Eddy Currents

Eddy currents are closed circulating currents induced inside a bulk conductor. A thin wire constrains charge motion to one obvious path. A metal plate, disk, or transformer core contains many possible closed paths, each of which can enclose changing magnetic flux. Faraday's law produces an emf around those paths; finite conductivity allows the emf to drive local current loops.

The induced loops follow Lenz's law. Their magnetic fields oppose the change in external flux through the local material. A changing field through a stationary plate can heat the plate. A plate moving into or out of a field region can experience a magnetic drag force. The electrical energy comes from the varying field source or from the mechanical agent that moves the conductor. The local current patterns can be complicated because current paths depend on conductor shape, field geometry, and conductivity.

Current Paths and Heating

Consider a wide metal plate in a magnetic field directed into the page. If the field magnitude increases, the into-page flux through every small loop drawn in the plate increases. The induced currents circulate so their magnetic contribution points out of the page. Viewed from the page, the conventional current is counterclockwise around each local loop. The loops overlap in a real sheet; the diagram represents the direction of the distributed current density rather than separate insulated wire rings.

If the into-page field decreases, the current circulation reverses. The induced magnetic contribution then points into the page and tends to preserve the waning external flux. The direction decision uses the field change, rather than the field value. A large steady field through a stationary plate can have zero eddy current after transients die out.

The circulation can also arise from conductor motion. Pull a metal sheet rightward out of the gap between magnet poles. A material loop carried with the sheet has decreasing flux through the field region. Its induced current produces into-page magnetic field and experiences a magnetic force leftward in the gap. The force opposes the sheet motion and converts mechanical work into Joule heating in the sheet.

Current density and resistive heating

At each point in an ordinary isotropic conductor, the local current density satisfies

where $\sigma$ is conductivity and ${\vec{E}}_{\mathrm{eff}}$ includes the induced driving field and, for moving material, the magnetic-force contribution per unit charge. A moving-medium expression is

when charge accumulation and material properties permit a local ohmic model. The current density closes through the conductor so that charge does not accumulate indefinitely at interior points.

Joule heating per unit volume is

The total eddy-current power is the volume integral of this nonnegative density. Large conducting loops, high conductivity, strong field changes, and low-resistance paths can therefore create substantial heat. A material's magnetic permeability can also alter the field distribution and change the induced-current pattern, especially in ferromagnetic cores.

The heating direction does not reverse with current direction. Reversing an eddy loop reverses $\vec{J}$, while $J^2/\sigma$ remains positive. An alternating magnetic field drives alternating eddy currents and deposits heat during both halves of the cycle. This property supports induction heating and also creates unwanted core loss in transformers and motors.

Magnetic Damping

Magnetic damping uses eddy currents to oppose motion without mechanical contact. A conducting plate attached to a pendulum can swing through the gap of a permanent magnet. As the plate enters the gap, material loops acquire increasing flux. As it exits, the same loops acquire decreasing flux. In both intervals, induced currents produce magnetic forces opposite the plate velocity. The plate loses mechanical energy, and the sheet warms by a small amount.

For sufficiently slow motion and a fixed geometry, the eddy-current drag often has the approximate form

where the coefficient $c$ depends on field strength, plate dimensions, conductivity, and the distribution of accessible current loops. The approximation follows because the flux rate is proportional to speed, the induced current is proportional to emf, and the magnetic force on that current is proportional to current. At larger speed or high frequency, field redistribution, inductance, skin effects, and nonuniform geometry can alter the simple linear relation.

The mechanical power removed by a linear magnetic damper is

The negative sign refers to mechanical energy of the moving object. The matching positive thermal power in the conducting plate is $c{v}^2$ under the ideal damping model. The sign relation prevents a magnetic brake from accelerating a freely moving plate; the induced-current force must oppose the relative motion that created it.

In a damped oscillator with mechanical restoring force, eddy-current damping decreases the amplitude on successive cycles. The damping force changes direction with velocity, so it removes energy on both halves of the motion. It does not permanently shift the equilibrium position of an ideal symmetric oscillator. A strong enough damper can make the return to equilibrium slow without oscillation; the exact underdamped or overdamped condition depends on the mass, restoring constant, and damping coefficient.

