Distribution Of Critical Current Density Research Articles

Effects of lateral critical current nonuniformity on stresses in dry-wound high-field REBCO coils

Abstract The distribution of critical current density (jc) in REBCO coated conductors affects the magnetization behaviors and subsequently screening-current-induced stresses, particularly for solenoid magnets in high fields. This paper studies numerically the correlation between lateral jc profile across conductor width and stress distribution in pancake coils. The modeling framework considers bending, winding, thermal contraction, and magnetic forces including coupled electromagnetic-mechanical behaviors, i.e., the deviation of the perpendicular field away from axial direction due to tilting deformation. The lateral nonuniformity is introduced using trapezoidal functions, emulating typical jc profiles originated in pristine tapes and those caused by slitting. First, parametric studies are carried out on the small test coils previously reported in the 'Little Big Coils' (LBC) paper. It is shown that while slitting edge defects have a moderate impact on peak strains, imperfections in the pristine tape with a larger shoulder width can accelerate the penetration process, shifting peak force to the structurally resilient middle section. Similar behaviors are found in the LBC3 case study, suggesting that lateral jc nonuniformity may have contributed to the observed degradation states in pancakes with different slit-edge orientations. Furthermore, the manipulation of lateral jc profile is proposed as a strategy to manage stresses in high field solenoids. This is demonstrated in the design study of a hypothetical REBCO insert. By adjusting the lateral jc distribution and the overall scaling factor, magnet designs with a reasonable current margin and moderate peak strain can be found. The multi-width concept is then applied to allow for a higher operating current and a larger margin for the end pancakes. Albeit being a generic case, this study highlights the sensitivity of peak stresses in high-field magnets to jc distribution. This feature may be taken into account to fine-tune magnet designs and adjust coil assemblies for better overall performance. It also emphasizes the need for careful characterization and effective control of lateral jc distributions in REBCO coated conductors.

Critical current density and AC magnetic susceptibility of high-quality FeTe0.5Se0.5 superconducting tapes

Iron telluride-selenium superconducting materials, known for their non-toxicity, ease of preparation, simple structure, and high upper critical fields, have attracted much research interest in practical application. In this work, we conducted electrical transport measurements, magneto-optical imaging, and AC magnetic susceptibility measurements on FeTe0.5Se0.5 superconducting long tapes fabricated via reel-to-reel pulsed laser deposition. Our transport measurements revealed a high critical current density that remains relatively stable even with increasing external magnetic fields, reaching over 1 × 105 A/cm2 at 8 K and 9 T. The calculated pinning force density indicates that normal point pinning is the primary mechanism in these tapes. The magneto-optical images demonstrated that the tapes show homogeneous superconductivity and uniform distribution of critical current density. The AC magnetic susceptibility measurements also confirmed their strong flux pinning nature of withstanding high magnetic field. Based on these characteristics, FeTe0.5Se0.5 superconducting tapes show promising prospects for applications under high magnetic field.

Spatial 3D correlation of flux pinning with porosity distribution in YBa2Cu3O7−δ using tensorial neutron tomography

Engineering devices from High Temperature Superconductors (HTS) for practical applications such as in transport and medical imaging requires an understanding of their critical current density (JC) distribution and how the material properties affect this. JC describes the maximum gradient of flux that can be found in a superconductor sustained by vortex pinning which is the preferential positioning of vortex in defective regions of weaker superconductivity. Local correlation of imperfections with high pinning has required destructive methods such as slicing then scanning with local magnetic probes. We describe the first case of non-destructive spatial correlation of flux pinning and consequently JC with bulk imperfections at different temperatures in top-seeded melt grown (TSMG) YBa2Cu3O7−δ chosen for its high pinning in addition to a rich structure of pores, twins and grain boundaries. This is facilitated by a combination of polarized neutron tomography to image trapped magnetic fields in a range around the material critical temperature and conventional neutron tomography to characterize potential pinning objects. The results indicate that there is indeed preferential trapping in the porous regions independent of the resolvable pore size. Polarized optical microscopy data suggests that the observed phenomenon is attributable to twin boundaries and crack defects located at the pore interface.

