation. At $B_e$ = 60 mT, tapes within the solenoid coil are fully subjected to the high $B_e$. Thus, the loss distributions do not change significantly. It should be pointed out that, due to the combined effect from both the transport current and the applied external field, the loss distributions are focused on the upper turns for the solenoid coils at both $B_e$ = 0 mT and $B_e$ = 60 mT at the first half current cycle.
Fig. 21. Loss distributions from the solenoid coil at different tape widths with (a) $B_e$ = 0 mT and (b) $B_e$= 60 mT. |
As shown in Fig. 22, the DP coil experiences a loss variation trend combined with the characteristics from both the spiral and the solenoid coil. At $B_e$ = 0 mT, the loss distributions from the DP coil skew to the innermost turns, with the AC losses slightly higher in the upper pancake. This phenomenon results from both the field cancelling effect and the loss deviation in the two copper stabilisers. With increasing tape width, this trend gets strengthened. At 60 mT, the shielding effect can be observed in the DP coils with larger tape widths due to their smaller aspect ratios. For example, when $w_{tape}$ = 2 mm, the losses from each turn are similar because the perpendicular magnetic field fully penetrates all individual turns. On the contrary, at $w_{tape}$ = 12 mm, the middle turns (2, 3, 6, 7) present a lower loss contribution compared to the side turns (1, 4, 5, 8) due to the shielding effect.
Fig. 22. Loss distributions from the DP coil at different tape widths with (a) $B_e$ = 0 mT and (b) $B_e$ = 60 mT. |
3.4. Case study with optimal coil design
3.4.1. Magnetic flux densities for three HTS coils with different inter-turn gaps
In WPT applications, the magnetic flux density generated by the resonant coils is essential as it reveals the strength of the electromagnetic resonance link and the power transfer capability between the transmitting coils and receiving coils. In this subsection, the magnetic flux density generated by three HTS coils at different inter-turn gaps is analysed. It is assumed that a receiving coil with a 0.1 m coil radius and a 0.1 m power transmission distance could collect the magnetic field generated by the HTS coil. The simulated region is in the air-gap centre plane as illustrated in Fig. 23. The magnetic flux density for three HTS coils is presented in Fig. 24.
Fig. 23. Cross-section view of the simulated region. The magnetic flux density is simulated along the air-gap centre between the two resonant coils. |
Fig. 24. Magnetic flux density along the r direction with $i_r$ = 0.5 at z = 50 mm. (a) Spiral coil, (b) solenoid coil (c) DP coil in Case 1 and (d) DP coil in Case 2. |
For the two-coil WPT system, the peak magnetic flux density $B_{peak}$ can be observed at the centre plane along the vertical direction. For the spiral coil, $B_{peak}$ stays at the same level as $d_r$ increased from 0.05$w_{tape}$ to 2$w_{tape}$. However, the flux density curve moves slightly to the right due to the increasing coil geometry in the r direction. Increasing centre diameter can also expand the magnetic flux density distribution with a lower amplitude compromise and can assist the misalignment problem [3]. The solenoid coil shows a noticeable reduction in the magnetic field at the increasing gap. In fact, $B_{peak}$ in the solenoid coil reduces about 1.5 times as $d_z$ increases to 2$w_{tape}$. This is because the electromagnetic interaction of the solenoid coil in the perpendicular direction decreases $d_r$amatically with an increasing gap. For the DP coil, two different cases show different variation trends. In case 1 with increasing $d_r$, the DP coil demonstrates a similar variation trend as the spiral coil due to their similar stacking origins. While in case 2, the magnetic flux density is almost constant at different $d_z$ owing to the multi-dimension coil structure of the DP coil.
3.4.2. Magnetic coupling efficiencies for three HTS coils with different inter-turn gaps
To estimate the impact of increasing inter-turn gaps on the Magnetic coupling efficiencies (MCEs) of the three HTS coils, Fig. 25 depicts the variation of the calculated coil-to-coil maximum magnetic coupling efficiencies ($η_t$) and the equivalent resistance ($R_{equ}$) of three HTS coils at different inter-turn gaps at $i_r$ = 0.2 and 0.5. The gap distance between the two HTS coils $d_{gap}$ is set at 0.1 m, with zero misalignment. The effect of the AC external magnetic field components perpendicular to the wide surfaces of HTS tapes has been considered in the simulation. The operating frequency is set at 85 kHz. The detailed calculation of the MCEs and equivalent resistance can be found in Appendix A (see Equation 18 for MCEs ($η_t$), Equation 11 and Equation 12 for $R_{equ}$) and the calculation of the mutual inductance between two resonant coils can be found in Appendix B.