Brakes and Core Loss

Eddy-current brakes use the same drag without brake pads pressing on a moving surface. A conducting disk passing between magnet poles, an aluminum track under a moving magnet, or a vehicle rail beneath an energized electromagnet can provide braking force. The brake becomes weak at low relative speed because the flux rate and induced current decrease. Mechanical friction brakes supply holding force near a complete stop, while eddy brakes supply contactless braking at moderate and high speed.

The dissipated heat appears in the conductor that carries the eddy loops. A rail brake can heat the rail; a disk brake can heat the disk. Thermal design must account for the heat capacity, conduction away from the active region, and repeated braking cycles. Slots or laminations that reduce unwanted transformer-core loss would also reduce the current paths needed for a strong eddy-current brake. The desired design depends on whether the current loops are intended to dissipate energy or must be suppressed.

Suppressing eddy-current loss in cores

Transformer cores, motor stators, and inductors often carry a changing magnetic flux through ferromagnetic material. The high permeability of the core guides magnetic flux, but the metal can also provide broad low-resistance paths for eddy currents. A solid core behaves like a large conducting plate: changing flux drives loops through its bulk, and their Joule heating wastes energy from the applied electrical source.

Laminating the core interrupts those broad loops. A laminated core is built from thin metal sheets separated by insulating coatings. Each sheet can still support small eddy loops within its own thickness, but the insulating layers block current that would cross from one sheet into the next. The effective resistance of a large circulating path rises sharply, and the path area available to a single loop falls. Both effects reduce the eddy-current magnitude.

Thin laminations are stacked parallel to the dominant magnetic field. That orientation keeps the desired magnetic flux largely within the sheets while placing insulation across the broad transverse current loops. A poor lamination orientation can increase magnetic reluctance or leave large current paths available. Core design balances magnetic permeability, saturation, mechanical strength, manufacturing cost, and the frequency of the applied field.

At higher frequencies, thinner sheets are required because the induced emf rises with the flux rate. Very high-frequency magnetic components may use ferrite or powdered-iron cores. Ferrites have much higher electrical resistivity than steel, which strongly limits eddy current. Powdered cores isolate small magnetic grains with insulating binder, again breaking long current paths. The material choice modifies both magnetic and electrical properties of the core.

Slots in a conducting plate produce a related effect. A disk with radial slots loses the wide circular paths that would carry large eddy currents. The slots raise path resistance and limit loop area. The same change reduces magnetic braking torque, so a slotted disk reduces heating and drag where those effects are unwanted, while it is unsuitable as the active element of an eddy-current brake.

Circuit Models

A simple calculation estimates an eddy loop by replacing a distributed current path with a representative loop of area $A$, resistance $R_{\mathrm{loop}}$, and approximately uniform perpendicular field. If the field changes at rate $\mathbf{d}B/\mathbf{d}t$, Faraday's law gives

The estimated heating is

Large low-resistance loops dissipate substantial power in a changing field. Doubling loop area raises the emf by two and the resistive heating by four when resistance is unchanged. Real eddy patterns contain many loops, shared current paths, inductance, and nonuniform field. The expression gives a scaling estimate, not an exact core-loss calculation.

A lamination of thickness $t$ has characteristic loop dimensions and resistance that change with $t$. Decreasing thickness generally reduces the loop area faster than it reduces resistance, lowering the loss. The detailed dependence includes material resistivity and magnetic field distribution, but the direction of the design change follows directly from the loop estimate.

Frequency and Penetration

An alternating source often produces a field that is approximately sinusoidal in a core region,

The field rate is

A representative current loop of area $A$ has induced-emf amplitude $\omega A{B}_0$. Doubling the frequency doubles that emf amplitude. In the resistive loop estimate, current amplitude also doubles, and average heating grows approximately with the square of frequency. Doubling the field amplitude has the same quadratic effect on this loss estimate.

The simple square-frequency relation applies only while the field penetrates the conductor and the local resistive response follows the source without large phase delay. At sufficiently high frequency, induced currents alter the interior field distribution. Current can concentrate near a surface, and the representative-loop estimate becomes inadequate. Magnetic devices then require electromagnetic field analysis, material data, and thermal measurements rather than a single core-loss coefficient.

At fixed frequency, a waveform with sharp transitions has large $\mathbf{d}B/\mathbf{d}t$ over short intervals. Such transitions can drive strong transient eddy currents even when the field's average value is small. Smooth sinusoidal excitation distributes the flux rate through the cycle. The field rate, not the mean field, sets the induced emf around a material path.

