Flux Trapping Research Articles

Effects of biofertilizers on the growth, leaf physiological indices and chlorophyll fluorescence response of spinach seedlings.

Chemcial fertilizer as the main strategy for improving the vegetable yields was excessively applied in recent years which led to progressively serious soil problems such as the soil acidification. According the situation, five different biofertilizer treatments [no fertilizer (CK), inoculations of Bacillus subtilis (Bs, T1), combination of Bs and Bacillus mucilaginosus (Bs+Bm, T2), Bs and Bacillus amyloliquefaciens (Bs+Ba, T3), and Bm+Ba (T4)] were conducted to investigate the effect of the growth, leaf physiological indices, and chlorophyll fluorescence of spinach seedlings in the growth chamber. The growth and physiological indices of the spinach seedlings attained a maximum under the T2 treatments. Under the T2 treatment, the ABS/RC (Absorption flux per RC), TR0/RC (Trapping flux per RC), and ET0/RC (Electron transport flux per RC) was significantly increased, while the DI0/RC [Dissipated energy flux per RC (at t = 0)] was decreased. The OJIP curve was improved under of the inoculations of fertilizers, and the increasing range was the largest under the T2 treatment. The leaf light response curve (LC) was also significantly increased under the T2 treatment. The plant growth characteristics [leaf length (LL), leaf weight (LW), plant height (PH)] were positively correlated with the J-I-P test chlorophyll fluorescence parameters [PIABS (Performance index for energy conservation from exciton to the reduction of intersystem electron acceptors), φP0 (Maximum quantum yield of primary photochemistry), φE0 (Quantum yield of electron transport), ψ0 (The probability that a trapped exciton moved an electron in electron transport chain further than QA-), TR0/RC, and ET0/RC] while negatively correlated with φD0 (Quantum yield of energy dissipation) and DI0/RC. The leaf physiological characteristics [SP (soluble protein concentrations), SC (soluble carbohydrate concentrations), Chl a (chlorophyll a), Chl b (chlorophyll b), Chl a+b, Chl a/b, and WP (water potential)] were positively correlated with the J-I-P test chlorophyll fluorescence parameters (PIABS, φP0, φE0, ψ0, ABS/RC, TR0/RC, and ET0/RC) while negatively correlated with φD0 and DI0/RC. These results indicated that the combination of Bs+Bm inoculations promoted the growth of the spinach and improved the adaptability of the vegetable to acid soil while Ba inoculation didn't have any effects to plants.

Statistical evaluation of the mechanical and flux trapping properties of standard and thin‐wall EuBCO(Ag) bulk superconductors

AbstractThe mechanical and flux trapping properties of 19 EuBCO(Ag) and 20 thin‐wall EuBCO(Ag) bulk superconductors were investigated statistically and compared to those observed for YBCO using Brazilian testing and high‐field magnetization. The average trapped fields of EuBCO(Ag) and thin‐wall EuBCO(Ag) bulk superconductors were significantly higher than those of YBCO bulk samples at 77 K measured under similar experimental conditions. However, due to the wide range of trapped field data for both materials, a statistical overlap was observed between the highest trapped fields observed for YBCO and the lowest values observed for EuBCO(Ag) bulk superconductors. All EuBCO(Ag) bulk materials measured in this study exhibit a superior tensile strength compared to YBCO bulk single grains. Unlike the case of YBCO, however, the introduction of artificial holes to EuBCO(Ag) bulk superconductors prior to melt processing was observed to decrease the average tensile strength of the single grain. Surprisingly, despite their superior mechanical and flux pinning properties, EuBCO(Ag) bulk superconductors do not exhibit enhanced performance when exposed to high fields at low temperatures.

