Electronics Guide

The Magnetic Circuit

The magnetic circuit is the central abstraction of wound-component engineering. It replaces a three-dimensional field problem with a lumped network that can be solved with the same algebra as a resistive circuit, and it is accurate enough to design most components before any field solver is opened.

Permeability Is Not a Single Number

A ferrite datasheet may quote initial, amplitude, effective, incremental, and complex permeability on the same page. They are different measurements of the same material, and using the wrong one is a common source of designs that miss their target inductance.

Initial permeability (μ_i) is measured at very low flux density, conventionally below 0.25 millitesla, with no bias. It is the number in the material name and the basis of small-signal catalog data, and it overstates what a power design will see. Amplitude permeability (μ_a) is the ratio of peak flux density to peak field strength at a stated large excitation; in manganese-zinc power ferrite it can exceed the initial value by a factor of two or more before falling away near saturation. Effective permeability (μ_e) is not a material property at all but a property of the assembled core, reduced from the material value by any air gap. Incremental permeability (μ_Δ) is the slope of a minor loop superimposed on a DC bias, and it is the correct figure for an output choke that carries direct current with a small ripple on top. Complex permeability resolves the response into an in-phase component μ′ and a loss component μ″; the ratio μ″/μ′ is the loss tangent, and its rise with frequency is what turns a suppression bead from an inductor into a resistor.

The practical discipline is to read the measurement conditions before the number. A permeability quoted at 10 kilohertz and 0.25 millitesla says nothing directly useful about a core running at 500 kilohertz with 100 millitesla of swing and 5 amperes of bias, and the gap between the two is where inductance tolerance budgets are lost.

Saturation, Bias, and Thermal Derating

Saturation is the point at which the magnetic domains in the core are substantially aligned and further field strength produces almost no additional flux density. In magnetic-circuit terms, the core reluctance rises toward that of air, so inductance collapses toward its air-core value.

Three points deserve emphasis. First, saturation is set by ampere-turns, not by current alone, so a design that adds turns to raise inductance moves closer to saturation at the same current unless the core or the gap changes to compensate. Second, saturation flux density falls substantially with temperature: typical manganese-zinc power grades give up roughly a fifth of their room-temperature saturation by 100 degrees Celsius, dropping from something in the neighborhood of half a tesla to something nearer 0.4 tesla, and the Curie temperature above which magnetic ordering collapses entirely is typically a little above 200 degrees Celsius for power grades. Both figures appear on the datasheet, and the hot one governs. A margin taken at room temperature is not a margin. Third, the shape of the approach to saturation differs by material class. Ferrite saturates abruptly, so the inductance falls off a cliff and converter current runs away within a switching cycle. Distributed-gap powder cores roll off gradually, retaining useful inductance well past the nominal rating, which buys tolerance to transient overload at the cost of an inductance that varies continuously with load.

Remanence, the flux density remaining when the drive returns to zero, matters wherever the flux excursion is unipolar. A forward converter must reset the core each cycle or the flux walks toward saturation over successive cycles; the available flux swing is the distance from the remanent point to saturation, not the full ±B_sat range. Gapping shears the loop over, lowering remanence and widening the usable swing, which is one more reason gapped cores dominate unipolar applications.

Air Gaps and Energy Storage

The most counterintuitive result in magnetics is that an inductor stores its energy in the air gap rather than in the core. Energy density in a magnetic field is B²/(2μ), so for a given flux density, a region of low permeability stores far more energy per unit volume than a region of high permeability. Since the flux is continuous through the magnetic circuit, the gap and the core carry essentially the same flux density, and a gap with a relative permeability of one stores two thousand times the energy density of a ferrite with a relative permeability of two thousand. The core's job is to guide flux with little magnetomotive force; the gap's job is to hold the energy.

This gives a direct sizing relation. To store energy W = ½LI² at a peak flux density B, the gap volume must be approximately 2Wμ₀/B². An energy-storage inductor therefore needs gap volume, and a component that must store more energy grows even when the winding could carry the current in a smaller package. Transformers in forward-mode topologies, which transfer energy rather than store it, want the opposite: minimum gap and minimum magnetizing current.

The gap also sets the effective permeability of the assembled core. For a gap of length l_g in a path of effective length l_e, the standard approximation is μ_e = μ_i / (1 + μ_il_g/l_e). Once μ_il_g/l_e greatly exceeds unity, the denominator dominates and μ_e approaches l_e/l_g, independent of the material. That is the mathematical statement of why a well-gapped inductor is insensitive to ferrite permeability tolerance and to the permeability drift of the material with temperature.

