A full-range formulation for dynamic loss of high-temperature superconductor coated conductors

Dynamic loss is significant for evaluating the performance of type-II high-temperature superconductor (HTS) coated conductors (CC), especially when used in electric machines. Although analytical expressions of dynamic loss have been proposed and verified for HTS CCs exposed to low external magnetic fields with a low current load rate, the non-linearity of dynamic loss at high current load ratios and simultaneous high magnetic fields is still unclear and cannot be predicted by the existing analytical equations. This paper proposes a completely new formulation to characterize the non-linearity of dynamic loss taking into account the Jc(B) dependence of CCs, which can be used for full ranges of both magnetic fields and current load ratios. The proposed analytical formulas have been verified by a T-formulation based numerical model and experiments. This paper comprehensively demonstrates the variation of dynamic loss and provides a meaningful reference of loss controlling for HTS CCs, which is vital for HTS machine windings to avoid quench.


Introduction
Dynamic loss has prompted widespread concern recently in the domains related to second generation (2G) high-temperature superconductor (HTS) coated conductors (CC) [1][2][3][4][5]. It is generated when HTS CCs carry a direct current (DC) in an alternating current (AC) magnetic field, which is tightly relevant for machine windings and other power applications. The widely adopted analytical expression for dynamic loss per unit time, Q dyn,l , can be written as [4] = - where w is the half-width of the HTS CC, f is the frequency of the AC magnetic field, L is the length of the CC, I t represents the transport current, and i denotes the load ratio between I t and Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. the self-field critical current, I c0 . B ext is the amplitude of the external AC magnetic field, and B th is the threshold field, defined by [6]  where μ 0 is the free space permeability, h is the thickness of the HTS thin film, and J c0 is the critical current density in selffield, determined by I c0 /(2wh).
The linearity of dynamic loss with respect to B ext , as shown in (1), has been well verified by many studies [7][8][9]. However, recent experimental measurements have shown that, when the HTS CC with a high i is exposed to a high external magnetic field, the dynamic loss exhibits a nonlinearity with the augmentation of field intensity and the CC is under the risk of quench [10][11][12]. A critical-state model for dynamic resistance has been proposed to describe this nonlinearity in [8]. When B ext is far greater than B th , dynamic loss can be written as [8] = where B 0 is a constant depending on the material property.
However, it appears that the measured dynamic resistance rises much faster than the analytical results obtained by (3) [8]. Therefore, for now, the existing analytical expressions are inapplicable to characterize the non-linearity of dynamic loss, and the magnetic field and current load ratio causing the sudden rise in loss are still unclear. In [9], a time-averaged DC flux flow resistance has been introduced to explain the non-linear fast rise of dynamic resistance, which provides a good description of the deviation from linearity at high currents. On the basis of this work, we have studied the current density and magnetic flux density distributions inside the HTS CC in detail, and propose a new formulation for dynamic loss, which considers a full-range of magnetic fields and load ratios. This paper helps better understand the variation properties of dynamic loss from the perspective of theoretical fundamentals and determine its influencing factors, which is significant for avoiding quench in HTS machine windings, etc.

New formulation of dynamic loss
When a HTS CC carries a DC transport current under an AC magnetic field, the transport current occupies the superconducting layer with width 2iw in the center of the CC (named dynamic region), leaving the rest with width 2(1−i)w free on both sides [10]. Therefore, the dynamic loss per unit time can be formulated by where E represents the electric field, and J is the current density along the width of the coated conductor.
Considering the field dependence of the critical current density J c (B) is the local magnetic field perpendicular to the wide surface of the CC. B ⊥ is decided by both B ext and the self-field B s . The negative sign indicates the direction of the field vector. Although the current density and magnetic distribution along the width of the CC has been studied by Brandt, it is not feasible to calculate the integration in (7) directly by analytical methods [13]. However, the non-linear fast rise of dynamic loss occurs when the real load ratio I t /I c (B) is approaching 1, i.e., when the CC is in a critical state. In this case, we can simplify (7) by investigating the current density and magnetic flux density characteristics in the dynamic region.
Firstly, the J properties need to be studied. According to the Bean Model, when the superconductor is in a critical state, the current density distribution along the width of the HTS CC should be characterized by the critical current ±J c . Here, concerning dynamic loss, we assume that the dynamic region is fully occupied by I t . To verify this assumption, the T-formulation based numerical modeling method has been adopted here [14,15]. We define the average current density during one cycle T in the dynamic region, J avg , as During the simulation, the transport current I t has been fixed as 60 A. When I c0 decreases from 160 A to 80 A (i.e. i increases from 0.375 to 0.75), the relationship between i and J avg /J t under different B ext is presented in figure 1. Here, J t is the average current density determined by the transport current, with J t =I t /(2wh). It can be seen that with the increase of i, J avg will get close to J t . In other words, the dynamic region with the width of 2iw is the effective region to carry transport current and in the critical state, the dynamic region has been filled with I t . Therefore, the current density distribution in the dynamic region should be decided by J t .
Next, the magnetic flux density properties in the dynamic region need to be explored. In Based on the above analysis, J(y) is determined by J t , so (10) can be simplified as Then the average self-field flux density in the dynamic region can be calculated by According to (2), it is evident that B th decreases with i. Therefore, at a high load rate, B s becomes much smaller compared to B ext , i.e. B ⊥ will be dominated by the external field. In this case, we have Based on the above formulas, (7) can be transformed to Based on binomial theorem and Euler's formula, through a series of derivations, the details of which can be found in appendix, (14) can be written as (16). It should be emphasized that Q dyn,nl in (16) is derived on the basis that the HTS CC is at relatively high i and simultaneous high B ext , which can only be used to characterize the non-linearly fast-rising part of the In (1), Q dyn,l is used to describe the linear correlation between dynamic loss and B ext . There will be a cross point between the Q dyn,l (B ext ) and Q dyn,nl (B ext ) curves as at low external magnetic field, B ext is much smaller than B 0 , thus the n-value does not play an important role and Q dyn,nl will approach 0. However, at high external field, B ext becomes comparable to B 0 , then under the influence of the n-value, Q dyn will increase rapidly in the form of a power function. In short, before the cross point, Q dyn,l is far greater than Q dyn,nl , and vice versa. On the basis of the above analysis, we have proposed a full-range analytical equation for calculating dynamic loss, as (17). It should be noted that, in (17) n is even. When n is odd, we need to modify the upper bound of summation accordingly. To verify the correctness of (17), both numerical modeling and experimental methods have been adopted. In addition, based on the time-averaged DC flux-flow resistance term mentioned in [9], Q dyn can also be fitted by .

