Magnetic Wind Research Articles

CALCULATION OF POWER TRANSFORMER INDUCTANCE MATRIX

Practical calculations of electric equipment and electric networks are performed using equivalent circuits. Without taking into account saturation phenomena, electromagnetic processes in a power transformer are described by a system of linear equations, which can be represented in matrix form. The transformer inductance matrix contains self and mutual winding inductances, whose values for one phase are determined directly from the passport data containing the results of no-load and short circuit experiments. Mutual inductances between phases depending on the type and size of the magnetic core are not usually given in the transformer documentation. It does not allow to simulate the operating modes that differ from the steady state with uniform phase load.
 
 The purpose of the work is to develop an algorithm for calculating elements of the power transformer inductance matrix according to its passport data and magnetic circuit parameters.
 
 The scientific novelty lies in use of information about magnetic circuit type and size to determine elements of the inductance matrix.
 
 Materials and methods. Methods of linear electric and magnetic circuit theory are used in the work.
 
 Results. The proposed algorithm for calculation of power transformer inductance matrix includes three stages: calculation of self and mutual inductances for one phase; construction of an equivalent circuit for a magnetic circuit and calculation of magnetic fluxes in order to take into account the mutual influence of windings on various phases; assembly of the inductance matrix of three-phase transformer. A method is proposed for constructing the transformer inductance matrix with an arbitrary type of magnetic circuit, phases and windings quantity. The method involves preliminary calculation of the coefficient matrix that characterizes the magnetic circuit and depends mainly on its type and, to a less, on the ratio of geometric dimensions in rods and yokes. The coefficient matrix can be calculated in advance and used to determine the inductance matrix of a wide range of transformers.
 
 Conclusions. 1. The passport data of the transformer do not allow to model transformer operation at unbalanced load validly. 2. An algorithm for determining the inductance matrix of transformers has been developed. 3. A method for compilation of a transformer inductance matrix with an arbitrary type of magnetic circuit, the number of phases and windings is introduced. The practical significance of the algorithm is determined by the simplicity of algorithmizing, as well as by the possibility of using it in computer simulation of electrical circuits with transformers, for example, using the modified nodal analysis method.

Determining the parameters of the dynamic magnetic hysteresis loop of the structural materials by attachable primary transducers

Coercive force, which is one of the structure-sensitive characteristics of ferromagnetic materials, is most often used for magnetic analysis. This approach requires the use of powerful magnetization systems, which leads to an increase in the size of the attachable magnetic probe and, accordingly, to a decrease in the locality of control. The above-mentioned disadvantages can be eliminated by using the parameters of the dynamic magnetic hysteresis loop (MHL), which involves the use of variable magnetic fields. A device for studying the dynamic MHL during cyclic remagnetization for the tasks of testing ferromagnetic materials, namely structural analysis, is proposed. The device consists of a generator of sinusoidal signals, an amplifier, a cur¬rent sensor, an attachable magnetic probe, a two-channel analog-to-digital converter (ADC), and a computer. The attachable magnetic probe is proposed in the form of a U-shaped core with magnetization windings on one leg and measurement winding on the other leg. The signal from the generator through the amplifier and the current sensor enters the magnetization winding. The ADC receives two signals: first one – from the measurement winding, and second – from the current sensor. Both signals from the ADC output in digital form are processed and analyzed by the computer using the developed specialized software in the LabVIEW language. Due to the developed software the filtering and integration of the signal from the measuring winding as well as the MHL plotting and MHL parameters calculation was realized. To reduce interference when determining MHL parameters, averaging over 50 magnetization cycles was performed. The attachable magnetic probe with the dimensions of the contact part 30x17 mm created a field with a strength of 300 A/m at a frequency of 50 Hz. The device was tested on specimens of sheet steel of 09G2S type with a thickness of 4 mm, in which uniaxial tensile stresses were created. The obtained results show the sensitivity of the dynamic MHL area to mechanical tensile stresses, as it increases from 14.8 to 16.8 mT A/m during loading.

Jets from Neutron-Star Merger Remnants and Massive Blue Kilonovae.