Anomalous field-temperature phase diagram of superconductivity in Sn–Pb solder

Sn–Pb solders are superconducting materials whose Sn and Pb are perfectly phase-separated. Recently, anomalous magnetic-flux trapping in a Sn45–Pb55 solder has been revealed, where the magnetic fluxes are selectively trapped in the Sn regions due to the supercurrents in the surrounding Pb regions. Here, we report on the observation of the anomalous critical field (Hc)-temperature (T) phase diagram of superconductivity in the Sn45–Pb55 solder. Although the Hc(T) for the Pb regions decreases with increasing field as in normal type-I superconductors and is consistent with the conventional trend with Hc(0) ∼ 800 Oe, the Hc(T) for the Sn regions exhibits anomalous behaviors. The most noticeable trend was observed in the field-cooled (under 1500 Oe) solder. The Hc-T phase diagram for the Sn regions largely varies when the applied external field is reversed, and Tc increases with increasing field amplitude when H < 0 Oe is applied. On the basis of the flux trapping and the observed anomalous Hc-T phase diagrams, we propose that the field-robust superconductivity in the Sn regions is related to the ferromagnetically aligned magnetic fluxes (or formation of vortices) in the Sn regions of the Sn45–Pb55 solder.

Cryogenic reconfigurable logic with superconducting heater cryotron: Enhancing area efficiency and enabling camouflaged processors

Superconducting electronics are among the most promising alternatives to conventional CMOS technology, thanks to the ultra-fast speed and ultra-high energy efficiency of the superconducting devices. Having a cryogenic control processor is also a crucial requirement for scaling the existing quantum computers up to thousands of qubits. Despite showing outstanding speed and energy efficiency, Josephson junction-based circuits suffer from several challenges such as flux trapping leading to limited scalability, difficulty in driving high impedances, and so on. Three-terminal cryotron devices have been proposed to solve these issues, which can drive high impedances (>100 kΩ) and are free from any flux trapping issue. In this work, we develop a reconfigurable logic circuit using a heater cryotron (hTron). In conventional approaches, the number of devices to perform a logic operation typically increases with the number of inputs. However, here, we demonstrate a single hTron device-based logic circuit that can be reconfigured to perform 1-input copy and NOT, 2-input AND and OR, and 3-input majority logic operations by choosing suitable biasing conditions. Consequently, we can perform any processing task with a much smaller number of devices. Also, since we can perform different logic operations with the same circuit (same layout), we can develop a camouflaged system where all the logic gates will have the same layout. Therefore, this proposed circuit will ensure enhanced hardware security against reverse engineering attacks.

Investigation of the thermodynamics of trapped magnetic flux in type-I and type-II superconductors using polarized neutron imaging

Polarized neutron imaging was used to observe field cooling (FC) and zero field warming (ZFW) processes in type-I and type-II superconductors. ZFW is an irreversible magnetic process that cannot be described by formulas and therefore it was studied experimentally with polarized neutron imaging to visualize and quantify the trapped magnetic flux as a function of temperature.

Understanding photochemical adaptations of microalgal ammonium extremophiles to elevated ammonium concentrations in abattoir anaerobic digestate

Identifying microalgal species with outstanding environmental resistance to NH4+ toxicity is critical to developing effective algal-based anaerobic effluent treatment. This is the case for two microalgal NH4+ extremophiles, Chlorella sp. MUR269 and Scenedesmus sp. MUR270 well suited to treat high nitrogen digestates. However, their exact tolerance mechanisms to high NH4+ conditions still need to be understood. A chlorophyll fluorescence approach based on high-throughput pulse amplitude-modulated fluorimetry was used to understand the photochemical adaptations of these species under increasing NH4+ concentrations (0, 100, 200 mgNL−1) derived from abattoir anaerobic digestate. Both species tolerated 200 mgN-NH4+L−1 in the digestate by exhibiting different photochemical mechanisms. Chlorella decreased its PSII primary photochemistry efficiency (Fq′/FM′) and maximum electron transport (rETRmax) immediately after exposure to the treatment conditions. In contrast, Scenedesmus maintained unaltered Fq′/FM′ regardless of NH4+-N concentration, while rETRmax was significantly reduced to a level of Chlorella at 200 mgNL−1. Energy transduction parameters showed Chlorella increased the size of its reaction centre antennae to capture more photons, which increased the trapping flux ratio and maintained efficient dissipation of excess energy. Conversely, Scenedesmus displayed reductions in the absorbed, trapped and dissipated energy fluxes, though its transported flux for photochemical energy production was similar to Chlorella sp. The energetic communication and connectivity between Chlorella PSII units went up by 47 % and 40 % when the NH4+-N level increased from 50 to 100 and N200 mgL−1, respectively, while Scenedesmus exhibited a dose-dependent decrease in the connectivity between its PSII components. The photosynthetic performance index revealed that the overall energetic flux processing efficiency by Scenedesmus was 86 % (50 mgNL−1), 73 % (100 mgNL−1) and 91 % (200 mgNL−1) higher than Chlorella sp. Thus, the two chlorophytes maintained their overall photochemical capacity by simple, economic biological mechanisms involving adjustments of antenna size and energetic communications between PSII neighbour units.

