Flux Distribution Research Articles

Design of Small Permanent-Magnet Linear Motors and Drivers for Automation Applications with S-Curve Motion Trajectory Control and Solutions for End Effects and Cogging Force

This paper designs and fabricates a small-type permanent-magnet linear motor and driver for automation applications. It covers structural design, magnetic circuit analysis, control strategies, and hardware development. Magnetic circuit analysis software JMAG is used for flux density distribution, back electromotive force (back-EMF), and electromagnetic force analysis. To address the lack of a complete closed magnetic circuit path at the ends of the linear motor, which causes magnetic field asymmetry, a phenomenon known as end effects, auxiliary core structures are proposed to compensate for the magnetic field at the ends. It successfully utilizes auxiliary cores to achieve the phase voltages of each phase, which are balanced at a phase voltage error of 0.02 V. To address the cogging force caused by variations in the magnetic reluctance of the core, this paper analyzes the relationship between electromagnetic force and mover position, conducting harmonic content analysis to obtain parameters. These parameters are applied to the designed cogging force control compensation strategy. It successfully achieves q-axis current compensation of around 1.05 A based on the mover’s position, ensuring that no jerking caused by cogging force occurs during closed-loop electromagnetic force control. The S-curve motion trajectory control is proposed to replace the traditional trapezoidal acceleration and deceleration, resulting in smoother position control of the linear motor. Simulations using JMAG-RT models in MATLAB/Simulink verified these control strategies. After verification, practical test results showed a maximum position error of approximately 5.0 μm. Practical tests show that the designed small-type permanent-magnet linear motor and its driver provide efficient, stable, and high-precision solutions for automation applications.

PICZL : Image-based photometric redshifts for AGN

Computing reliable photometric redshifts (photo-z) for active galactic nuclei (AGN) is a challenging task, primarily due to the complex interplay between the unresolved relative emissions associated with the supermassive black hole and its host galaxy. Spectral energy distribution (SED) fitting methods, while effective for galaxies and AGN in pencil-beam surveys, face limitations in wide or all-sky surveys with fewer bands available, lacking the ability to accurately capture the AGN contribution to the SED, hindering reliable redshift estimation. This limitation is affecting the many tens of millions of AGN detected in existing datasets, such as those AGN clearly singled out and identified by SRG/eROSITA. Our goal is to enhance photometric redshift performance for AGN in all-sky surveys while simultaneously simplifying the approach by avoiding the need to merge multiple data sets. Instead, we employ readily available data products from the 10th Data Release of the Imaging Legacy Survey for the Dark Energy Spectroscopic Instrument, which covers > 20,000 deg of extragalactic sky with deep imaging and catalog-based photometry in the grizW1-W4 bands. We fully utilize the spatial flux distribution in the vicinity of each source to produce reliable photo-z. We introduce PICZL a machine-learning algorithm leveraging an ensemble of convolutional neural networks. Utilizing a cross-channel approach, the algorithm integrates distinct SED features from images with those obtained from catalog-level data. Full probability distributions are achieved via the integration of Gaussian mixture models. On a validation sample of 8098 AGN PICZL achieves an accuracy $ NMAD $ of 4.5<!PCT!> with an outlier fraction eta of 5.6<!PCT!>. These results significantly outperform previous attempts to compute accurate photo-z for AGN using machine learning. We highlight that the model's performance depends on many variables, predominantly the depth of the data and associated photometric error. A thorough evaluation of these dependencies is presented in the paper. Our streamlined methodology maintains consistent performance across the entire survey area, when accounting for differing data quality. The same approach can be adopted for future deep photometric surveys such as LSST and Euclid, showcasing its potential for wide-scale realization. With this paper, we release updated photo-z (including errors) for the XMM-SERVS W-CDF-S, ELAIS-S1 and LSS fields.

On the Peculiarities of Wire-Feed Electron Beam Additive Manufacturing (WEBAM) of Nickel Alloy–Copper Bimetal Nozzle Samples