HTS Cross-Shaped Strand With Copper Former: Improvement of HTS Quasi-Isotropic Strand for Better Twisting Characteristics

Twisting strain is a main reason that leads to the degradation of the critical current of high temperature superconducting (HTS) quasi-isotropic strand (Q-IS) under torsional loads. In this article, an HTS cross-shaped strand (C-SS) is proposed in order to improve the twisting characteristics of Q-IS. We first simulate and measure the critical current of Q-IS and C-SS under self-field at 77 K. Subsequently, the arrangement direction of REBCO tapes in Q-IS and C-SS that can minimize the twisting strain is analyzed and obtained. Finally, the twisting strain distribution and twisting characteristics of Q-IS and C-SS under different torsional loads are analyzed numerically and experimentally. The results show that the critical current of C-SS is slightly less than that of Q-IS without torsional load. However, the critical current density and magnetic field distribution of C-SS show better isotropic properties than that of Q-IS. When torsional loads are exerted on both strands, C-SS presents better twisting characteristics than Q-IS and its normalized critical current degrades slower than that of Q-IS.

Beyond the effective length: How to analyze magnetic interference patterns of thin-film planar Josephson junctions with finite lateral dimensions

The magnetic-field-dependent critical current ${I}_{\text{c}}(B)$ of a Josephson junction is determined by the screening currents in its electrodes. In macroscopic junctions, a local vector potential drives the currents; however, in thin-film planar junctions, with electrodes of finite size and various shapes, they are governed by nonlocal electrodynamics. This complicates the extraction of parameters such as the geometry of the effective junction area, the effective junction length, and the critical current density distribution from the ${I}_{\text{c}}(B)$ interference patterns. Here, we provide a method to tackle this problem by simulating the phase differences that drive the shielding currents and use those to find ${I}_{\text{c}}(B)$. To this end, we extend the technique proposed by Clem [Phys. Rev. B 81, 144515 (2010)] to find ${I}_{\text{c}}(B)$ for Josephson junctions separating a superconducting strip of length $L$ and width $W$ with rectangular, ellipsoid, and rhomboid geometries. We find the periodicity of the interference pattern ($\mathrm{\ensuremath{\Delta}}B$) to have geometry-independent limits for $L\ensuremath{\gg}W$ and $L\ensuremath{\ll}W$. By fabricating elliptically shaped superconductor--normal-metal--superconductor junctions with various aspect ratios, we experimentally verify the $L/W$ dependence of $\mathrm{\ensuremath{\Delta}}B$. Finally, we incorporate these results to correctly extract the distribution of critical currents in the junction by the Fourier analysis of ${I}_{\text{c}}(B)$, which makes these results essential for the correct analysis of topological channels in thin-film planar Josephson junctions.

Study on field-based superconducting cable for magnetic energy storage devices

During the design of superconducting magnets, the Ic(B) curve is widely used to evaluate the critical current density distribution in the magnet, safety factor is determined by the lowest value of critical current density and operating current. However, other parts of the magnets will work in over-safe state, which leads to a huge waste of superconducting properties as well as budget problem. This article presents a Field-based cable to improve the utilizing rate of superconducting magnets in SMES system. The quantity of HTS tapes are determined by the magnetic field distribution. By this approach, the cost of HTS materials can be potentially reduced. Firstly, the main motivation as well as the entire design method are introduced. Then, a typical no-insulation coil is designed and fabricated based on Field-based stacked cable to validate the feasibility of this method. Both charging/discharging test and the magnetic field test are carried out to explore the working mechanism of this coil. The unique working performance on time constant and field distribution are analyzed in detail.