Fig. 25. Equivalent resistances and MCEs for three HTS coils at different inter-turn gaps. |
Here, $η_t$ is the maximum coil-to-coil magnetic coupling efficiency and can be expressed as:
$ \eta_{\mathrm{t}}=\frac{P_{\text {out }}}{P_{\text {in }}}=\frac{\text { Energy stored in the receiving coil }}{\text { Input energy of the transmitting coil }} $
and $P_{in}$ can be expressed as:
$ P_{\text {in }}=P_{\text {out }}+\mathrm{AC} \text { losses. } $
It is to be noted that, for calculating the coil-to-coil magnetic coupling efficiency $η_t$, the cryostat losses and the current lead losses are not considered. The discussion regarding the system efficiency considering the cryocooling losses and the current lead losses will be presented at the end of this section. For simplification, the switching losses from the power electronics are not considered in this model.
From Fig. 25 it can be seen that, the variation trends of the equivalent resistances follow the variation trend of AC losses because the AC losses determine the equivalent resistances within the HTS coils. With smaller $R_{equ}$, the maximum MCEs for the spiral coil are much higher at larger inter-turn gaps. For example, as $d_r$ increases from 0.2 mm to 4 mm, the maximum MCE increases from 96.7% to 99.2% at $i_r$ = 0.2. The variation of $η_t$ is minimal as $d_r$ increases further due to the convergences of the AC power losses.
For the solenoid coil, its maximum MCE reaches the highest value at $d_z$ = 2 mm. As $d_z$ increases further, the maximum MCE decreases with $d_z$ even though the equivalent resistances are still decreasing. This reduction of $η_t$ is because of the reduction of the mutual inductance. At larger $d_z$, the reduction in the AC losses for the solenoid coil cannot compensate for the reduction in the magnetic coupling (mutual inductance) between the two resonant coils. Therefore, the maximum MCE reduces with $d_z$ at $d_z$ >2 mm. Comparing Fig. 25 (c) and (d), one can find that increasing $d_r$ has a bigger impact on $η_t$ of the DP coil. With increasing $d_r$, the variation of$R_{equ}$ and ηt in the DP coil is closer to that in the spiral coil. While with increasing $d_z$, the variation of Requ and $η_t$ is closer to that in the solenoid coil.
Table 2 illustrates the parameters for three HTS coils in the model, the calculation of the Self inductance L, mutual inductance M and the coupling coefficient k can be found in the Appendix B. The value of the capacitors are chosen to keep the coils operating at resonant frequency. Table 3 summarises the performance of three HTS coils at a non-optimised inter-turn spacing (0.2 mm) and an optimised inter-turn spacing at two different input currents. Q is the quality factor, and $P_{in}$ is the input power for the WPT system that the maximum MCEs can be achieved.
Table 2. Simulation model parameters of three HTS coils at different inter-turn gaps. |
| Coil shapes Units | $d_r$ (mm) | $d_z$ (mm) | L (μH) | M (μH) | C (nF) | k | $l_{coil}$ (m) |
|---|---|---|---|---|---|---|---|
| Spiral | 0.2 | N/A | 56.1 | 4.02 | 62 | 0.07 | 5.06 |
| Spiral | 4 | N/A | 25.8 | 5.12 | 136 | 0.20 | 5.73 |
| Solenoid | N/A | 0.2 | 17.8 | 2.63 | 197 | 0.15 | 5.03 |
| Solenoid | N/A | 2 | 15.2 | 2.20 | 230 | 0.15 | 5.03 |
| DP | 0.2 | 0.2 | 47.8 | 3.51 | 73 | 0.07 | 5.04 |
| DP | 4 | 4 | 25.2 | 3.82 | 139 | 0.15 | 5.32 |
Table 3. Comparison of three HTS coils at different inter-turn gaps. |
| Coil shapes Units | $i_r$ | $d_r$ (mm) | $d_z$ (mm) | Volume (×10-4m3) | AC losses (W/m) | Requ (mΩ) | Q | $η_t$ | $P_{in}$ (kW) | $P_{out}$ (kW) |
|---|---|---|---|---|---|---|---|---|---|---|
| Spiral | 0.2 | 0.2 | N/A | 1.3 | 8.3 | 115.2 | 484 | 96.7% | 2.55 | 2.47 |
| Spiral | 0.2 | 4 | N/A | 2.1 | 1.5 | 23.6 | 1104 | 99.2% | 2.55 | 2.53 |