In order to gain insight into the unique characteristics of manufacturing large-scale products with intricate geometries, experimental nozzle-shaped samples were created using wire-feed electron beam additive technology. Bimetal samples were fabricated from nickel-based alloy and copper. Two distinct approaches were employed, utilizing varying substrate thicknesses and differing fabrication parameters. The two approaches were the subject of analysis and comparison through the examination of the surface morphology of the samples using optical microscopy, scanning electron microscopy, and X-ray diffraction analysis. It has been demonstrated that the variation in heat flux distributions resulting from varying the substrate thicknesses gives rise to the development of disparate angles of grain boundary orientation relative to the substrate. Furthermore, it is demonstrated that suboptimal choice of the fabrication parameters results in large disparities in the crystallization times, both at the level of sample as a whole and within the same material volume. For example, for the sample manufacturing by Mode I, the macrostructure of the layers is distinguished by the presence of non-uniformity in their geometric dimensions and the presence of unmelted wire fragments. In order to characterize the experimental nozzle-shaped samples, microhardness was measured, uniaxial tensile tests were performed, and thermal diffusivity was determined. The microhardness profiles and the mechanical properties exhibit a higher degree of strength than those observed in pure copper samples and a lower degree of strength than those observed in Inconel 625 samples obtained through the same methodology. The thermal diffusivity values of the samples are sufficiently close to one another and align with the properties of the corresponding materials in their state after casting or rolling. The data discussed above indicate that Mode II yields the optimal mechanical properties of the sample due to the high cooling rate, which influences the structural and phase state of the resulting products. It was thus concluded that the experimental samples grown by Mode II on a thinner substrate exhibited the best formability.

Numerical Study of Complex Heat Transfer and Aerodynamics in a Tube Furnace with a Flat Fuel Combustion Mode

The object of the study is a tubular furnace for steam reforming of natural gas. The channel walls are formed by a refractory lining and a tubular screen with a known temperature. The chamber volume is filled with a selectively radiating, absorbing and weakly scattering medium of gaseous fuel combustion products. The research method is based on the numerical solution of a system of two-dimensional integro-differential equations of radiative gas dynamics, closed by a two-parameter model of turbulent convection. It is shown that when burners are located on the side walls of the radiation chamber, a significant restructuring of the heat flux distribution occurs. A decrease in the chamber width is accompanied by an increase in both radiant and convective heat fluxes to the tubular screen.

Process optimization and finite element modeling in magnetorheological abrasive finishing of SS-316L with surface characterization

This study explores the potential of Magnetorheological Abrasive Finishing (MRAF) for ultra-precision finishing of SS-316L using a high-performance Nd-Fe-B (N-35 grade) magnetic tool. Key parameters, including the relationship between normal force (Fn) and working gap, magnetic flux distribution, and finishing spot area, were analyzed using ANSYS 2020 R1® Magnetostatic module simulations. Material removal rate (MRR) and surface roughness (Ra) were optimized through response surface methodology (RSM) using a central composite design (CCD) with two primary input factors: SiC abrasive volume fraction (5%, 10%, 15%) and abrasive mesh size (800, 1000, 1200). Statistical analysis via ANOVA revealed a significant influence of process variables, achieving a remarkable surface finish with a minimum Ra of 52 nm in just 60 min, utilizing an MR fluid with 10% SiC abrasives. Microscopic and spectroscopic evaluations confirmed enhanced surface morphology and microstructural refinement. Moreover, bioactivity studies in simulated body fluid highlighted improved hydroxyapatite layer formation on the finished SS-316L samples, underscoring the process’s potential for biomedical applications.

Analysis of the air gap magnetic field in cylindrical magnetic couplings based on mathematical and finite element approach

The air gap magnetic field of a magnetic coupling plays a crucial role in the examination of its static torque and eddy current losses. Precise characterization of the air gap magnetic field is essential for ensuring the accuracy of performance assessments of the magnetic coupling. This study focuses on the cylindrical magnetic coupling as the subject of investigation, employing mathematical analysis and finite element methods to evaluate and quantify the magnetic field properties. The study establishes a mathematical model of the magnetic field of a magnetic coupling based on electromagnetic field principles and the superposition theorem. An analytical formula for the magnetic flux density distribution of the air-gap magnetic field is derived. Subsequently, a three-dimensional magnetic field finite element model is created using the finite element method to numerically calculate the air gap magnetic field and validate the analytical formula. The research also explores the periodic distribution characteristics of the magnetic field in the air gap, analyzing axial differences in magnetic flux density value and direction distribution influenced by end effects.