Adopting a Standard Track Routing Architecture for Next-Generation Hybrid AC/DC-Biased Logic Circuits

The adiabatic quantum-flux-parametron (AQFP) and rapid single flux quantum (RSFQ) logic families both provide distinct advantages for data processing using superconductor electronics. The development of next-generation AQFP-RSFQ hybrid logic circuits provides the opportunity to combine the benefits of both of these logic families. Adopting a standard track routing architecture for both AQFP and RSFQ layouts provides a unified framework for physical layout synthesis. We demonstrate a standard track routing architecture developed under the ColdFlux project and its implementation within AQFP. An additional advantage of the standard track routing architecture is that it provides a systematic way to overcome metal density requirements for modern-day fabrication processes. It also provides a methodical approach towards strategic moat placement for protection against unwanted flux trapping. Implementations of passive transmission line (PTL) connections and their physical placement for hybrid AQFP-RSFQ circuits are discussed. Several AQFP layouts for the ColdFlux project are redesigned to adopt the standard track routing architecture. Experimental simulation of moat performance is also presented.

Stripline Topology for Flux Mitigation

The increasing complexity of modern single flux quantum (SFQ) circuits has increased the importance of flux trapping and trapped magnetic fields within SFQ systems. This trapped flux reduces margins while damaging the operability of superconductive circuits. In this paper, an area efficient stripline topology is introduced to prevent flux from being trapped within striplines. The topology is composed of coupled narrow lines rather than wide striplines. The proposed topology uses a fingered narrow line configuration. The fingered narrow line topology enhances the scalability of SFQ systems while not requiring additional area. The proposed topology decreases the length of the striplines by exploiting the mutual inductance between narrow parallel lines. The topology requires less area while preventing flux from being trapped within wide superconductive striplines. Due to the configuration of the proposed stripline, residual current is eliminated in VLSI complexity SFQ circuits. The fingered narrow line topology also reduces coupling capacitance between striplines. The proposed topology is compatible with automated routing of large scale SFQ integrated circuits.

Optimisation of stacked, bulk high temperature superconductors for trapped-field magnet applications

It is necessary to fabricate (RE)BCO bulk high temperature superconductors in the form of individual single grains in order to maximise the length scale over which current flows, and hence the trapped magnetic field. However, inherent difficulties in the grain growth process place limitations on the diameter and height of the single grain that may be achieved by existing melt processes. A practical approach to increase the height of the sample and the trapped field at its surface is by assembling individual single grains in a stack formation with their ab planes aligned parallel, primarily to avoid the expensive process of fabricating large, individual monoliths. The trapped fields observed at the top and bottom surfaces of a single grain sample are frequently different since both the superconducting and physical properties of single grain (RE)BCO samples are generally non-uniform. This leads to challenges in determining how to spatially arrange stacks of single grain samples to generate the largest and most uniform trapped field overall. In this study, we report the optimisation of two-stack configurations involving a total of 8 individual GdBCO/Ag single grains. The samples were arranged in four pairs and configured with different surfaces in contact in the assembly of the stack. The primary superconducting properties for trapped field and total flux distributions were measured at 77 K and compared for each stack arrangement. The initial results indicate that surfaces with inferior flux trapping properties (measured in terms of the overall trapped field value) of a two-sample stack should be positioned at the middle of the assembly to achieve the best overall trapped field and higher total flux at the external, and therefore, usable surface of the stack sample. A numerical modelling method that incorporates different Jc-B characteristics for the top and bottom layers of a single grain to take account of the variability in physical properties and spatial non-uniformity confirmed the optimised experimental arrangement of the stacked bulk samples. Furthermore, the optimisation of single grains of ring geometry to achieve a longer and wider uniform magnetic field zone inside the bore was also performed.