| Solenoid | 0.2 | N/A | 0.2 | 10.5 | 3.2 | 44.2 | 402 | 96.0% | 0.80 | 0.77 |
| Solenoid | 0.2 | N/A | 2 | 14.5 | 1.7 | 23.4 | 646 | 97.9% | 0.80 | 0.79 |
| DP | 0.2 | 0.2 | 0.2 | 2.6 | 5.7 | 78.8 | 602 | 94.9% | 1.13 | 1.07 |
| DP | 0.2 | 4 | 4 | 4.8 | 1.6 | 23.4 | 1082 | 98.5% | 1.13 | 1.11 |
| Spiral | 0.5 | 0.2 | N/A | 1.3 | 72.4 | 160.8 | 346 | 90.8% | 7.87 | 7.15 |
| Spiral | 0.5 | 4 | N/A | 2.1 | 14.9 | 37.4 | 695 | 98.1% | 7.87 | 7.72 |
| Solenoid | 0.5 | N/A | 0.2 | 10.5 | 23.3 | 31.4 | 345 | 95.3% | 4.96 | 4.72 |
| Solenoid | 0.5 | N/A | 2 | 14.5 | 12.7 | 28.0 | 540 | 97.4% | 4.96 | 4.83 |
| DP | 0.5 | 0.2 | 0.2 | 2.6 | 64.4 | 142.6 | 333 | 90.7% | 6.92 | 6.28 |
| DP | 0.5 | 4 | 4 | 4.8 | 21.5 | 50.2 | 504 | 96.7% | 6.92 | 6.69 |
The optimised $d_r$ for the spiral coil and the DP coil is chosen at 4 mm because ηt cannot be improved significantly as $d_r$ increases further, while the size of the coil can increase $d_r$amatically. As shown, the maximum MCEs are high for the low current scenario ($i_r$ = 0.2) due to the low equivalent resistances and high quality factors. The spiral coil with an optimal inter-turn gap has the highest $η_t$ at both low and high current as it experiences lower AC power losses and thus higher k and Q. Overall, increasing the inter-turn gap has stronger impact on the spiral coil and the DP coil compared to the solenoid coil. It is because increasing the gap between the stacked HTS coil structure can both improve the coupling coefficient and reduce their AC power losses. For the solenoid coils, however, the mutual inductance between the resonant coils and the coupling coefficient can reduce significantly with increasing inter-turn gap.
3.4.3. Magnetic flux densities for three HTS coils with varying tape widths
The magnetic flux density generated by three HTS coils at the air–gap centre are presented in Fig. 26 (a), (b), and (c), respectively. The input current is set at$I_t$ = 50 A, with f=85 kHz. An evident reduction in the magnetic flux density can be seen in the solenoid coil with larger tape width as the size of the solenoid coil increases $d_r$amatically. $B_{peak}$ at 2 mm tape width is about 1.7 times $B_{peak}$ at 12 mm tape width for the solenoid coil. The DP coil shows a similar phenomenon as the tape width increases, with $B_{peak}$ reduced by about 20%. The magnetic flux density is kept constant for the spiral coil at different tape widths, indicating that the impact of increasing the tape width to the spiral coil is negligible.
Fig. 26. Magnetic flux density along the r direction with $I_t$ = 50 A at z = 50 mm. (a) Spiral coil, (b) solenoid coil, (c) DP coil. |
3.4.4. Magnetic coupling efficiencies for three HTS coils with varying tape widths
The impact of the reduced AC power losses on the equivalent resistances and the coil-to-coil maximum MCEs is also studied. Fig. 27 depicts the variation of Requ and ηt at different tape widths for three HTS coils at $I_t$ = 20 A and 50 A. It can be seen that increasing tape width can be more effective for the high current scenario as it $d_r$amatically reduces the input current ratio. For three coil configurations, the equivalent resistance $R_{equ}$ reduces with tape width due to the smaller AC power losses generated. Meanwhile, the improvement in the maximum MCEs is noticeable as $w_{tape}$ increases from 2 mm to 6 mm. As tape width increases further, the increment for Requ and $η_t$ are relatively small. With stronger coupling, the spiral coil has the highest $η_t$ among the three HTS coils. On the contrary, the solenoid coil shows the lowest $η_t$ even though it has the minimum AC power losses due to the weak coupling performance. The efficiency of the DP coil ranks between the spiral and the solenoid coils, while it also experiences the highest losses among the three coils.