Finite element simulation and structure optimization of HTS solenoid

When the current passing through a high temperature superconducting (HTS) coil that exceeds a critical value, the properties of the materials which make up the coil will fail, generating large amounts of heat and even causing serious accidents. Aiming at the above safety problem, this paper took three solenoid magnets with different structures as the object, and conducted a simulation study on their electromagnetic performance through finite element method (FEM). The magnetic field intensity H was taken as the dependent variable of the control equations of the physical field. With the aid of the partial differential equation (PDE) interface of the simulation software used, the control equations were easily constructed. The pancake coil wound by many turns of ribbon conductors was abstracted as a bulk-like conductor with the same cross-sectional area. The main idea of this equivalent replacement is to simplify the internal structure of the device without affecting its electromagnetic behavior, which can accelerate the convergence speed of the simulation process and reduce the CPU burden. Models of solenoid magnets with rectangular, trapezoidal and inverted trapezoidal cross sections were established by stacking many pancake coils. The simulation results corresponding to these models show that the solenoid magnet with trapezoidal cross-section has the largest critical current and most uniform density distribution. Such advantages not only reduce the risk of superconducting material failure due to overheating at both ends, but also fully exploit the current carrying capacity of the coil in the middle area of the solenoid.

Study of Quench Behavior of No-Insulation REBCO Pancake Considering Complex Critical Current Density Distribution

This study analyzes the quench behavior of No-insulation (NI) REBCO pancake. An quench analysis simulation model considering the complex critical current density distribution was developed and validated experimentally. The developed model includes an equivalent circuit model, a thermal model, a magnetic model, and a magnet critical current estimation model. The simulation model is based on a co-simulation using MATLAB and COMSOL, considering the balance of accuracy and computation time. The lumped parameter of the NI magnet was used to solve the governing equations of the equivalent circuit module and thermal module in MATLAB. To accurately calculate the non-uniform electromagnetic field distribution over the coil, a magnetic finite element method was used in COMSOL. A neural network was used to predict the temperature-field-angle dependent magnet critical current. Considering the numerical convergence at different over-current phases, adaptive simulation was used to reduce the total simulation time. The quench behavior was analyzed using the proposed model, and the results were verified experimentally. The simulation results showed that when the azimuthal current is greater than the magnet critical current, it will gradually saturate and approach the magnet critical current, and then decrease at a similar speed. The rate of decrease is equal to the rate of decrease of the magnet critical current. Through the analyses, the corresponding sudden-discharging behavior was simulated and analyzed. The proposed model can be used to perform the post-quench and sudden discharging analyses of pancake coil and has promising applications in multi-pancake post-quench studies.

The effect of facet lines on critical current density and trapped field in bulk RE–Ba–Cu–O single grains

Bulk, single grain RE–Ba–Cu–O (where RE = rare earth or yttrium) [(RE)BCO] high temperature superconductors could potentially be used to generate stable magnetic fields for magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR). In these applications, however, the homogeneity of the magnetic field is of critical importance. As a result, the spatial distribution of critical current density, J c, within the bulk single grain and the effects of the magnetisation process, which are primary drivers of the uniformity of the achievable trapped magnetic field, are fundamental to assessing the performance of these technologically important materials. This paper reports the systematic measurement of the distribution of J c–B at 77 K over a vertical cross-section of a single grain along a facet line and through the seed crystal [(110)-F] at 20 positions within a 20 mm diameter Gd–Ba–Cu–O sample in an attempt to understand and assess the distribution of J c along this microstructural feature. A comparison of the data within the whole vertical plane across the seed measured along the a or b direction within the [(100)-a] plane shows that J c–B at 77 K at the facet line is more than 10% higher for applied fields between 0.2 T and 2.5 T. The effect of the J c–B relationship of the facet line on the overall trapped field measured in an individual bulk sample was investigated by measuring the magnitudes of trapped fields and their contour maps for sections cut from four single grain samples of GdBCO–Ag at different sizes and shapes parallel to the ab-plane from the top to the bottom of the bulk sample. Based on the results reported here, we demonstrate a method to achieve more uniform trapped fields through an optimal arrangement of an assembly of sections of individual GdBCO single grains.