We perform three-dimensional general-relativistic magnetohydrodynamic simulations with weak interactions of binary neutron-star (BNS) mergers resulting in a long-lived remnant neutron star, with properties typical of galactic BNS and consistent with those inferred for the first observed BNS merger GW170817. We demonstrate self-consistently that within ≲30 ms postmerger magnetized (σ∼5-10) incipient jets emerge with asymptotic Lorentz factor Γ∼5-10, which successfully break out from the merger debris within ≲20 ms. A fast (v≲0.6c), magnetized (σ∼0.1) wind surrounds the jet core and generates a UV/blue kilonova precursor on timescales of hours, similar to the precursor signal due to free neutron decay in fast dynamical ejecta. Postmerger ejecta are quickly dominated by magnetohydrodynamically driven outflows from an accretion disk. We demonstrate that, within only 50ms postmerger, ≳2×10^{-2}M_{⊙} of lanthanide-free, quasispherical ejecta with velocities ∼0.1-0.2c is launched, yielding a kilonova signal consistent with GW170817 on timescales of ≲5 d.

Stator Core Loss Analysis and Suppression of Permanent Magnet Vernier Machines

Permanent magnet vernier machine is a new type of permanent magnet machine, which is operated based on flux modulation effect. In this paper, the stator core loss of the permanent magnet vernier machine is investigated based on the relationship of the stator core loss to harmonic components of the airgap magnetic flux density crossing airgap and leakage flux not crossing airgap. In this manner, the stator core loss can be analyzed with consideration of flux modulation effect and flux modulation pole structure. First, stator core losses generated by magnetic fields of permanent magnet and armature winding are analyzed. Then, stator tooth fluxes coming from airgap magnetic flux and leakage flux are derived. Also, non-working stator tooth flux contributing stator core loss but not contributing torque is identified. Based on this, two approaches to suppress the stator core loss of the permanent magnet vernier machine are proposed. Finally, the effectiveness of the stator core loss analysis and suppression is verified by finite element analysis and experiments.

Stars Bisected by Relativistic Blades

We consider the dynamics of an equatorial explosion powered by a millisecond magnetar formed from the core collapse of a massive star. We study whether these outflows—generated by a priori magneto-centrifugally driven, relativistic magnetar winds—might be powerful enough to produce an ultrarelativistic blade (“lamina”) that successfully carves its way through the dense stellar interior. We present high-resolution numerical special-relativistic hydrodynamic simulations of axisymmetric centrifugally driven explosions inside a star and follow the blast wave propagation just after breakout. We estimate the engine requirements to produce ultrarelativistic lamina jets and comment on the physicality of the parameters considered. We find that sufficiently collimated—half-opening angle θ r ≤ 0.°2—laminas successfully break out of a compact progenitor at ultrarelativistic velocities () and extreme isotropic energies (E k,iso ∼ 5 × 1052 erg) within a few percent of the typical spin-down period for a millisecond magnetar. The various phases of these ultrathin outflows, such as collimation shocks, Kelvin–Helmholtz instabilities, and lifetime, are discussed, and we speculate on the observational signatures echoed by this outflow geometry.

Torque Ripple and Mass Comparison between 20 MW Rare-Earth and Ferrite Permanent Magnet Wind Generators

This article investigates the comparison between two configurations of 20 MW offshore synchronous wind generators using ferrite and rare-earth permanent magnets. The optimization-based comparison concerns the torque ripple and active mass, which are two crucial criteria for offshore wind generators. Both generators adopt surface-mounted permanent magnet type with direct-drive technology to avoid problems associated with the gearboxes. The result shows that at the full-load condition, the ferrite permanent magnet generator can reduce the torque ripple to as much as 0.12%, while the rare-earth counterpart can be about 2.5 times lighter than the former one.

The magnetized and thermally unstable tails of jellyfish galaxies

ABSTRACT Jellyfish galaxies are promising laboratories for studying radiative cooling and magnetic fields in multiphase gas flows. Their long, dense tails are observed to be magnetized, and they extend up to 100 kpc into the intracluster medium (ICM), suggesting that their gas is thermally unstable so that the cold gas mass grows with time rather than being fully dissolved in the hot wind as a result of hydrodynamical interface instabilities. In this paper, we use the arepo code to perform magnetohydrodynamical windtunnel simulations of a jellyfish galaxy experiencing ram-pressure stripping by interacting with an ICM wind. The ICM density, temperature and velocity that the galaxy encounters are time-dependent and comparable to what a real jellyfish galaxy experiences while orbiting the ICM. In simulations with a turbulent magnetized wind, we reproduce observations, which show that the magnetic field is aligned with the jellyfish tails. During the galaxy infall into the cluster with a near edge-on geometry, the gas flow in the tail is fountain-like, implying preferential stripping of gas where the rotational velocity vectors add up with the ram pressure while fall-back occurs in the opposite case. Hence, the tail velocity shows a memory of the rotation pattern of the disc. At the time of the nearest cluster passage, ram-pressure stripping is so strong that the fountain flow is destroyed and instead the tail is dominated by the removal of gas. We show that gas in the tail is very fragmentative, which is a prediction of shattering due to radiative cooling.