Fig. 27. Equivalent resistances and maximum MCEs for three HTS coils at different tape widths. |
Table 4 illustrates the parameters for three HTS coils in the model, and Table 5 summaries the performance of three HTS coils with non-optimised tape widths and optimised tape widths at$I_t$ = 20 A and 50 A. With increasing tape width, all three HTS coils show improved maximum MCEs. At $I_t$ = 20 A the spiral coil has the highest ηt among all three coils with the minimum coil volume. Even with minimum AC losses, the solenoid coil requires a coil size around seven times the size of the spiral coil. Meanwhile, the solenoid coil also presents the minimum $η_t$ due to the weak magnetic coupling between resonant coils. The DP coil has a medium coil size; however, it generates the highest AC power losses.
Table 4. Simulation model parameters of three HTS coils at different tape widths. |
| Coil Units | $w_{tape}$ (mm) | L (μH) | M (μH) | C (nF) | k | $l_{coil}$ (m) |
|---|---|---|---|---|---|---|
| Spiral | 4 | 25.8 | 5.13 | 136 | 0.20 | 5.73 |
| Spiral | 12 | 25.8 | 5.13 | 136 | 0.20 | 5.73 |
| Solenoid | 4 | 15.2 | 2.20 | 230 | 0.15 | 5.03 |
| Solenoid | 12 | 9.4 | 1.17 | 371 | 0.12 | 5.03 |
| DP | 4 | 25.2 | 3.80 | 139 | 0.15 | 5.32 |
| DP | 12 | 22.8 | 3.08 | 154 | 0.14 | 5.32 |
Table 5. Comparison of three HTS coils at different tape widths. |
| Coil shapes Units | $I_t$ (A) | $w_{tape}$ (mm) | Volume (×10-4m3) | AC losses (W/m) | $R_{equ}$ (mΩ) | Q | $η_t$ | $P_{in}$ (kW) | $P_{out}$ (kW) |
|---|---|---|---|---|---|---|---|---|---|
| Spiral | 20 | 4 | 2.1 | 0.50 | 14.2 | 1828 | 99.4% | 0.96 | 0.95 |
| Spiral | 20 | 12 | 6.3 | 0.31 | 8.9 | 2905 | 99.6% | 0.96 | 0.95 |
| Solenoid | 20 | 4 | 18.9 | 0.38 | 8.7 | 1343 | 98.8% | 0.32 | 0.31 |
| Solenoid | 20 | 12 | 39.0 | 0.16 | 4.1 | 2045 | 99.4% | 0.32 | 0.31 |
| DP | 20 | 4 | 4.8 | 0.66 | 17.6 | 1440 | 98.9% | 0.64 | 0.63 |
| DP | 20 | 12 | 11.1 | 0.39 | 10.3 | 2220 | 99.4% | 0.64 | 0.63 |
| Spiral | 50 | 4 | 2.1 | 6.21 | 28.4 | 916 | 98.8% | 4.90 | 4.84 |
| Spiral | 50 | 12 | 6.3 | 3.82 | 17.4 | 1492 | 99.1% | 4.90 | 4.86 |
| Solenoid | 50 | 4 | 19.8 | 4.64 | 18.7 | 701 | 97.6% | 1.94 | 1.89 |
| Solenoid | 50 | 12 | 39.0 | 1.54 | 6.2 | 1359 | 99.2% | 1.94 | 1.92 |
| DP | 50 | 4 | 4.8 | 8.80 | 37.6 | 675 | 97.8% | 4.19 | 4.10 |
| DP | 50 | 12 | 11.1 | 4.15 | 17.7 | 1289 | 98.9% | 4.19 | 4.15 |
3.4.5. Discussion regarding the system efficiency
When considering a WPT system using HTS coils, a refrigeration system is always needed to keep the operating temperature below critical temperature. Considering the refrigeration system, the revised electrical efficiency is presented as Eq. 20. The term electrical efficiency in this paper refers to the power transmission efficiency when both AC losses and cryostat power dissipation are considered. Then the electrical efficiencies ($\eta_{\mathrm{t}^{\prime}}$) for three HTS coils are illustrated in Table 6. Here, three HTS coils with optimised coil design are chosen, with a gap of 100 mm. The operating frequency is chosen at 85 kHz according to the standard. Also, it is assumed that the current amplitudes in both coils are similar. The load is set at the optimal load resistance to achieve maximum electrical efficiency.