A Reel-to-Reel Scanning Hall Probe Microscope for Characterizing Long REBCO Conductor in Magnetic Fields Up to 5 Tesla

We have developed and routinely used a reel-to-reel (R2R) scanning Hall-probe microscopy (SHPM) system to characterize REBCO conductors developed at the University of Houston. By mapping the screening-current-induced field of a magnetized REBCO conductor, the SHPM system enables 2-dimensional imaging of critical current density (<i>J</i>_c) distribution in a contact-free, non-destructive approach. Most existing SHPM systems investigate the magnetized REBCO conductors at remnant state without external background field. However, because of the inhomogeneity in the distribution of nanoscale artificial pinning centers in advanced REBCO conductors, there is a demand for reel-to-reel in-field examination over long conductor lengths. We have been developing a new R2R SHPM, capable of measuring the <i>J</i>_c profile of long REBCO conductors at the temperature of 63.5&#x2013;77 K and in an applied field up to 5 T. With optimal data acquisition and processing techniques, we overcame the challenge of measuring weak screening-current-field signals in a strong background field. R2R <i>J</i>_c mapping has been performed on long REBCO conductors with a spatial resolution as low as 0.15 mm² and a longitudinal scanning speed comparable with our existing system. Our previous study revealed a good correlation between the REBCO conductors&#x2019; 65-K in-field <i>J</i>_c and their <i>J</i>_c over a wide span of temperatures and fields. Using this correlation, the in-field R2R SHPM will serve not only as a means of quality control, but also provide insight into high-field performance of advanced REBCO conductors at low temperatures.

J c and Composition Distribution of YBCO Coated Conductors Fabricated by the TFA-MOD Process

The Trifluoroacetate-Metal Organic Deposition (TFA-MOD) process is one of the most promising fabrication processes for low-cost superconducting coated conductors (CCs) because of its high yield and non-vacuum system. However, in the doping and multi-coating process for the enhancement of critical current (<i>I</i>_c), degradation of <i>I</i>_c often occurs. In this study, to clarify the origin of <i>I</i>_c degradation, we have investigated the local critical current density (<i>J</i>_c) distribution, in-field <i>J</i>_c properties and the variation of composition for TFA-MOD processed YBa₂Cu₃O_y (YBCO) CCs. The local <i>J</i>_c distributions were measured by Scanning Hall-Probe Microscopy for confirming the location of the high&#x002F;low-<i>J</i>_c regions. The CCs were then cut into 3 mm&#x00D7;3 mm or 3 mm&#x00D7;4 mm for dc magnetization measurement. The <i>T</i>_c of the all samples were about 87 K to 90 K. In-field <i>J</i>_c at 77 K was obtained from <i>M</i>-<i>H</i> measurements and all the samples were compared with each other. The in-field <i>J</i>_c of high-<i>J</i>_c sample showed about 100 times higher value than that of the low-<i>J</i>_c sample. Then we have observed the microchemistry and the variation of composition by SEM-EDS. In the high-<i>J</i>_c sample, the grains of the superconducting phase were large in size and were connected to each other. This means that superconducting current could form through the whole sample area and generate a large magnetic moment. On the other hand, in the low-<i>J</i>_c sample, though the superconducting phase could be observed, the grains were small in size and sparse. Therefore, the magnetic moment measured was small. However, it was also suggested that the intra-grain <i>J</i>_c would be almost the same as that of high-<i>J</i>_c sample.

Transverse distribution of critical current density in Ag-sheathed Ba1− x K x Fe2As2 tapes

The Ag-sheathed Ba1−x K x Fe2As2 (Ba-122) monofilamentary tapes were prepared by the ex-situ powder-in-tube method. The variation of the microstructure and superconducting properties with the thickness of the superconducting core in the cross-section of sintered tapes is studied. At the same time, the reason is studied in comparison with the unsintered tapes. The research results show that the magnetic J c of the iron-based superconducting tapes increases continuously with distance from the core-sheath interface, which is the complete opposite of the Bi-based superconductor. The magnetic J c of central layers for final Ba-122 tapes is about 33% higher than the Jc of the whole tape at 4.2 K and 7 T. We have found that the center of the superconducting core shows higher hardness and better texture. In addition, it is also found that there is a reaction layer at the Ag-superconductor interface. These reasons may result in the reduction of the critical current density near the interface in the tapes. Moreover, we also found the presence of a reaction layer in the hot-pressed (HP) high-performance samples. However, no unevenness was found in the unsintered samples. Therefore, the superconductivity of Ba-122 tapes will be better by reducing the reaction layer and eliminating inhomogeneity at the core-sheath interface of the sintered tapes.