PCB-Based Magnetic Integration and Design Optimization for Three-Phase LLC

The LLC converter is the most efficient topology in server and telecom applications. Furthermore, three interleaved LLC converters have been shown to yield efficiency advantages at power levels of several kilowatts. The magnetic components of a multiphase LLC, on the other hand, are complex, laborious, and difficult to build cost-effectively. To solve these issues, a high-frequency GaN-based three-phase LLC converter is used in this research. With GaN working at 500 kilohertz, all magnetic components, including six inductors and six transformers, can be housed in a single structure, while all magnetic windings are accommodated in a 4-layer PCB with 3 oz. of copper. Furthermore, EMI can be improved by virtue of three-phase by 20 dB and additional two layers is added to gain additional 20-30 dB reduction from a low frequency up to 30 MHz. The proposed construction is simple and economically mass-producible. A 6kW 400V/48V 3-phase prototype is implemented, with a peak efficiency over 99% and a power density of 1000 W/in3 (61 kW/L).

EVOLUTION OF THE X-RAY BINARY SYSTEM Sco X-1

The possible evolution of a bright low-mass X-ray binary system Sco X-1 is numerically investigated within the framework of a model assuming that the donor of the system (a satellite of a neutron star) fills its Roche lobe. The calculations take into account a strong induced stellar wind (ISW) of the donor, which occurs due to irradiation by hard radiation of an accreting relativistic star. At the same time, using the example of Sco X-1, three hypotheses are investigated, within the framework of which a high rate of mass exchange can be obtained for semi-separated X-ray binary stars. The first hypothesis is the presence of a strong ISW of the donor with standard magnetic braking. Calculations have shown that in this case it is possible to obtain a high rate of mass exchange, but at the same time the donor cannot fill the Roche lobe – it “goes under it”. The second hypothesis is an increase of magnetic braking, that is, an increase of the loss of angular momentum from the system due to the magnetic stellar wind of the donor (MSW). Such an amplification may be associated with the intense ISW of the donor in the presence of a strong magnetic field. Numerical modeling shows that with an increase of MSW by ~20 times, a high rate of mass exchange is possible when the donor fills the Roche lobe. The third hypothesis suggests the possibility of canceling the direct exchange of angular momentum between the orbital moment of the system and the moment of accreted matter passing from a low-mass donor to a more massive accretor. With such cancellation, the main process, increasing the semi-axis of the orbit, disappears. Calculations show that in this case it is possible to obtain a sufficiently high rate of mass exchange. However, the most likely reason for the increase of the rate of mass exchange in low-mass X-ray binary systems is probably the increase of magnetic braking.

Investigation of the consistency of changes in the wavelet phase characteristics of heliocosmic and climatic variables and changes in the components of the word water balance. Part 3