Table 6. HTS-WPT system with optimal coil structure. |
| $I_t$ (A) | $P_{in}$ (W) | $P_{out}$ (W) | $P_{cool}$ (W) | $\eta_{\mathrm{t}^{\prime}}$ | $\eta_{\mathrm{s}^{\prime}}$ |
|---|---|---|---|---|---|
| Spiral coil | |||||
| 20 | 2550 | 2470 | 1679 | 58.4% | 57.2% |
| 50 | 7870 | 7150 | 14653 | 31.7% | 30.7% |
| Spiral coil-optimal | |||||
| 20 | 960 | 950 | 71 | 92.1% | 91.5% |
| 50 | 4900 | 4860 | 871 | 84.2% | 83.8% |
| Solenoid coil | |||||
| 20 | 802 | 770 | 644 | 53.2% | 52.1% |
| 50 | 4960 | 4720 | 4688 | 48.9% | 47.8% |
| Solenoid coil-optimal | |||||
| 20 | 320 | 318 | 32 | 90.3% | 88.2% |
| 50 | 1940 | 1920 | 310 | 85.3% | 84.0% |
| DP coil | |||||
| 20 | 1130 | 1070 | 1149 | 47.0% | 45.8% |
| 50 | 6920 | 6280 | 12983 | 31.6% | 30.6% |
| DP coil-optimal | |||||
| 20 | 640 | 632 | 83 | 87.4% | 86.9% |
| 50 | 4190 | 4150 | 909 | 81.5% | 81.1% |
It should be pointed out that the current lead losses are not considered in the calculation of the electrical efficiency as it is difficult to accurately quantify the current lead losses: they are dependent on the chosen materials (e.g., a purer copper with a higher residual resistance ratio has a very high conductivity), the operating temperature, and the form of materials (Litz wires vs. solid bars). However, considering the conclusions $d_r$awn in [14], the current lead losses can be comparable to the coil AC losses with a ratio approaching 1:1. Therefore, the estimated entire system efficiency ($\eta_{\mathrm{s}^{\prime}}$) is also presented in Table 6. It can be seen that when considering the current lead losses here equal to the AC losses of the HTS coils, $\eta_{\mathrm{s}^{\prime}}$ will not decrease significantly compared to the electrical efficiency because the majority of the losses are cooling losses. Nevertheless, when the current lead losses are much larger than the AC losses, then the entire system efficiency will $d_r$op significantly. For example in the spiral coil with optimal coil design and 50 A input current, when considering the current lead losses as equal to the coil AC losses, the efficiency decreases from 84.2% to 83.8%. When considering current lead losses much larger (say 10 times) than the AC losses, the efficiency will decrease $d_r$amatically from 84.2% to 77.6%.
Similar results from simulations and experiments regarding the power transmission efficiencies can be found in the literature [33], [46], [56]. Researchers has simulated the system efficiencies at non-optimal HTS coil conditions and the system achieves efficiencies up to 62.17% at different power level considering the cooling penalty [33]. In [46], a whole system DC-to-DC power transfer efficiency of 97.44% has been achieved for the two-coil HTS-WPT system when the cooling penalty is not considered. Generally, the numerically analysed results presented in this paper comply well with the experimentally measured HTS-WPT system efficiency range.
From Table 6 it can be seen that, with optimal coil design, the electrical efficiency for all three systems increases $d_r$amatically. With $I_t$ = 20 A, the highest $\eta_{\mathrm{t}^{\prime}}$ at is observed in the 12 mm tape-width spiral coil with a 4 mm inter-turn gap. Nevertheless, when cooling penalty is taken into account, the efficiency is greatly affected for the proposed system. In general, the electrical efficiency decreases rapidly due to the cooling penalty and the power consumption by the refrigeration system. According to the SAE standard, the minimum target efficiency of the WPT system should be higher than 85%, which is hard to achieve for high power applications (50A). For the low power applications (20A) however, system need to be designed carefully to minimise other system losses such as ferrite losses and power electronics losses.
4. Conclusion
In this paper, AC loss mitigation for HTS coils exploited in WPT has been studied in the typical frequency range, by increasing the inter-turn gap and tape width. For each loss mitigation method, three typical coil layouts, namely the spiral coil, the solenoid coil and the double pancake coil, were simulated and analysed based on an experimentally validated 2D axisymmetric multi-layer HTS coil model. The study has been focused on the AC loss characteristics, loss distributions in each turn, the electromagnetic performances, as well as the overall efficiency of the HTS-WPT system.
The simulation results have demonstrated the effectiveness of the two proposed approaches for AC power loss reduction in the HTS coils for WPT applications. Sufficient loss reduction can be achieved for HTS coils when subject to small external magnetic fields at f = 85 kHz for both methods. However, when subjected to larger external magnetic fields ($B_{e}>B_{\text {thre }}$), simulation results have shown that these two methods could increase the total power losses generated in the HTS coils. With the proposed 0.1 m power transmission distances, the low external magnetic field was attainable under WPT conditions. Among the three coils, the spiral coil has achieved the highest loss reduction at 84.4% as the inter-turn gap increased from 0.2 mm to 8 mm at half current ratio. With tape width increasing from 2 mm to 12 mm, the solenoid coil showed the highest loss reduction at 86.2% with a 50 A input current.