Extended 2-D Magnetic Field Modeling of Linear Motor to Investigate the Magnetic Force Parameters of High-Speed Superconducting Maglev

A 2D numerical finite element model of a linear synchronous motor (LSM) is extended based on the magnetic field and moving mesh properties to investigate the magnetic flux and magnetic force characteristics of the high-speed EML Maglev system, by using HTS coils rather than lower current carrying copper coils on the onboard unit and PMs on the ground. In this 2D LSM model, the transient, time dependent solver is used to obtain the magnetic flux densities. Both the propulsion and levitation forces increased with increasing supercurrent J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c0</sub> values indicating the advantage of using the superconducting windings in PM-LSM of Maglev vehicles as compared to the copper wires. It was also determined that although the propulsion force is obtained on a comparable level with the studies in literature, the lower levitation force issue than the EDS, despite the better propulsion to levitation force ratio, can be overcame by using the high flux trapping capacity bulk HTSs in the onboard unit together with the superconducting coils. The determined higher propulsion force in this PM-LSM Maglev model indicates that the Maglev vehicle can reach higher velocities in a short-distance and thus this vehicle can be effectively used in short-distance travels in addition to the long-distance transportation. On the other hand, since the EDS system to be accelerated in a certain time interval via conventional wheels to achieve sufficient levitation, PM-LSM system can be integrated into the EDS Maglev system to ensure higher acceleration in a short time interval in addition to the higher levitation and propulsion force performances.

Insight to the Duration of 120 $^{\circ }$C Baking on the Performance of SRF Niobium Cavities

Low-temperature baking at 100 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{\circ}$</tex-math></inline-formula> C–140 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{\circ}$</tex-math></inline-formula> C in ultrahigh vacuum for several hours is the final cavity processing technique to achieve the highest accelerating gradient cavity by eliminating the high field Q-slope, typically on cavities made of high purity fine grain niobium and subjected to electropolishing. The temperature and duration of baking control the onset of the high field Q-slope and overall quality factor over the baseline measurements. The increase in quality factor is generally related to the reduction in the temperature-dependent surface resistance, which refers to BCS surface resistance due to the reduction in the electronic mean free path and elimination of high field Q-slope is still being actively investigated. Here, we present the results of a series of measurements on 1.3 GHz TESLA shape single-cell cavities with successive low-temperature baking at 120 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{\circ}$</tex-math></inline-formula> C up to 96 h. The rf loss related to the trapping of residual magnetic field refers to flux trapping sensitivity measured with respect to the duration of 120 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{\circ}$</tex-math></inline-formula> C bake.

Numerical study on flux-jump occurrence in a cup-shaped MgB2 bulk for magnetic shielding applications

MgB2 is one of the most promising materials for superconducting bulk applications. However, thermomagnetic instabilities can arise in the material because of its low heat capacity and thermal conductivity as well as its high critical current density. Being able to predict these phenomena, can guide and optimize MgB2-based devices for magnetic flux shielding or trapping applications. In this work, the flux-jump occurrence in an MgB2 cup-shaped shield is numerically studied using the finite element method by means of the commercial software COMSOL 6.0 Multiphysics®. To this aim, we developed a 2D axial-symmetric model coupling the heat diffusion equation and the magnetic equations based on a magnetic vector-potential () formulation. The comparison of the computed shielding curves with the experimental ones evidenced a good agreement between the two sets of data at different temperatures and positions along the shield’s axis. The as-validated model was then exploited to investigate possible optimization routes via the improvement of both the thermal conductivity of the material and the thermal exchange between the device and the cooling stage.

Modelling and mitigating flux jumps in bulk high-temperature superconductors during quasi-static, high-field magnetisation

(RE)-Ba2Cu3O bulk superconductors acting as permanent trapped field magnet analogues have been shown to trap fields in excess of 17 T. However, the occurrence of thermomagnetic instabilities during their magnetisation process can limit their performance. In 2019, Naito et al trapped 15.1 T in a (Y)-BaCuO bulk pair under applied fields of up to 22 T (2019 J. Appl. Phys. 126 243901), whilst also documenting the mechanical failure of the bulks due to a flux jump. Here, we accurately numerically model this experiment and the observed flux jump, providing insight into the experimental results, such as the role of heating, the presence of thermal gradients, and a possible reason for the location of the bulk-pair fracturing. Extending the study with an investigation into the role of enhanced cooling of the bulk-pair, the influence of cooling power on the calculated trapped field and flux jump is explored. Finally, we propose and numerically investigate a new composite bulk configuration, consisting of stacks of (Y)-BaCuO bulk and interspaced metallic discs, to enhance the thermal stability of the bulk pair, to ultimately improve the trapped field performance.