Numerical investigation on boiling crisis characteristic of a 7-rod HCF assembly in hexagonal lattice

Helical cruciform fuel (HCF) has the advantages of larger heat transfer area, enhanced coolant mixing and self-supporting, which contribute to increasing power density and safety margins. Compared with the square lattice configuration, the hexagonal arrangement of HCF assembly is more compact, which can help achieve a higher power density. In this paper, the flow characteristics and heat transfer behaviors of HCF in hexagonal lattice were predicted at high and low vapor quality during boiling crisis based on Eulerian two-fluid model. The influence of twist pitches and cross-sections of the fuel rod on heat transfer efficiency and fuel temperature was also studied. The cross-flow intensity changed periodically with a 30° cycle at low vapor quality, and did not fluctuate periodically at high vapor quality, which decreased with the increase of flow resistance. The highest heat flux of HCF rod was the at the blade root and the lowest was at the blade tip, and the maximum to average heat flux ratio was about 1.8. The peak vapor fraction and temperature occurred at leeside side of the fuel rods. The increase of the twist pitch reduced the critical heat flux (CHF), and the increase of blade length enhanced the non-uniformity of heat flux distribution. During boiling crisis, the maximum temperature of the fuel was lower than the phase transition temperature of U-50 wt%Zr alloy, which means the cladding meltdown caused by boiling crisis will occur before phase transition of the fuel.

Effect of soft magnetic particles content on multi-physics field of magnetorheological composite gel clutch with complex flow channel excited by Halbach array arrangement

In this paper we investigate the influence of soft magnetic particle mass fraction on the multi-physics field of proposed magnetorheological (MR) gel clutch. A novel MR gel clutch with complex cup-shaped gap is described, whose performance is based on the relative placement between Halbach array arrangement and MR gel. Smart MR gel with 40 wt%, 50 wt%, 60 wt%, 70 wt% and 80 wt% of soft magnetic particles used as the transfer medium and Halbach array is adopted as magnetic excitation system. Its magnetic/mechanical analysis is carried out based on Bingham-plastics model using COMSOL Multiphysics software, which takes into account the variation of dynamic viscosity with magnetic flux density. The distribution of the magnetic flux density, shear yield stress, post-yield viscosity, shear stress and torque in the four flow channels during the transition from engagement state to disengagement state are obtained and analyzed in detail. Multi-physics field characteristics of proposed MR gel clutch with five kinds of MR gels are studied and compared in order to give some useful suggestions in the design phase. Finally, the dynamic torque of the MR clutch with different MR gel is experimentally evaluated.

Electron temperature and ion density distribution on a vertical section in a weakly magnetized inductively coupled plasma

In this study, the distributions of electron temperature and ion density on a vertical section in a weakly magnetized inductively coupled plasma were measured using radially movable floating probes placed at different axial positions. The chamber used in this experiment included two cylindrical parts: a smaller radius top part with a planar antenna on the top quartz window and a larger radius downstream part. A magnet coil around the chamber top part maintained a divergent magnetic field in the discharge region. As the current in the magnet coil increased, the magnetic field also increased. Due to the variations of the radio frequency electric field in the plasma, the increase in electron temperature can be divided into different stages. At the higher magnetic field, the electric field of the electrostatic wave can increase electron temperature at the chamber center axial. Also, since the electron cyclotron resonance (ECR) heating in the chamber downstream part changed with the magnetic field, the maximum ion density was observed when the magnetic field around the bias electrode was slightly larger than the ECR magnetic condition. The reasons for these variations were verified in the plasma numerical simulations. The ion flux distribution measured on the bias electrode can change from a center-high distribution to an M-shape distribution with the increased magnetic field.

FLOW AND HEAT TRANSFER IN ROTATING COMPRESSOR CAVITIES WITH INVERTED SHROUD-THROUGHFLOW TEMPERATURE DIFFERENCES

Abstract In an aero-engine compressor, co-rotating discs form cavities that interact with an axial throughflow of secondary air at low radius. In the high-pressure (HP) compressor the shroud is hotter than the throughflow (directed downstream to the turbine) and the radial temperature gradient creates buoyancy-induced flow at Grashof numbers ∼ 1013. Such flows can be unstable and typically take the form of counter-rotating vortex pairs separated by radial hot and cold plumes. However, in low pressure (LP) and intermediate pressure (IP) compressors the secondary air is directed upstream. In this inverse scenario the axial throughflow is hotter than the compressor discs, reversing the disc temperature gradient and eliminating the fundamental driver for buoyancy. Despite its practical application and importance, this inverse scenario has not been previously investigated. The University of Bath Compressor Cavity Rig has been uniquely designed to simulate such flows, measuring temperature and unsteady pressure in the frame of reference of the rotating discs. Bayesian and spectral analysis have determined the radial distribution of disc heat flux, as well as the asymmetry of the rotating vortex structures and their slip relative to the discs. Unexpectedly, the new data reveal the flow structure in cavities with positive and inverted temperature differences are fundamentally similar (albeit with reversed radial-temperature profiles). Isothermal cases identified a critical Rossby number (Ro), above which the flow structure in the cavity was dominated by a toroidal vortex. At sub-critical Ro, the flow structure for the inverted temperature gradient continued