The relevance of research is due to the need to establish the true causes and patterns of changes in the hydrometric characteristics of the global water balance as a whole and its components, natural environments on Earth.
 Purpose of research: to establish patterns and causes of changes in the components of the global water balance on Earth: trends in the increase in water mass reserves, dynamics of the water balance of the South Seas of Europe, the Black Sea, dynamics of levels of the Indian Ocean, Western and Central Pacific Ocean, Eastern Pacific and Atlantic Oceans, level fluctuations of the World Ocean, changes in sea ice cover, river resources, reservoirs, variability of sea levels in the Arctic Ocean according to observations at coastal stations, changes in the balance of ice masses in glacial areas, changes in interseasonal thawing in the permafrost zones of the continent.
 Objects of research: time series of heliocosmic, climatic variables, hydrometric characteristics of the components of the global water balance.
 Methods of research: developed by the author, a method for interaction of observations of a variable or variables with groups of specified heliocosmic and climatic factors in the frequency and time domains, obtained using wavelet transformations of observations with the calculation of phase-frequency and phase-time characteristics in equal time intervals; comparative analysis of changes in the obtained phase characteristics of groups of variables with the calculation of their consistency matrices and the construction of graphs in the frequency and time domains.
 Main results of research: in the observed time intervals, in the trend of periodic changes in the reserves of water masses (glaciers, groundwater, lakes, reservoirs) in the frequency domain, significant negative unidirectional influences of changes in heliocosmic and multidirectional influences of climatic variables are observed; changes in the reserves of the World Ocean are more influenced by multidirectional changes in climatic factors compared to the unidirectional influences of heliocosmic factors; the frequency bands of changes in water mass reserves are less than the frequency bands of heliocosmic factors, due to the influence of climatic factors; The variability of water mass reserves on Earth is significantly and unidirectionally consistent with changes in evaporation, evaporation and precipitation on land, and in different directions with changes in cloudiness on the continents. Similar patterns are observed in changes in the water balance of the South Seas in Europe, the Black Sea, sea ice cover, river resources, reservoirs on land, ice mass balances in glacial areas, and interseasonal thawing in the permafrost zones of the continent. In the dynamics of the levels of the Indian, Western and Central Pacific, Eastern Pacific and Atlantic, fluctuations in the level of the World Ocean, significant multidirectional changes in levels in the frequency domain are observed on the effects of heliocosmic variables with the greatest influences of solar activity and long-term solar insolation, the Ap-index of geomagnetic activity, the parameter the ratio of plasma pressure to magnetic solar wind, changes in the ozone layer in the ionosphere, changes in the speed of rotation of the Earth, the influence of climate variables in the frequency domain with the greatest influences of solar radiation, precipitation, atmospheric processes, PTV is significant. The variability of sea levels in the Arctic Ocean, as observed at coastal stations, is characterized by significant multidirectional changes in the frequency domain due to the influence of heliocosmic and climatic factors, multidirectional mutual consistency of the component levels of the ocean seas, caused by anomalous changes in the Earths magnetic field in areas of the Northern Hemisphere. In changes in the phase-time characteristics of groups of factors, lattice structures of differently and unidirectional changes in variables are observed, characterizing the cyclical nature of climate change on Earth.

Power electronic transformers in wind power generation systems

Power electronic transformers (PET) have received extensive attention due to their advantages of compensating reactive power, improving power quality, limiting fault current, and reducing the number of traditional transformer materials. At present, it is also an important research topic to replace conventional power frequency transformers in wind power generation systems. This paper first analyzes the topology of the primary circuit of the power electronic transformer. Several common PET topologies are introduced, and their advantages and disadvantages are analyzed. The module cascade topology and modular multilevel topology are mainly introduced, and the topological circuit diagram is given. The paper also comprehensively introduced the research status quo of wind power generation systems based on the interface of PET, permanent magnetic wind turbines and dual-feed fan wind turbines. The traditional structure of the permanent magnet synchronous generator (PMSG) system and the doubly fed induction generation (DFIG) system is shown. Finally, the problems in PET and the future development direction are summarized.

Superconducting joints of reacted monofilament MgB2 wires sintered by hot uniaxial pressing system

Successful superconducting joints of reacted magnesium diboride (MgB2) monofilament wires are reported in this paper. The absence of a reliable method to develop superconducting joints between reacted MgB2 wires presents a major obstacle to the wider adoption of MgB2 as a material for magnet winding. A hot uniaxial pressing (HUP) system was exploited for sintering purposes since it can facilitate the formation of condensed in situ bulk on the wire filament. The wires were manufactured with an extra thick barrier material to protect the filament from damage during HUP sintering. The sintering temperature and pressure of the HUP system were varied to comprehend the best-performing joint. The performance of joints could be improved by depreciating the pores within the intermediate bulk of the joint. To prove this point, joints were cut to study their morphology. However, due to sintering in pressurised conditions, the reaction of the in situ intermediate bulk was not completed. The x-ray diffraction result detected a significant unreacted magnesium phase in the intermediate bulk. This work obtained joints of reacted MgB2 wires which can be considered for industrial MgB2 magnetic resonance imaging magnets fabrication.

Computation of Winding-Based Magnetic Helicity and Magnetic Winding Density for SHARP Magnetograms in Spherical Coordinates

Magnetic helicity has been used widely in the analysis and modelling of solar active regions. However, it is difficult to evaluate and interpret helicity in spherical geometry since coronal magnetic fields are rooted in the photosphere and helicity is susceptible to gauge choices. Recent work extended a geometrical definition of helicity from Cartesian to spherical domains, by interpreting helicity as the average, flux-weighted pairwise winding of magnetic-field lines. In this paper, by adopting the winding-based definition of helicity, we compute helicity and winding in spherical coordinates for SHARP (Spaceweather HMI Active Region Patches) magnetograms. This is compared with results obtained in Cartesian coordinates to quantitatively investigate the effect of spherical geometry. We find that the Cartesian approximations remain mostly valid, but for active regions with large spatial extents or strong field strengths (usually leading to flares and coronal mass ejections) there are significant deviations due to surface curvature that must be accounted for.