Induction heating versus core loss

The local heating density follows from the current distribution,

• Induction heater: a driven coil places a changing field inside the selected workpiece volume. Coil shape, frequency, resistivity, thickness, and thermal contact determine where $J$ and deposited power are greatest.
• Transformer core or motor housing: the same circulation becomes a loss channel. Temperature rise changes resistance, magnetic properties, and insulation lifetime. Laminations, high-resistivity material, reduced field rate, and cooling reduce this contribution.
• Thermal calculation: combine the electrical loss with conduction through the object, convection and radiation at its surface, heat capacity, and duty cycle. A short high-power pulse and a continuous low-power drive can have different peak temperatures at the same average energy.
• Direction and magnitude: Lenz's-law direction fixes the local current sense and the force direction on moving material. The field solution provides $J$, heating, and force density. These are separate outputs of the same changing-flux source term.

Geometry of distributed current paths

An eddy-current drawing should not be read as a circuit diagram with a single resistor. The metal contains a continuum of possible paths. Current density must satisfy charge conservation,

inside a region where charge density is steady. At an insulating outer surface, the normal component of current density is zero. These conditions bend current paths so they close within the available conductor. Sharp corners, holes, slots, thin necks, and changes in conductivity all reshape the loops.

A circular plate in a spatially uniform changing perpendicular field has circular symmetry. The induced electric field and eddy current run along circles centered on the plate axis. A rectangular plate breaks that symmetry: current follows rounded loops that turn before reaching the boundaries. A hole forces the current to circulate around the hole, often increasing local current density near its edge. The qualitative direction still follows Lenz's law, while the magnitude requires a solution of the field and current distribution.

Thickness adds a second spatial coordinate to the current model.

• Thin sheet at low frequency: current density can be approximately uniform through the thickness, so a planar material-loop estimate can establish scale and direction.
• Thick conductor or rapid drive: the eddy-current field changes the internal applied-field distribution. Surface and interior regions require a spatial electromagnetic solution.
• Moving sheet: loops near the leading and trailing field boundaries have different flux histories. Current density follows the flux rate through each material loop.
• Stationary sheet in a uniform ramp: the geometry can support a more symmetric circulation, although holes, slots, and material variations still redirect paths.

Energy and Force

Take a conducting disk of moment of inertia $I_{\mathrm{rot}}$ rotating with angular speed $\Omega$ through a fixed magnetic gap. Suppose the low-speed braking torque is

Newton's rotational law gives

so the angular speed decays as

The rotational kinetic energy is $K=I_{\mathrm{rot}}{\Omega }^2/2$. Its time derivative is

The positive heat rate in the disk equals $\gamma {\Omega }^2$ in the ideal model. A brake that brings a disk from one angular speed to another converts the corresponding loss of rotational kinetic energy into internal energy of the conducting disk and nearby parts. No friction pad needs to touch the rotating surface.

The low-speed coefficient in ${\tau }_{\mathrm{eddy}}=-\gamma \Omega$ belongs to the complete geometry rather than to the conductor alone.

Record the field map, disk temperature, speed range, and magnet position before fitting $\gamma$. A value measured at one gap or speed range cannot be transferred to a different brake geometry without a new current-path model.

Eddy currents versus other magnetic losses

Eddy-current loss is resistive heating from induced charge circulation. It changes when laminations, slots, or high-resistivity material interrupt the available current paths. Magnetic material can dissipate energy through a separate magnetization process. A measured device loss can contain both channels together with winding resistance:

Lamination is a differential test for the eddy-current term when the field waveform, core geometry, and winding state are held fixed. Residual heating after lamination points to magnetic-material loss, winding loss, thermal paths, or an incomplete electrical isolation of the sheets.

Force distribution near a field boundary

The local force density is

Magnetic braking concentrates where material loops enter or leave a field region, or where field strength changes across the conductor. A rigid plate fully inside a spatially uniform region can contain local charge redistribution and circulating current patterns with a small net translational force.

A wide sheet moving rightward out of an into-page field has a local loop near the trailing boundary that loses into-page flux. The induced circulation produces into-page field through that loop. The portion of the loop inside the gap has leftward $\vec{J}\times {\vec{B}}_{\mathrm{ext}}$ force. Away from the boundary, paired local contributions can reduce the net force density.

The same analysis applies to a moving magnet above a stationary conducting sheet. The magnet's field pattern moves relative to the metal, so the flux through nearby material loops changes. The induced currents produce fields that oppose that changing pattern. The resulting interaction can exert a retarding force on the magnet, a force on the sheet, or equal and opposite forces on both when the apparatus permits motion.