Furthermore, the loss distribution studies have illustrated that increasing inter-turn spacings reduced the turn-to-turn loss deviation. While with increasing tape width, the turn-to-turn loss differences increased due to the decreasing aspect ratio. It is observed that both loss reduction methods decrease the magnetic coupling between resonant coils for the solenoid and DP configurations due to the increasing coil structure in the power transmission(vertical) direction. In contrast, the spiral coil is less affected because each turn is tacked in the horizontal direction. This result indicates that properly increasing the inter-turn gaps or tape width for the spiral coil can help reduce its loss dissipation while maintaining the coupling between the resonant coils at the same time.
In terms of the optimal coil design, the estimated maximum electrical efficiencies considering both the cryostat losses and HTS coil AC losses have been demonstrated. For all three coils, using the optimal coil structure can significantly increase their electrical efficiencies due to the huge reduction of cooling losses. Results showed that the spiral coil could achieve the highest $\eta_{\mathrm{t}^{\prime}}$′ (92%) at 20 A input current with minimum coil size subjected to 12 mm tape width and 4 mm inter-turn gap. The AC losses and the required cooling power from the solenoid coil were the smallest. Nevertheless, at optimal structure condition, the solenoid coil also has the maximum volume. The DP coil is not recommended here as it experiences the highest AC power losses and lowest $\eta_{\mathrm{t}^{\prime}}$ among the three HTS coils. Results also show that the application of HTS coils in high power level is not practical considering the cooling penalty can increase rapidly with the AC loss at high power level. When considering the lead current losses, the entire WPT system efficiency can $d_r$op significantly: when the current lead losses are 10 times higher than the HTS coil AC losses, the total system efficiency will $d_r$op below 80% even for the optimal HTS coil group design.
In summary, it is believed that this paper can provide useful guidance for the loss reduction and efficiency improvement of an HTS-WPT system. It also opens the way to more practical and efficient wireless energisation applications aimed at future net zero transport.
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 supported by the IEEE Council on Superconductivity Graduate Study Fellowship in Applied Superconductivity.
Appendix A. Efficiency model for the HTS-WPT system
For the two coil HTS-WPT system as shown in Fig. 28, the total impedance of the transmitting coil and the receiving coil can be expressed as:
$Z_{\mathrm{t}}=R_{\mathrm{t}}+j \omega L_{\mathrm{t}}-j / \omega C_{\mathrm{t}}$
$Z_{\mathrm{r}}=R_{\mathrm{r}}+R_{\mathrm{L}}+j \omega L_{\mathrm{r}}-j / \omega C_{\mathrm{r}}$
where ω is the angular frequency and $\mid \omega=2 \pi f$.
Fig. 28. Equivalent circuits model for two coils WPT system with series - series compensation topology. |
With the AC power losses calculated from the FEM model, equivalent resistance for the transmitting and receiving coil can be expressed as:
$R_{\mathrm{t}}=\frac{P_{\mathrm{t}} l_{\text {coil }}}{I_{\mathrm{t}}^{2}} \approx \frac{\left(P_{\mathrm{UTS}}+P_{\text {Cu }}\right) l_{\text {coil }}}{I_{t}^{2}},$
$R_{\mathrm{r}}=\frac{P_{\mathrm{x}} l_{\text {coil }}}{I_{\mathrm{r}}^{2}} \approx \frac{\left(P_{\mathrm{HTS}}+P_{\mathrm{Cu}}\right) l_{\text {coil }}}{I_{t}^{2}}$
Here $c_{oil}$ is the length of the HTS tape used to build the coil, $P_{HTS}$ is the power losses from the HTS layer and $P_{Cu}$ is the power losses from copper layers. The losses are in unit W/m. The Ag and substrate layer losses are ignored here.