Analisa kerusakan bearing mesin kompresor sekrup boge tipe bs 102 nr 1023728

A compressor is a machine for compressing air or gas. Air compressors usually suck airfrom the atmosphere. However, there are also those who suck in air or gas at a pressurehigher than atmospheric pressure. In this case the compressor works as a booster. On theother hand, there are also compressors that suck in gas at a lower pressure than atmosphericpressure. One of the problems with screw type compressor machines is overload, where thepower required by the compressor is less than the normal requirement, which forces themachine to work more than its capacity limits. In the end, this results in the electric motorgetting hotter (overheating) and can even cause a fire in the magnetic winding. The aim ofthis research is to analyze the causes of bearing damage to the Boge type BS 102 screwcompressor machine outside the maintenance schedule which disrupts the productionprocess. This research used field observation methods, reviewing machine manuals andliterature studies. The results of the research show that the cause of damage to bearings inbogie screw compressor machines is due to an incorrect lubrication system and installationand repair of bearings that do not comply with procedures

Magnetic field effects on nucleosynthesis and kilonovae from neutron star merger remnants

ABSTRACT We investigate the influence of parametric magnetic field configurations of a hypermassive neutron star (HMNS) on the outflow properties, nucleosynthesis yields, and kilonova light curves. We perform three-dimensional dynamical space–time general-relativistic magnetohydrodynamic simulations, including a neutrino leakage scheme, microphysical finite-temperature equation of state, and an initial poloidal magnetic field. We find that varying the magnetic field strength and falloff impacts the formation of magnetized winds or mildly relativistic jetted outflows, which in turn has profound effects on the outflow properties. All of the evolved configurations collapse to a black hole ∼38–40 ms after coalescence, where the ones forming jetted outflows seem more effective at redistributing angular momentum, which result in earlier collapse times. Larger mass ejecta rates and radial velocities of unbound material characterize the systems that form jetted outflows. The bolometric light curves of the kilonovae and r-process yields that are produced by the post-merger remnant system change considerably with different magnetic field parameters. We conclude that the magnetic field strength and falloff have robust effects on the outflow properties and electromagnetic observables. This can be particularly important as the total ejecta mass from our simulations (≃10−3 M⊙) makes the ejecta from HMNS a compelling source to power kilonova through radioactive decay of r-process elements.

Design optimization of a stepped HTS magnet for electrodynamic suspension train

High-temperature superconducting (HTS) magnets are promising candidates for transportation and power systems, such as the electrodynamic suspension (EDS) train, ultra-high field magnet and magnetic resonance imaging, because of their large current-carrying capacity and low power loss. The critical current depending on magnetic flux density is an essential factor in assessing the application performance of HTS magnets. As usual, the existing HTS magnet is wound with rectangular cross-section, which results in magnetic field concentration inside winding. In this paper, we propose a novel HTS magnet structure with stepped cross-section to alleviate the magnetic field concentration, and resultantly improve the critical current. From this point, this paper aims to design and optimize a stepped HTS magnet with the critical current maximized. Firstly, the structure design of the stepped HTS magnet is performed with a consideration of the application scenario of EDS train. Then, the critical current of the HTS magnet is estimated with a homogenized self-consistent model. Afterwards, an HTS magnet with stepped cross-section is optimized, fabricated and finally tested. The critical current was experimentally measured to verify the simulation results, followed by the electromagnetic investigations of the stepped HTS magnet in the EDS train.

Towards a holistic magnetic braking model from the evolution of cataclysmic variables to stellar spin-down – I: the spin-down of fully convective M-dwarfs

ABSTRACT We extend a magnetic braking (MB) model, which has been used earlier to address the evolution of cataclysmic variables, to address the spin period Pspin evolution of fully convective M-dwarf (FCMD) stars. The MB mechanism is an α–Ω dynamo, which leads to stellar winds that carry away angular momentum. We model our MB torque such that the FCMDs experience an MB torque, approximately scaling as $P_\mathrm{spin}^{-1}$ at shorter periods, before transitioning into a Skumanich-type MB torque, scaling as $P_\mathrm{spin}^{-3}$. We also implement a parametrized reduction in the wind mass-loss owing to the entrapment of winds in dead zones. We choose a set of initial conditions and vary the two free parameters in our model to find a good match of our spin trajectories with open clusters containing FCMDs such as NGC2547, Pleiades, NGC2516, and Praesepe. We find that our model can explain the long spin periods of field stars and that a spread in spin distribution persists till over 3 Gyr. An advantage of our model is in relating physically motivated estimations of the magnetic field strength and stellar wind to properties of the stellar dynamo, which other models often remain agnostic about. We track the spin dependence of the wind mass-losses, Alfvén radii, and surface magnetic fields and find good agreement with observations. We discuss the implications of our results on the effect of the host FCMD on any orbiting exoplanets and our plans to extend this model to explain solar-like stars in the future.