Effect of Successive Heat Treatment on the Performance of Superconducting Radio Frequency Niobium Cavities

One of the primary sources of radio frequency residual losses leading to lower quality factor is trapped residual magnetic field during the cooldown of superconducting radio frequency cavities. It has been reported that non-uniform recrystallization of niobium cavities after the post fabrication heat treatment leads to higher flux trapping during the cooldown, and hence the lower quality factor. Here, we fabricated two 1.3 GHz single cell cavities from high purity fine grain niobium and processed with successive heat treatments in the range 800-1000 °C to measure the flux expulsion and flux trapping sensitivity. The result indicates that although flux expulsion improves with increased heat treatments, there is a noticeable difference between the flux trapping sensitivity depending on the cavity. Evaluation of microstructure maybe crucial to understand the impact of flux trapping sensitivity on cavity performance.

Homeostatic influence of calcium and kinetin: Regulation of defense strategies against NaCl stress in cyanobacterium Nostoc muscorum ATCC 27893

• NaCl negatively affected the growth and growth related parameters i.e. proteins and exopolysaccharides contents, light harvesting pigments and photosynthesis as well as PS II photochemistry by increasing the oxidative stress. • Calcium and kinetin alleviated the toxicity induced by NaCl on cyanobacterial physiology. • Calcium is essentially needed for kinetin mediated regulation of NaCl toxicity. In the present study, the possible roles of kinetin (KN, 20 nM) and calcium (Ca 2+ , 0.8 mM) in the alleviation of the salinity stress (50 mM NaCl) induced damage on the growth, and metabolism of cyanobacterium Nostoc muscorum ATCC 27893 was explored. The results revealed that salinity stress decreased the growth, light-harvesting pigments (chlorophyll a , carotenoids and phycobiliproteins) and exopolysaccharides (EPS) contents, photosynthesis (whole-cell oxygen evolution), and the performance of PS II photochemistry [Chl a fluorescence kinetics: maximum photosynthetic quantum yield (Fv/Fm or Phi_Po), quantum yield of electron transport flux (Phi_Eo), yield of electron transport per trapped exciton (Psi_o), size and number of active photosynthetic reaction centers (Fv/Fo) and performance index (PI ABS )]. Reverse to this, enhancement of respiratory rate and values of PS II related energy flux parameters [absorption flux per reaction center (ABS/RC), trapping flux per reaction center (TRo/ RC), dissipation flux per reaction center (DIo/RC), and electron transport flux per reaction center (ETo/RC)], and rise in levels of oxidative biomarkers (superoxide radical; O 2 • , hydrogen peroxide; H 2 O 2, and malonaldehyde; MDA equivalents) even under accelerated activity of enzymatic antioxidants (superoxide dismutase; SOD, peroxidase; POD, catalase; CAT and glutathione-s-transferase; GST) were observed. Kinetin in presence of Ca 2+ improved the growth, pigments contents and performance of PS II photochemistry by down-regulating the oxidative stress biomarkres. Application of Ca 2+ scavenger (EGTA) in KN treated samples increased the damaging effect on growth and photosynthesis by aggravating oxidative stress under NaCl stress, thereby indicating the positive role of Ca 2+ in NaCl-induced toxicity alleviation by KN. The overall results suggest that Ca 2+ together with KN could be applied in paddy fields to up-regulate the growth of N 2 –fixing cyanobacterium Nostoc muscorum ATCC 27893, an inhabitant of paddy field under NaCl stress, hence they may be considered for sustainable agriculture.