When the HTS coil is applied with AC input current $I_t$, the total active power supplied by the power source and the active power consumed by the load $R_L$ can be expressed as:
$P_{\mathrm{t}}=\left|I_{\mathrm{t}}\right|^{2} \operatorname{Re}\left(Z_{\mathrm{t}}^{\prime}\right)=\left|I_{\mathrm{t}}\right|^{2}\left(R_{\mathrm{t}}+(\omega M)^{2}\left(R_{\mathrm{r}}+R_{\mathrm{L}}\right) /\left|Z_{\mathrm{r}}\right|^{2}\right),$
$P_{\mathrm{L}}=\left|I_{\mathrm{r}}\right|^{2} R_{\mathrm{L}}=\left|I_{\mathrm{t}}\right|^{2}(\omega M)^{2} R_{\mathrm{L}} /\left|Z_{\mathrm{r}}\right|^{2}.$
Here M is the mutual inductance between the two HTS coils. Divided (15) by (16), the power transfer efficiency from transmitting coil to the load can be expressed by:
$\eta_{\mathrm{t}}=\frac{P_{\mathrm{L}}}{P_{\mathrm{t}}}=\frac{R_{\mathrm{L}}(\omega M)^{2}}{R_{\mathrm{t}}\left|Z_{\mathrm{I}}\right|^{2}+\left(R_{\mathrm{I}}+R_{\mathrm{L}}\right)(\omega M)^{2}}$
By using resonant turning units, the receiving side can be adjusted to pure resistive component and thus $Z_{\mathrm{r}}=R_{\mathrm{r}}+R_{\mathrm{L}}$. Then (17) becomes:
$\eta_{\mathrm{t}}=\frac{P_{\mathrm{L}}}{P_{\mathrm{t}}}=\frac{R_{\mathrm{L}}(\omega M)^{2}}{R_{\mathrm{t}}\left(R_{\mathrm{t}}+R_{\mathrm{L}}\right)^{2}+\left(R_{\mathrm{t}}+R_{\mathrm{L}}\right)(\omega M)^{2}}$
Another quick way to estimate the maximum power transmission efficiency between the resonant coils is
$\eta=\frac{k^{2} Q_{1} Q_{2}}{\left(1+\sqrt{1+k^{2} Q_{1} Q_{2}}\right)^{2}} \approx 1-\frac{2}{k Q}$
where Q is the quality factor of the individual coils and k is the coupling coefficient [13].
In a real HTS-WPT system, a cooling module is necessary to maintain the operating temperature of the HTS coil below the critical temperature. According to the Carnot-cycle theory, a typical refrigerator efficiency around the nitrogen temperature zero is $\eta_{\text {Carnot }}=34.5 \%$. Consider other power losses from a small-midsize cryostat with efficiency at 10% - 20%, a typical total efficiency of η=5% can be achieved [21]. The 5% total efficiency indicates that a power of 20 W (defined as Specific cooling power, $k_{\mathrm{c}}=1 / \eta$) is required to remove 1 Watt heat in the cryogenic environment. When the cooling power $P_{cool}$ is considered, the revised efficiency is given by:
$\eta_{\mathrm{t}}^{\prime}=\frac{P_{\mathrm{L}}}{P_{\mathrm{t}}+P_{\mathrm{copll}}}=\frac{R_{\mathrm{L}}(\omega M)^{2}}{R^{\prime}\left(R_{\mathrm{r}}+R_{\mathrm{L}}\right)^{2}+\left(R_{\mathrm{r}}+R_{\mathrm{L}}\right)(\omega M)^{2}}$
where $R^{\prime}=R_{\mathrm{t}}+k_{\mathrm{c}}\left(R_{\mathrm{t}}+R_{\mathrm{r}}\right)$.
Appendix B. Inductance calculation
B.1. Mutual inductance
This section presents the calculation of the mutual inductance between the transmitting and the receiving coil. Two single-turn HTS coils are illustrated in Fig. 29. Set the centre of coil$C_1$ as the origin of the coordinate, the radius for coil C1 and $C_2$ are defined as $R_1$ and $R_2$,$d_{gap}$ is the gap distance between two coils. $D_l$ is the lateral misalignment between two coils. θ represents the direction of the secondary coil in the xy plane.
Fig. 29. Schematic diagram of two single-turn coils for mutual inductance calculation. |
The mutual inductance between two coils can be calculated with Neumann’s equation as
$M=\frac{\mu_{0}}{4 \pi} \oint_{C_{1} C_{2}} \oint_{\mathrm{dl}_{c 1} \cdot \mathrm{dl}_{c 2}}^{R}$
where $\mathrm{dl}_{\mathrm{cl}}$ and $\mathrm{dl}_{\mathrm{c2}}$ are the wire elements for coil $C_1$ and $C_2$. Define the centre point of $C_2$ as ($D_{1} \cos \theta, D_{1} \sin \theta, d_{\text {gap }}$), $\mathrm{dl}_{\mathrm{cl}}$ and $\mathrm{dl}_{\mathrm{c2}}$ can be expressed by (22), (23), respectively. The distance between the wire element R can be expressed by (24). Angle α and β represent the directions for two wire elements.