Characteristics of divertor heat flux distribution with an island divertor configuration on the J-TEXT tokamak

On J-TEXT, the temporal evolution of heat flux distribution on the high-field side (HFS) divertor plate has been measured by an infrared (IR) camera during the plasma operation with an island divertor configuration. In experiments, the island divertor configuration is an edge magnetic island chain structure surrounded by stochastic layers, which can be induced by resonant magnetic perturbations (RMPs). The experimental results show that the heat flux distribution on the HFS target plate depends significantly on the edge magnetic topology. Furthermore, the impact of hydrogen fueling using supersonic molecular beam injection (SMBI) on the divertor heat flux distributions is studied on J-TEXT with an island divertor configuration. It has been observed that power detachment can be achieved when the radiation front approaches the last closed flux surface (LCFS) after each SMBI pulse. This result may provide a method of access for divertor detachment on a fusion device with a three-dimensional (3D) boundary magnetic structure.

13 C-MFA helps to identify metabolic bottlenecks for improving malic acid production in Myceliophthora thermophila

BackgroundMyceliophthora thermophila has been engineered as a significant cell factory for malic acid production, yet strategies to further enhance production remain unclear and lack rational guidance. 13C-MFA (13C metabolic flux analysis) offers a means to analyze cellular metabolic mechanisms and pinpoint critical nodes for improving product synthesis. Here, we employed 13C-MFA to investigate the metabolic flux distribution of a high-malic acid-producing strain of M. thermophila and attempted to decipher the crucial bottlenecks in the metabolic pathways.ResultsCompared with the wild-type strain, the high-Malic acid-producing strain M. thermophila JG207 exhibited greater glucose uptake and carbon dioxide evolution rates but lower oxygen uptake rates and biomass yields. Consistent with these phenotypes, the 13C-MFA results showed that JG207 displayed elevated flux through the EMP pathway and downstream TCA cycle, along with reduced oxidative phosphorylation flux, thereby providing more precursors and NADH for malic acid synthesis. Furthermore, based on the 13C-MFA results, we conducted oxygen-limited culture and nicotinamide nucleotide transhydrogenase (NNT) gene knockout experiments to increase the cytoplasmic NADH level, both of which were shown to be beneficial for malic acid accumulation.ConclusionsThis work elucidates and validates the key node for achieving high malic acid production in M. thermophila. We propose effective fermentation strategies and genetic modifications for enhancing malic acid production. These findings offer valuable guidance for the rational design of future cell factories aimed at improving malic acid yields.

Optimizing Interface Chemistry with Novel Covalent Molecule for Highly Sustainable and Kinetics-Enhanced Sodium Metal Batteries

Metallic sodium has attracted increasing attention as an ideal anode material for next-generation high energy density and low-cost secondary batteries. However, it is highly desired yet remains challenging to improve their cycling stability and safety due to unstable solid electrolyte interphase and dendrite growth. Herein, a hybrid interface layer composed of Na2Se and Na3P is constructed on the surface of Na (Na@NPS) via in situ spontaneous reaction. The hybrid interface layer with merits of high sodiophilicity and high Na-ion conductivity can effectively induce homogeneous Na-ion flux distribution, accelerate the reaction kinetics and suppress decomposition of electrolyte components. Benefitting from the above advantages, the Na@NPS symmetric cell delivers a long cycle life (1000 h at 1 mA cm–2 and 1 mAh cm–2). Furthermore, the full cell coupling with Na3V2(PO4)3-based cathode provides an exceptionally long lifespan (1500 cycles) at 20 C with a capacity retention of 98.2% and high energy density (226 Wh kg–1). Therefore, the enhanced electrochemical performance illustrates the feasibility of the covalent molecule derived hybrid multifunctional interfaces in solving the irregular deposition of Na-ion and expediting reaction kinetics in Na metal batteries.