Numerical modelling
The numerical model of the HTS CC was developed by use of finite element method based on the T-formulation [14,15]. The diagram of the modelled HTS CC is shown as figure 2.
The governing equation can be written as [4] where l en is the width of each element, T en e is the current vector potential in each element, r en is the distance between the current source element and the calculation point, nʹ is the normal vector at the current source element, n is the normal vector at the calculation point, and σ is the equivalent conductivity determined by the E-J power law, σ=J/E.

Experiment
Two measurement systems were used for the experimental results [3,9]. One of the experimental setups for measuring Q dyn is shown in figure 3. It is composed of a custom-built AC magnet, which can produce a flux density up to 100 mT, a DC power supply which provides I t varying between 0-240 A, a nano-voltage meter for averaged voltage measuremnt across the voltage taps attached on the middle of the sample conductor, and a cryostat containing liquid nitrogen. Q dyn is obtained by multiplying the measured voltage across the sample with the transport current.
Two different YBCO CCs have been tested, of which the specifications are shown in table 1.

Results and analyses
For the HTS CCs with different I c0 varying from 80 to 160 A while carrying I t =60 A, they are exposed to AC external magnetic fields with increasing amplitude between 10-500 mT. The simulated Q dyn (solid symbols) and the analytical results calculated by (1)     It shows that there exists a cross-field between the Q dyn,l (B ext ) and Q dyn,nl (B ext ) curves, as mentioned above. Before the cross-field, Q dyn (B ext ) agrees well with (1), which shows a linear correlation. After the cross-field, Q dyn (B ext ) is in good accordance with (16), which is in the form of a summation of power functions. In general, the simulated Q dyn (B ext ) agrees well with (17) in the full magnetic field range. Although (18) also describes the non-linearity of Q dyn , it appears to be much larger in terms of value. (17) has been derived through a rigorous process, providing a more accurate result. Furthermore, (18) is given based on the estimation of the time-average DC flux flow resistance, which cannot intuitively account for the non-linearity of Q dyn in the form of a summation of power functions. With regard to the cross-field, it increases inversely with i. In fact, for a lower i the CC has a larger capacity to withstand the induced current by the external magnetic field, thus it will be harder for I t to go beyond the real critical current I c (B). Therefore, the crossfield increases with I c0 for a fixed I t , which complies with the J c (B) dependence and the conclusions drawn in [4].
The simulated Q dyn based on T-formulation, the measured experimental data, and the analytical results obtained by (1), (3), (16) and (17) for the two tested CC samples are presented together in figure 5. It can be found that the experimental data is in good agreement with the simulation results and the proposed equation (17). The Q dyn (B ext ) curve exhibits an evident nonlinearity with the increase of the external magnetic field, which is significantly different from the previously widely-adopted analytical equations (1) and (3). It should be pointed out that the relatively poor agreement between the simulation and experiment at low magnetic fields for the CC sample with I c0 =87.67 A is due to its effective width, which is different from its physical width [3]. However, it does not affect the validity of (17), in that it describes well the non-linearity of Q dyn under high magnetic fields.
With respect to computational complexity, analytical methods generally require far less computing time and resources than the finite element method (FEM) based numerical modelling. In our case, (17) can be simply calculated in a few seconds using MATLAB. However, it requires more time and effort to build and run a FEM model. For example, the computational time to obtain Q dyn of a single HTS layer with the T-formulation based numerical model is tens of seconds or longer, depending on the solving accuracy and number of mesh elements. Therefore, the proposed formulation (17) is a more efficient and convenient way to calculate dynamic loss compared with numerical models.

Conclusion
Based on the J c (B) dependence and E-J power law, through investigation of the characteristics of current density and magnetic flux density distributions in the dynamic region of HTS CCs, a new analytical expression for dynamic loss has been proposed. This expression considers a full-range of magnetic fields and load ratios. At low external fields and simultaneous low load ratios, dynamic loss is in a linear correlation with the external magnetic field. At both high magnetic fields and high load ratios, it is in the form of a summation of power functions. This new formulation can be used to characterize the non-linear variation of dynamic loss, which cannot be accurately predicted and explained by the existing analytical methods. The proposed formulas have been well verified by the T-formulation based numerical modeling method and experiments, which can be achieved in MATLAB  According to [3], the effective width of the CC with I c0 =87.67 A has been chosen as 2w=3.51 mm. and dramatically save computational effort compared with numerical modelling methods. The cross-field has also been studied, of which the variation complies well with the J c (B) dependence. This paper comprehensively demonstrates the variation of dynamic loss at different load ratios and magnetic fields. It can serve as a guide for loss controlling and avoiding quench in the application of HTS CCs, especially with regards to machine windings.