$\mathrm{dl}_{c 1}=\left(-R_{1} d \alpha \sin \alpha, R_{1} d \alpha \cos \alpha, 0\right)$
$\mathrm{dl}_{c 2}=\left(-R_{2} d \beta \sin \beta, R_{2} d \beta \cos \beta, 0\right) \text {, }$
$R=\sqrt{A^{2}+B^{2}+d_{\text {gap }}^{2}}$
where $A=D_{1} \cos \theta+R_{2} \cos \beta-R_{1} \cos \alpha$ and $B=D_{1} \sin \theta+R_{2} \sin \beta-R_{1} \sin \alpha$. By substituting (22)–(24) into (21), the mutual inductance between the two coils is obtained as:
$M=\frac{\mu_{0}}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{2 \pi} \frac{R_{1} R_{2} \cos (\alpha-\beta)}{\sqrt{K+T-2 R_{1} R_{2} \cos (\alpha-\beta)}} d \alpha d \beta$
where $K=D^{2}+R_{1}^{2}+R_{2}^{2}+d_{\text {gap }}^{2}$ and $T=2 R_{2} D \cos (\beta-\theta)-2 R_{1} D \cos (\alpha-\theta)$.
For two coils with multiple turns, the mutual inductance should be the sum of the mutual inductance between each pair of turns from the transmitting coil to receiving coil. The equation can be expressed as:
$M_{\text {total }}=\sum_{i=1}^{I} \sum_{j=1}^{J} M_{i, j}$
where I is the turn number for the transmitting coil, and J is the turn number for the receiving coil.
B.2. Self-inductance
This section demonstrates the calculation of the self-inductance for the circular coil based on Bio-savart’s law. For one turn coil as shown in Fig. 29, the perpendicular magnetic field in random point P inside the coil can be calculated using Bio-savart’s law as
$B_{\text {perp }}\left(a_{\mathrm{d}}, \varphi\right)=\frac{\mu_{0} I_{1}}{4 \pi} \int_{0}^{2 \pi} \frac{R_{1}^{2}-R_{1} \alpha_{\mathrm{d}} \cos (\theta-\varphi)}{\left[R_{1}^{2}+a_{\mathrm{d}}^{2}-2 R_{1} a_{\mathrm{d}} \cos (\theta-\varphi)\right]^{\frac{3}{2}}} d \theta$
where $a_d$ is the distance between P and the coil centre, φ is the angle of Point P.
The magnetic flux linkage within the coil Φ can be calculated as
$\Phi=\iint_{S} B_{\text {perp }}\left(a_{\mathrm{d}}, \varphi\right) \cdot d \vec{S}$
Substituting (27) into (28), the self-inductance for a single-turn coil can be calculated as
$L=\frac{\mu_{0}}{4 \pi} \int_{0}^{R_{1}} \int_{0}^{2 \pi} \int_{0}^{2 \pi} \frac{a_{\mathrm{d}} R_{1}^{2}-R_{1} a_{\mathrm{d}}^{2} \cos (\theta-\varphi)}{\left[R_{1}^{2}+a_{\mathrm{d}}^{2}-2 R_{1} a_{\mathrm{d}} \cos (\theta-\varphi)\right]^{\frac{3}{2}}} d \theta d \varphi d a_{\mathrm{d}}.$
For coil with multiple turns, the equivalent self-inductance is calculated as the summation of the self-inductance of each turn and the mutual inductance between each pair of turn
$L_{\text {coil }}=\sum_{i=1}^{I} L_{i}+\sum_{i=1}^{I} \sum_{j=1}^{J} M_{i j}.$
The mutual inductance between each turn within the coil in (30) can be calculated by (25).
B.3. Coupling coefficient and Quality factor
The value of coupling coefficient k and mutual inductance M are fundamental parameters that indicate the degree of the coupling strength between two resonant coils. The coupling coefficient k is a value between zero and one. In a WPT circuit, a larger mutual inductance M facilitate the effectiveness of the system. The coupling coefficient k and mutual inductance M can be defined as
$k=\frac{M}{\sqrt{L_{\mathrm{t}} L_{\mathrm{x}}}}$
$M=k \sqrt{L_{\mathrm{t}} L_{\mathrm{r}}}$
where $L_t$ is the self-inductance of the transmitting coil and $L_r$ is the self-inductance of the receiving coil.
The quality factor Q for resonant coils can be described as:
$Q=\frac{\omega L}{R}=\frac{2 \pi f L}{R}$
where Q is the quality factors of the coil, R is the self-resistances of the coil.