Characterization of Spatial Distribution of Local Critical Current Density in a Co-Doped BaFe2As2 Film Based on Magnetic Microscopy

We have investigated spatial distribution of local critical current density (J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ) in a Co-doped BaFe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> As <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (Ba122:Co) film based on scanning Hall-probe microscopy (SHPM). In-plane J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> distribution was derived from the magnetic field distribution. Thanks to the high-resolution characterization, local obstacle can be detected clearly, as well as a local spot with extremely high J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> over 7 MA/cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , which is among the highest value at remanent state reported for this series of films so far. The SHPM allows us to separate the influence of current blocking obstacles and intrinsic flux pinning properties on J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> . Validity of the J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> analysis was checked by a magnetization measurement based on a SQUID magnetometer. Moreover, we investigated the J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> distributions at different temperatures by SHPM and studied their probability distribution functions (PDFs). PDFs are found to be scaled by a Weibull function with the same shape parameter, while J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> s are decreasing with higher temperature. These results indicate that the intrinsic in-plane pinning is controlled by the same mechanism.

Electromagnetic Analysis of HTS DC Cables Based on Critical State Model

High-temperature superconducting (HTS) DC cables have the advantages of low transmission loss, high current density and large transmission capacity, which have gained more and more attentions in recent years. The typical structures of HTS DC cables include unipolar and bipolar coaxial that doubles the cable's transmission capacity. Based on the critical state model, this article analyses and contrasts the critical current density distributions and degradations due to magnetic field in the HTS tapes of the unipolar and coaxial bipolar DC cables. The influences of the bipolar cables with different voltage levels are also compared. The results show that the critical current density degradations due to magnetic field are almost the same in both unipolar and coaxial bipolar structures, which are mainly caused by the self-field. For the bipolar cables, the current density distribution in the outer layer HTS tapes changes with the voltage level. To evaluate the influence of the axial magnetic field, a 3-D model of the unipolar cable is established with a helical wound pitch of the HTS tapes. The results show that the axial magnetic field is relatively low and the critical current density would not degrade if the helical angle of the HTS tapes is reasonably small.

Crack-Inclusion Problem in a Superconducting Cylinder with Exponential Distribution of Critical Current Density

In this paper, the crack-inclusion interaction considering the exponential distribution of critical current density is carried out for a functionally graded superconducting cylinder during the zero-field cooling (ZFC) process. By determining the stress intensity factors (SIFs) at the crack tips and performing magnetoelastic stress analysis, the result indicates that the increase of the gradient parameter of critical current density could effectively reduce the stress intensity factors based on the effect of the gradient parameter on the elastic constants ratio of inclusion/matrix, sizes of the inclusion and crack, and the distance between them upon electromagnetic force.

Effects of Zn doping on the pinning potential and the glass-liquid transition temperature of YBa2Cu3O6+δ films

A pristine YBa2Cu3O6+δ film as well as two Zn-doped YBa2(Cu1−x Zn x )3O6+δ films with doping levels x of 0.2 % and 0.4 %, respectively, grown epitaxially on SrTiO3 single crystalline substrates by pulsed laser deposition, were investigated regarding the effect of the doping on the glass-liquid transition line and the pinning potential distribution. Zn doping did not show any effect on the shape of the transition line but caused a suppression of the transition temperature T GL of about 16 K%−1. Although the doping did not lead to an additional pinning effect, the increase of the critical current density j c towards lower temperatures is mainly driven by the variation of the threshold critical current density, j cm. The width of the critical current density distribution j 0 saturates at low temperatures and only drops above ~ 0.9T GL.