At very small separation between a magnet and a conductor, the field gradient can be large and the current paths confined to a shallow region. The force becomes sensitive to gap distance, plate thickness, and conductivity. A simple $F=-cv$ relation may hold over a limited low-speed range, but the coefficient is a property of the full geometry rather than a universal constant for the material.

Energy loss in a damped pendulum

Suppose a pendulum plate has mass $m$, moves through a magnetic gap, and experiences an approximately linear tangential damping force $-cv$. Its mechanical energy is

The rate of change of that energy from the eddy-current force is

The energy lost during one swing through the gap is

The corresponding positive thermal energy appears in the plate and magnet assembly. The integral emphasizes that damping depends on the speed history. A fast swing loses more energy than a slow swing through the same field region. A narrow magnet gap limits the time within the field but can have a large field strength; the net damping requires the actual velocity and force profile along the path.

Identify the source that maintains the flux rate before assigning the heating budget.

• Freely decaying pendulum or disk: mechanical energy becomes heat.
• Stationary conductor in a driven electromagnet: electrical input to the field source becomes heat.
• Motor-driven disk in a permanent-magnet gap: motor work becomes heat at constant speed.

The current paths and force distribution depend on geometry. The energy source follows the actuator or field supply that maintains the changing flux.

Field penetration and effective resistance

Eddy currents produce fields that oppose changes in the applied field inside a bulk conductor. The penetration scale for a homogeneous linear material under sinusoidal excitation is

where $\omega$ is angular frequency, $\sigma$ is conductivity, and $\mu$ is the relevant permeability. The relation provides a regime check, not a complete geometry solution.

An effective eddy-path resistance summarizes one geometry and frequency. Material conductivity, permeability, dimensions, boundary conditions, and field waveform can change its fitted value. Use the lumped loop for sign and scale; use a field solution when penetration or gradients control the result.

Measurement and Design

Eddy currents are inferred from their power, force, or impedance consequences because the circulation has no single wire terminal.

• Force comparison: measure the additional pull force on a conducting plate or the decay envelope of a damped vane, then subtract friction and air drag from a matched field-off control.
• Thermal comparison: record temperature or calorimetric power under a stated field waveform, duty cycle, cooling condition, and conductor geometry.
• Electrical comparison: measure coil voltage and current, or complex probe impedance, while controlling target spacing, conductivity, and drive frequency.
• Geometry comparison: compare solid, slotted, and laminated specimens at the same field map. Reversing field direction reverses local circulation while preserving the heating and drag magnitudes for an otherwise matched run.

Measurement channels

Use at least two independent records for a quantitative eddy-current result. Force, thermal, and electrical records constrain different parts of the same energy balance and diagnose different systematic errors.

• Force channel. Measure force at several positive and negative speeds under matched field-on and field-off conditions. The subtraction removes bearing friction, aerodynamic drag, and a static offset only when those contributions remain unchanged. Reverse the motion direction and retain the sign convention. A drag force that reverses with velocity and remains opposite the motion supports the Lenz-law assignment. Plotting $F_{\mathrm{eddy}}/v$ over the intended speed range tests whether a linear damping coefficient is defensible.
• Thermal channel. A short early-time temperature record can be reduced with a lumped heat-capacity model,

When the temperature rise is small and heat transfer is initially weak, $P_{\mathrm{eddy}}\approx C_{\mathrm{th}}\mathbf{d}T/\mathbf{d}t$. Longer runs require the thermal conductance $G_{\mathrm{th}}$, the mounting path, sensor location, and ambient condition. Surface temperature can differ from the hottest current-carrying region, especially when penetration concentrates loss near one face.

• Electrical channel. For a driven coil and target, record synchronized terminal voltage and current rather than rms magnitudes alone. The cycle-average input is

Subtract measured winding loss, known mechanical output, and calibrated background loss before attributing the remaining power to the target. A phase shift between voltage and current changes average power even when both rms values remain fixed. Repeat the empty-coil measurement with the same drive waveform and coil temperature.