Ensemble Kalman Filter Data Assimilation into the Surface Flux Transport Model to Infer Surface Flows: An Observing System Simulation Experiment

Knowledge of the global magnetic field distribution and its evolution on the Sun’s surface is crucial for modeling the coronal magnetic field, understanding the solar wind dynamics, computing the heliospheric open flux distribution, and predicting the solar cycle strength. As the far side of the Sun cannot be observed directly and high-latitude observations always suffer from projection effects, we often rely on surface flux transport (SFT) simulations to model the long-term global magnetic field distribution. Meridional circulation, the large-scale north–south component of the surface flow profile, is one of the key components of the SFT simulation that requires further constraints near high latitudes. Prediction of the photospheric magnetic field distribution requires knowledge of the flow profile in the future, which demands reconstruction of that same flow at the current time so that it can be estimated at a later time. By performing Observing System Simulation Experiments, we demonstrate how the ensemble Kalman filter technique, when used with an SFT model, can be utilized to make “posterior” estimates of flow profiles into the future that can be used to drive the model forward to forecast the photospheric magnetic field distribution.

Comparative Numerical and Experimental Analyses of Conical Solar Collector and Spot Fresnel Concentrator

This paper aims to compare the thermal performances of the conical solar collector (CSC) system and the spot Fresnel lens system (SFL) using water and CuO nanofluid as the working fluids. The studied CFD models for both systems were validated using experimental data. At an optimal flow rate of 6 L/min, the SFL system showed higher optical and thermal performance in comparison with that of the CSC system. In the case of the SFL system, the availability of a greater amount of solar energy per unit collector area caused an increase in thermal energy. Moreover, in the case of the CSC system, the non-uniform distribution of solar flux on the absorber’s outer surface leads to an increase in temperature gradient and heat losses. As a heating medium, the CuO nanofluid outperformed the water in terms of higher thermal conductivity and heat capacity. The average thermal efficiencies of 64.7% and 61.2% were achieved using SFL with and without CuO nanofluid, respectively, which were 2.4% and 0.5% higher than those of the CSC with and without nanofluid. CFD simulations show a 2.80% deviation for SFL and 2.92% for CSC, indicating acceptable accuracy compared to experimental data.

Greenhouse gas concentrations and diffusive fluxes in the middle reach of the Lancang River before and after damming

With increasing concerns of greenhouse gas (GHG) emissions from reservoirs, the debate surrounding whether hydropower qualifies as green energy has become increasingly contentious. However, quantifying the impact of dam construction on river GHG emissions still poses challenging due to the absence of direct observational data on river GHG emissions pre- and post-damming. In the present study, the differences in GHG diffusive fluxes from pre- and post-dammed river channels in the middle reaches of the Lancang River were quantified through three field campaigns in 2016, 2018, and 2023. The results indicate that reservoir construction increases the diffusive fluxes of GHG from the natural river, the total GHG emissions (expressed as carbon dioxide equivalent: CO2eq) from post-dammed river ranged from 234.42 to 527.55 mg CO2eq m−2 d−1 and were 1.4–5.2 times higher than that from pre-dammed river. However, the GHG diffusive fluxes from the reservoirs in the Lancang River remain below the average of global reservoirs. Moreover, river damming enhances the methane (CH4) emissions, the average value of CH4/CO2 ratio (in mg m−2 d−1) was 0.003 post-dam and was 13 times higher than that of pre-dam. Our results revealed that alterations in sediment distribution following reservoir construction emerge as a pivotal factor contributing to the variations in spatial distributions of GHG diffusive fluxes in the studied river channel. Overall, the findings of the study provide a primary and direct evidence of river damming impact on GHG emissions and underscore the necessity of long-term continuous in-situ measurements of GHG fluxes in the rivers undergoing hydropower development.

Evaluation of local erosion and deposition on the W-monoblock of JA-DEMO divertor

We developed and performed a 3D Monte-Carlo simulation on local erosion and deposition in outer divertor target of JA-DEMO. The local erosion and deposition were evaluated considering the divertor plasma parameters and the roof shape of W-monoblocks. First, for local erosion simulation, the trajectories of D ions and impurity (He and Ar) ions were traced before irradiating the monoblock surface. The physical sputtering was calculated by these ions irradiation, considering their charge states, incident-energies and -angles. The highest sputtering flux was produced by multiply-charged Ar ions (Ar3+–Ar5+). The sputtering flux strongly depended on the poloidal positions along the divertor target. Although almost no erosion (sputtering) was obtained in low-temperature zone (∼1 eV), the erosion occurred in mid- (∼10 eV) and high-temperature (>10 eV) zone. Second, for local deposition simulation, the trajectories of W atoms emitted from the monoblock surface were traced. After transport (including ionization and recombination) in the plasma, W atoms and ions were deposited on the monoblock surface. The deposition occurred even on the area where plasmas did not irradiate (shadow). This was due to perpendicular diffusion to magnetic field lines, gyro-orbit effects for W ions and straight-line motion for neutral W atoms. The poloidal distribution of the deposition fluxes was similar to that of the sputtering fluxes. In low-temperature zone, no deposition flux occurred due to very low sputtering flux. The deposition flux in mid-temperature zone increased significantly and that in high-temperature zone remained high.