A LOFAR prompt search for radio emission accompanying X-ray flares in GRB 210112A

ABSTRACT The composition of relativistic gamma-ray burst (GRB) jets and their emission mechanisms are still debated, and they could be matter or magnetically dominated. One way to distinguish these mechanisms arises because a Poynting flux dominated jet may produce low-frequency radio emission during the energetic prompt phase, through magnetic reconnection at the shock front. We present a search for radio emission coincident with three GRB X-ray flares with the LOw Frequency ARray (LOFAR), in a rapid response mode follow-up of long GRB 210112A (at z ∼ 2) with a 2 h duration, where our observations began 511 s after the initial Swift-BAT trigger. Using time-sliced imaging at 120–168 MHz, we obtain upper limits at 3σ confidence of 42 mJy averaging over 320 s snapshot images, and 87 mJy averaging over 60 s snapshot images. LOFAR’s fast response time means that all three potential radio counterparts to X-ray flares are observable after accounting for dispersion at the estimated source redshift. Furthermore, the radio pulse in the magnetic wind model was expected to be detectable at our observing frequency and flux density limits which allows us to disfavour a region of parameter space for this GRB. However, we note that stricter constraints on redshift and the fraction of energy in the magnetic field are required to further test jet characteristics across the GRB population.

Black Hole Accretion with Saturated Magnetic Pressure and Disk Wind

We construct an analytical black hole accretion disk model that incorporates both magnetic pressure and disk wind, which are found to be important from numerical simulations. A saturated magnetic pressure that relates the Alfvén velocity with local Keplerian velocity and gas sound speed is assumed in addition to radiation and gas pressures. The mass accretion rate is assumed to have a power-law form in response to mass loss in the wind. We find three sets of self-consistent solutions that are thermally stable and satisfy the model assumptions. At high accretion rates, the disk is geometrically and optically thick, resembling the slim disk solution. At relatively low accretion rates, our model predicts an accretion flow consisting of a geometrically thin and optically thick outer disk (similar to the standard disk), and a geometrically thick and optically thin inner disk (similar to the advection-dominated accretion flow, or ADAF). Thus, this is a natural solution for a truncated disk connected with an inner ADAF, which has been proposed to explain some observations. The magnetic pressure plays a more important role than the outflow in shaping the disk structure. The observed disk luminosity tends to saturate around 8 times the Eddington limit, suggesting that supercritical accretion onto black holes can be used for a black hole mass estimate, or a standard candle with known black hole masses.

Living with a Red Dwarf: The Rotation–Age Relationships of M Dwarfs

Age is a fundamental stellar property, yet for many stars, it is difficult to reliably determine. For M dwarfs, it has been notoriously so. Due to their lower masses, core hydrogen fusion proceeds at a much slower rate in M dwarfs than it does in more massive stars like the Sun. As a consequence, more customary age determination methods (e.g., isochrones and asteroseismology) are unreliable for M dwarfs. As these methods are unavailable, many have searched for reliable alternatives. The M dwarfs comprise the overwhelming majority of the nearby stellar inventory, which makes the determination of their fundamental parameters even more important. Further, an ever-increasing number of exoplanets are being found to orbit M dwarfs, and recent studies have suggested that they may have a relatively higher number of low-mass planets than other spectral types. Determining the ages of M dwarfs then allows us to better study any hosted exoplanets as well. Fortunately, M dwarfs possess magnetic activity and stellar winds like other cool dwarf stars. This causes them to undergo the spin-down effect (rotating with longer periods) as they age. For this reason, stellar rotation rate has been considered a potentially powerful age determination parameter for over 50 yr. Calibrating reliable age–rotation relationships for M dwarfs has been a lengthy process, but here we present the age–rotation relationships for ∼M0–6.5 dwarfs, determined as part of the Living with a Red Dwarf program. These relationships should prove invaluable for a wide range of stellar astrophysics and exoplanetary science applications.