Electromagnetic Response in an Expanding Quark–Gluon Plasma

The validity of conventional Ohm’s law is tested in the context of a rapidly evolving quark–gluon plasma produced in heavy-ion collisions. Here, we discuss the electromagnetic response using an analytical solution in kinetic theory. As conjectured previously, after switching on an electric field in a nonexpanding plasma, the time-dependent current is given by J(t)=(1−e−t/τ0)σ0E, where τ0 is the transport relaxation time and σ0 is the steady-state electrical conductivity. Such an incomplete electromagnetic response reduces the efficiency of the magnetic flux trapping in the quark–gluon plasma, and may prevent the observation of the chiral magnetic effect. Here, we extend the study to the case of a rapidly expanding plasma. We find that the decreasing temperature and the increasing transport relaxation time have opposite effects on the electromagnetic response. While the former suppresses the time-dependent conductivity, the latter enhances it.

A low-cost approach for recycling large GdBCO bulk superconductors using an infiltration technique with different liquid sources

The high-temperature superconductor has become a research hotspot, because of its high critical temperature, strong trapped flux density, stable suspension characteristics and large magnet levitation force. Single-domain REBa2Cu3O7−δ (REBCO) superconductors, where RE is a rare-earth element such as Y, Gd, Sm or Nd, have wide and potential applications in high-tech fields, such as micro-magnet superconducting maglev trains, superconducting motors and superconducting magnetic separation systems. However, a large number of multi-domain samples are easily produced in the preparation process, resulting in a significantly lower success rate and a significant increase in cost, which restricts their practical applications. Thus, we successfully recycle a series of GdBCO samples by re-supplementing the liquid phase lost in the primary growth process and pre-treating the failed sample as solid-phase source billets, which was proposed by Shi. The growth morphology and superconducting properties of the recycled GdBCO bulk superconductors are investigated in detail in this study. The results show that the key superconductivity properties have been significantly improved, which provides the scientific basis and new ideas for the development of low-cost and highly efficient fabrication yields of REBCO bulk superconductors.

Observation of quenching-induced magnetic flux trapping using a magnetic field and temperature mapping system

Highly efficient superconducting radio-frequency cavities exhibit low heat loss and are used as components in accelerators and superconducting devices due to their high $Q$-values. The precise location of magnetic flux trapping in cavities is necessary to identify its effects on the performance of superconducting cavities. In this study, we report a new combined mapping system to measure the temperature and magnetic field on the equator. The proposed system comprehensively maps local magnetic field changes as magnetic flux trapping due to quenching. Our experimental results show that magnetic flux trapping due to quenching increases the local surface resistance of superconducting cavities at 2 K. Thus, the proposed system elucidates the relationship between local flux trapping due to quenching and surface resistance in superconducting cavities and highlights the effect of quenching on the surface resistance. This system can aid in the development of superconducting cavities with higher $Q$ values.

Flux Mitigation in Wide Superconductive Striplines

The increasing complexity of modern superconductive circuits, and single flux quantum (SFQ) circuits in particular has made the issue of flux trapping of growing importance. The use of wide superconductive striplines for signal routing has exacerbated this issue. Trapping residual magnetic fields in these striplines degrades performance while reducing margins, damaging the operability of superconductive circuits. In this article, an area efficient topology for striplines is introduced to manage flux trapping in large scale SFQ circuits. This topology is composed of narrow parallel lines in series with small resistors. The proposed topology decreases the length of the striplines by exploiting the mutual inductance between the narrow parallel lines. The topology requires significantly less area while preventing flux trapping within wide superconductive striplines. The narrow parallel line topology also reduces coupling capacitance between striplines. The proposed approach is compatible with automated routing of large scale SFQ integrated circuits.

Optimizing trapped field in superconductors with perforations

The use of artificial holes can improve the performance of high-temperature bulk superconductors. We report the results of Monte Carlo simulations of the trapped magnetic flux in superconducting samples with different configurations of perforated holes. Vortex trapping and emission at the hole boundary were modeled to account for flux pinning on the holes. It was found that the trapped flux can reach its maximum value, which does not depend on the number of holes. The dependence of the trapped flux on the diameter and number of holes is explored and described by the suggested relations that account for pinning on the holes and the effective hole area. Recommendations are provided to attain the maximum trapped flux for a fixed number or diameter of holes.