Effective Measuring Position of Hall Probe and &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$j_\mathrm{c}$&lt;/tex-math&gt; &lt;/inline-formula&gt; Characterization of Rectangular HTS Thin Films

Scanning Hall probe (SHP) system is a powerful instrument to investigate the planar distribution of areal critical current density (j c ) and defects of high temperature superconducting (HTS) thin film samples. The distance between the effective measuring position and probe bottom (h 1 ) is usually unknown for the commercialized Hall probes. However, h 1 value has a significant influence on the measurement results, according to the calculation of the magnetic flux density (B 0 ) above the planar center of a rectangular HTS thin film sample. In this paper, a method was proposed to obtain h 1 by using a series of copper strips and analyzing the relationship of B 0 against the parameters, such as sample width and current. Then, SHP system was employed to characterize the j c distribution of an HTS thin film fabricated via chemical solution deposition. The relationship of B 0 and h 1 could also be utilized to obtain the average j c value (j c B0 ). A series of commercialized coated conductor samples were measured with both SHP and the traditional four-probe method. The feasibility of preliminary evaluation for rectangular HTS thin film samples by using j c B0 was discussed based on its correlation with the results of four-probe method.

Effects of critical current density on mechanical properties of cylindrical superconductors

High-temperature superconductor has high critical temperature, high transport current capacity and low energy consumption, which correspondingly offer the wide applications in the field of electric power. As an important concern, the mechanical properties of superconductor carried with transport current have received extensive attention. Still, its mechanical properties in various electromagnetic environments are under study. Most of previous studies are based on the assumption of uniform distribution of critical current density, and only few researches based on the non-uniform distribution of critical current density are carried out. In this work, the mechanical flux pinning response of cylindrical superconducting structures is studied. Considering the non-uniform features of critical current density along the radial direction, the distribution law of induced magnetic field and current for the cylindrical superconducting structure is obtained based on the Bean model. Combined with the plane strain method, the analytical expression of magnetic flux pinning force, stress and magnetostriction in the superconducting structure are obtained. The results show that the uneven distribution of critical current density causes the flux pinning force to change, which further leads the superconductor`s local radial stress to vary with the critical current density. When the transport current flowing through the superconductor is increased in the ascending field, the radial stress and the hoop stress both appear compressive. The non-uniform distribution of critical current density has no significant effect on the overall trend of the internal stress, but displays an obvious influence on the stress distribution, and the superconducting structure is compressed and deformed. The results are consistent with those in the uniform case. When the transport current decreases during field descending, the critical current starts to reverse from the outermost part, then the compressive stress and tensile stress exist simultaneously. The hoop stress has a discontinuous point at the discontinuous portion of the critical current density, thus the damage probability is higher than that of the uniform distribution. In other words, the shear strength of superconductor is required to be higher for application. Also, the degree of magnetostriction is higher when the distribution of critical current density is set to be uniform, that is, the non-uniform distribution of the critical current density causes the superconducting structure to undergo greater deformation. Therefore, in engineering applications, the structural strength of the superconducting material must be enhanced to cope with the challenge from the uneven distribution of critical current density.

Effect of Inclusions on Magnetostriction in Superconducting Cylinder with Exponential Distribution of Critical-Current Density

The magnetoelastic behaviour subjected to the electromagnetic body force induced by flux pinning is studied with Kim model for zero-field cooling (ZFC) magnetization process, in which the non-uniform parameter η is considered for inclusion-superconducting matrix system. For superconducting composites, the effect of inhomogeneous distribution of critical current density on effective magnetostriction is also obtained based on the plane strain approach. The results show that the exponential distribution of critical-current density will lead to a larger trapped field and magnetostriction inside the inhomogeneous sample, which also means that it is worthwhile to investigate the magnetoelastic problem of bulk superconductors with inhomogeneous distribution of critical-current density.

Influence of Critical Current Density Distribution on Transport AC Losses in Superconducting Wire in a DC Magnetic Field

In typical application-like conditions, the inhomogeneous distribution and anisotropy of critical current density must be considered simultaneously in transport AC loss calculation. In this paper, we derive the analytical formulas of transport AC losses for high-temperature superconductors (HTSs) with linear and quadratic distribution of critical current density under applied DC magnetic field. The influence of the inhomogeneous distribution and anisotropy of critical current density has been analyzed. The results show that the impact of the distribution form on transport AC loss is more obvious under applied DC magnetic field. And the influence of applied DC magnetic field will increase as the distribution form becomes steeper.