• Impedance channel. A probe-coil measurement should retain amplitude and phase: $Z=V/I$ is complex under sinusoidal drive. Target lift-off, orientation, thickness, conductivity, and crack geometry can each change the real and imaginary parts. Use reference specimens that vary one parameter at a time. A single amplitude reading cannot distinguish a material change from a spacing error.
• Flux and geometry channel. Record the field waveform at the conductor location, the region over which it acts, and the material speed or angular speed. A Hall probe, search coil, or calibrated drive coil can supply the field record. The relevant quantity is the local loop flux rate; a nominal magnet grade or source-current value alone does not determine the induced emf in the specimen.
• Closure test. At steady speed in a brake, compare $F_{\mathrm{eddy}}v$ with the measured thermal or electrical loss after background corrections. During a decay, integrate the force-power record and compare it with the loss of mechanical kinetic energy. Agreement within stated uncertainty supports the current-path and field-map model. A discrepancy can arise from unmeasured friction, temperature-dependent conductivity, an incomplete field map, or a sensor boundary that differs from the modeled object.
• Uncertainty record. Report force-sensor calibration, velocity uncertainty, field repeatability, temperature, specimen dimensions, target spacing, and background-loss subtraction. Evaluate the result over the same speed, frequency, and temperature range used for the stated model. A fitted damping coefficient without those conditions has limited predictive value because current paths and conductivity can change between measurements.
• Control sequence. Begin with a field-off run, then repeat with the same mechanical trajectory and field on. Reverse the field or motion for a sign check, repeat after thermal equilibrium for a temperature check, and compare a slotted or laminated specimen for a path-geometry check. The sequence separates a genuine induction response from a change in friction, alignment, or drive waveform.

Design choices for cores, plates, and brakes

Select the current-path geometry from the intended effect.

Conductivity, thickness, and cooling have opposite roles across these devices. A high-conductivity brake disk can provide strong torque but may need a heat path. A core uses high resistivity and short transverse loop dimensions to reduce loss. A thicker conductor adds heat capacity and available current area while also permitting nonuniform penetration. State the operating frequency, field map, and thermal boundary before selecting material thickness.

The placement of a conductor relative to a magnet changes the available flux gradient. A disk centered far from the gap may have only a small portion within strong field. A larger radius increases rim speed and can increase braking torque at the same angular speed. A magnet pair placed symmetrically on both sides of a disk can provide a more uniform gap field and reduce unwanted side forces. The resulting torque must still be measured or modeled from the actual current distribution.

A direction checklist

An eddy-current question becomes manageable when the material loop is selected before the current direction is drawn.

1. Choose a material loop or a small surface patch. Mark its normal and whether it moves with the conductor.
2. Determine the external-flux change through that loop. State whether the signed contribution becomes more positive or more negative.
3. Draw the induced magnetic contribution. It must oppose the signed change through the same local loop.
4. Use the right-hand rule for conventional circulation. In a bulk conductor, interpret the result as a local current-density loop.
5. Apply a force check for moving material. The current inside the external field must give force or torque opposite the imposed relative motion.
6. Apply an energy check. The induced-current heating comes from mechanical work or from the source that changes the magnetic field.

The current direction can reverse from one region of a sheet to another when local flux histories differ. A single arrow around the outer boundary is insufficient for a large object spanning a nonuniform field. Divide the sheet into regions, identify the local field change, and then assemble the distributed current pattern.

Limits and Checks

The quasistatic model uses a local constitutive relation,

after the induced electric field has been obtained from the changing magnetic flux. It is appropriate when the field pattern and material response adjust on a time scale short compared with the imposed mechanical motion or source waveform. The model gives current direction, heating density, and magnetic-force direction without requiring a single wire-like current path.

The induced-current mechanism has an unambiguous energy record: changing flux drives circulation, resistivity converts part of that response to heat, and the field-current interaction produces force or torque. Geometry, conductivity, permeability, and the field history determine the measured magnitudes. A large separation between magnet and conductor can reduce the flux rate below a measurable level. A close gap can demand a spatial field map rather than a uniform-field estimate.

Solid versus laminated core

Compare a solid core and a laminated core at the same magnetic operating point. The solid cross section permits broad circulation; insulated sheets break the transverse electrical path while retaining a continuous magnetic path. Under a controlled sinusoidal test, an energy balance for the measured assembly is

The residual includes eddy-current and magnetic-material loss. A lower residual for the laminated core at matched flux amplitude and frequency supports the expected reduction in eddy-current loss; it does not by itself isolate every microscopic loss mechanism.