Experimental and Numerical Studies of Heat Transfer Through a Double-Glazed Window with Electric Heating of the Glass Surface

This paper presents experimental and theoretical studies of heat transfer through single- and double-glazed windows with electrical heating of the internal surfaces. Heating is achieved by applying a voltage to the low emissivity coating of the inner glass. A thermophysical model has been developed to simulate the heat transfer through these units, allowing us to determine their thermal characteristics. Experimental data are used to validate the numerical model. The resulting heat flux and temperature distributions on the external and internal surfaces of electrically heated double-glazed units are analysed. According to the results of experimental and numerical studies, it was found that the adopted electric heating scheme allows 83–85% of the heat to enter the room and 15–17% is removed to the outside. This makes it possible to increase the radiation component of the heat flow from the window to the room and improve the thermal comfort in the room. In general, this article shows that existing industrial windows with low-emissivity glass surface coating can be upgraded with simple and inexpensive modernisation, without compromising the main function of the window—efficient transmission of visible light—and create an additional (backup) heating device that can work effectively together with the existing heating system in the event of a sudden cold snap at low temperatures (below −20 °C), to prevent condensation of water vapour in the windows, and to prevent condensation on the surface of the window facade wall. Formally, a back-up (emergency) heating system is created in the room, which contributes to the energy sustainability of the building and therefore to energy security in general.

Non-equilibrium plasma distribution in the wake of a slender blunted-nose cone in hypersonic flight and its effect on the radar cross section

This paper presents numerical results in hypersonic aerothermodynamics, a critical field within aerospace engineering, traditionally focusing on reentry capsules and more recently, slender hypersonic vehicles. While capsules typically undergo near-field analysis to assess heat flux and pressure distributions for Thermal Protection System sizing, slender bodies demand additional attention to wake dynamics due to plasma presence. Plasma can significantly affect the Radar Cross Section (RCS) of these vehicles, which require tracking during flight due to their maneuvering and sustained flight capability.Utilizing Computational Fluid Dynamics tools, we explore plasma distribution around a slender blunted cone, considering altitudes and Mach number regimes that may characterize gliding or sustained atmospheric hypersonic flight, emphasizing its impact on RCS. Our study integrates an aerothermodynamic model incorporating non-equilibrium relaxation equations for gas composition and energy. By evaluating characteristic plasma quantities, we underscore the importance of wake plasma for subsequent electromagnetic wave analysis, crucial for understanding RCS. Furthermore, we highlight the necessity for collaborative efforts between Computational Fluid Dynamics (CFD) and Computational Electro-Magnetics (CEM) disciplines to address this challenging interdisciplinary problem.

Lensing Point-spread Function of Coherent Astrophysical Sources and Nontrivial Wave Effects

Most research on astrophysical lensing has been conducted using the geometric optics framework, where there exists a clear concept of lensing images. However, wave optics effects can be important for coherent sources, e.g., pulsars, fast radio bursts, and gravitational waves observed at long wavelengths. There, the concept of lensing images needs an extension. We introduce the concept of the “lensing point-spread function” (LPSF), the smoothed flux density distribution of a coherent point source after being lensed, as a generalization of the lensing image concept at finite frequencies. The frequency-dependent LPSF captures the gradual change of the flux density distribution of the source from discrete geometric images at high frequencies to a smooth distribution at low frequencies. It complements other generalizations of lensing images, notably the imaginary images and the Lefschetz thimbles. Being a footprint of a lensing system, the LPSF is useful for theoretical studies of lensing. Using the LPSF, we identify a frequency range with nontrivial wave effects, where both geometric optics and perturbative wave optics fail, and determine this range to be ∣κ∣−1 ≲ ν ≲ 10, with κ and ν being the dimensionless lens amplitude and the reduced observing frequency, respectively. Observation of LPSFs with nontrivial wave effects requires either very close-by lenses or very large observing wavelengths. The potential possibilities are the lensing of gravitational waves, the plasma lensing of Milky Way pulsars, and lensing by the solar gravitational lens.