• Match the magnetic state. Use the same winding, core gap, excitation frequency, and flux waveform. A search-coil record or a calibrated coil-voltage integral verifies the flux amplitude.
• Measure input and copper loss separately. Record the synchronized voltage and current waveforms, coil resistance at the test temperature, and average input power. A temperature change in the winding alters the copper-loss subtraction.
• Inspect insulation and clamping. Lamination coating must block cross-layer current while the sheets remain magnetically well coupled. Damaged coating, a conducting bolt, or a conductive clamp can restore a broad loop and raise the loss.
• Use temperature as a secondary record. Temperature rise integrates loss over time and depends on cooling. It confirms a power trend only when the thermal boundary conditions and run duration are matched.

Eddy currents in measurement and control

Three applications use the same local flux-rate calculation with different measured outputs.

• Damped indicator: a conducting vane moves through a magnet gap. The eddy-current torque removes oscillation energy while the restoring and drive torques set the equilibrium position. At rest, the material-loop flux rate is zero and this torque vanishes.
• Probe coil: an alternating field drives currents in a nearby conducting target. The target response changes the coil terminal voltage-current relation. Crack size, thickness, conductivity, lift-off distance, and drive frequency require calibration against reference specimens.
• Magnetic brake: a disk or rail passes through a field gradient. The available flux rate falls with relative speed, so drag and heating fall as the vehicle or disk slows. A separate holding brake exerts force at zero speed.

Each apparatus needs an observable matched to the model: a decay envelope for the indicator, complex coil impedance for the probe, or force and speed for the brake. The current distribution remains internal to the conductor, so the measurement must close through mechanical, thermal, or electrical power rather than a single series ammeter.

Sign, unit, and magnitude checks

Magnetic flux has unit weber, and its rate has unit volt. A representative eddy-loop emf $A\mathbf{d}B/\mathbf{d}t$ therefore has unit

Dividing by loop resistance gives ampere. Multiplying a current density by an electric field gives watts per cubic metre. Multiplying a drag force by speed gives watts. These unit checks connect the induction calculation to the mechanical or thermal energy calculation and catch errors such as using field magnitude where field rate is required.

A direction check proceeds locally. Increasing into-page flux calls for an out-of-page induced contribution and counterclockwise conventional circulation as viewed. Decreasing into-page flux calls for the reverse circulation. A moving sheet entering a field region and a stationary sheet exposed to a strengthening field can have the same local current direction if their material loops experience the same signed flux change.

The induced current may be strong while the net force on a whole symmetric object is small. Opposite regions can experience forces that cancel in translation while producing a torque, or they can carry counter-rotating currents whose external fields partly cancel. Force, torque, heating, and current pattern must be evaluated as separate quantities for the stated geometry.

Material and geometry record

Record the material, geometry, and operating conditions for every eddy-current calculation or experiment.

• Field history: magnitude, direction, spatial extent, and time dependence of the external field.
• Conductor geometry: thickness, holes, slots, laminations, current-path dimensions, and motion relative to the field.
• Electrical properties: conductivity or resistivity, temperature range, and electrical contacts if present.
• Mechanical state: speed, angular speed, applied force or torque, mass or moment of inertia, and other damping forces.
• Thermal boundary conditions: heat capacity, cooling path, ambient temperature, and duty cycle.

Faraday's law and Lenz's law determine the induced-current direction; electrical and mechanical power identify the energy source and destination. Use the material and geometry record as a calculation input rather than a descriptive appendix.

• Static-field control. Hold the conductor at rest in a static field. Every material loop then has constant external flux, so induced emf and resistive eddy-current heating vanish after transient charge redistribution. Motion through a field boundary or a time-varying field restores a flux rate and identifies the mechanical drive or electrical source that provides the dissipated energy.
• Scale comparison. The response depends on available conducting-loop area, loop resistance, and local field rate. Larger or faster-changing configurations therefore need separate thermal and mechanical limits even when their current direction is unchanged. Record the selected material loop, field orientation, and motion direction before assigning current arrows.
• Repeated operating conditions. Retain reference field map, conductor temperature, motion history, sensor spacing, and system boundary for every comparison. A change in any one of these inputs can alter current paths or measured force. Compare power, force, temperature rise, and decay rate against the same geometry.
• Independent closure. Agreement among the power, force, thermal, and decay records supports the assigned current pattern and energy balance. A disagreement identifies an omitted current path, thermal loss, field-gradient error, or a measurement boundary that differs from the model.