Gaps have a cost. Flux fringes outward around a discrete gap and cuts through any winding conductor nearby, inducing eddy currents that create a local hot spot the average loss calculation will not predict. Keeping the winding set back from the gap region, splitting the gap into several smaller gaps along the center leg, or choosing a distributed-gap powder core in which the gap is dispersed microscopically throughout the material all reduce fringing loss, at the price of manufacturing complexity or higher core loss respectively.

Core Loss

Core loss is the energy dissipated in the magnetic material as the flux swings. It is conventionally decomposed into three mechanisms. Hysteresis loss is the area of the B-H loop traversed each cycle, so it scales with frequency to the first power and rises steeply with flux swing. Classical eddy-current loss comes from circulating currents induced in the conducting core and scales with the square of frequency and with the square of lamination or particle thickness, which is why line-frequency cores are laminated and why the high resistivity of ferrite makes it usable at hundreds of kilohertz. Residual or relaxation loss, associated with domain-wall dynamics that cannot follow the excitation, becomes significant as frequency rises.

In practice the three are never separated. Charles Proteus Steinmetz, in his 1892 paper on the law of hysteresis, found that hysteresis loss per cycle followed a power law in peak flux density with an exponent near 1.6. The relation that now carries his name generalizes that result by fitting a frequency exponent as well, so manufacturers reduce measured loss per unit volume to P_v = k f^αB^β, where B is the peak flux density of a sinusoidal excitation. The coefficients are a curve fit rather than physics. For power ferrites the frequency exponent α is commonly quoted between about 1.2 and 1.7 and the flux exponent β between about 2.2 and 3.0, and both drift across the frequency range, which is why datasheets fit separate coefficient sets to separate bands and why extrapolating a fit beyond the range it was taken from is unsafe.

Two limitations follow. Converter waveforms are rarely sinusoidal, and the Steinmetz coefficients are fitted to sinusoids; extensions such as the improved generalized Steinmetz equation reapply the fitted parameters to the instantaneous rate of change of flux density, capturing the duty-cycle asymmetry that the original form misses. And core loss depends on temperature in a way that is not monotonic: manganese-zinc power ferrites are formulated so that loss passes through a minimum near the expected operating temperature, often in the range of 80 to 100 degrees Celsius, which means a bench measurement at room temperature can be pessimistic while a hotter-running design can be better than predicted, up to the point where the loss curve turns back upward and thermal runaway becomes possible.

For comparing materials rather than computing a specific design, the performance factor—the product of frequency and peak flux density evaluated at a fixed core-loss density—is the standard figure of merit. Plotted against frequency it peaks where a material is most productive, and the peak identifies the frequency band in which a given grade is worth using. The reference loss density is a choice rather than a standard, and published curves do not all use the same one, so two performance-factor plots may be compared only when their references agree. Read that way, it is the fastest answer to whether a design should change material or change switching frequency.

Winding Design and AC Resistance

The copper is the other half of the loss budget, and above audio frequencies its resistance bears little relation to the DC value on the datasheet.

The skin effect confines alternating current toward the conductor surface, with the current density falling exponentially over a characteristic skin depth δ = √(ρ/(πfμ)). In copper at room temperature this is approximately 2.1 millimeters at 1 kilohertz, 0.21 millimeters at 100 kilohertz, and 0.066 millimeters at 1 megahertz, falling as the inverse square root of frequency. Conductor material thicker than roughly twice the skin depth adds weight and cost without carrying proportionate current.

The proximity effect is usually the larger problem in a multilayer winding. Each layer sits in the magnetomotive force built up by the layers before it, and that external field drives its own eddy currents in the conductor, crowding the current into narrow bands. Loss grows rapidly with layer count, and a winding of many layers can exhibit an AC resistance ten times or more its DC value at a frequency where a single layer would be barely affected.

The standard analytical treatment is the one-dimensional solution published by P. L. Dowell in 1966 (Proceedings of the IEE, volume 113, number 8), which gives the ratio of AC to DC resistance as a function of the number of layers and the conductor thickness expressed in skin depths. It remains the first calculation to run, with finite-element analysis reserved for geometries where fringing flux or unequal layer currents violate its assumptions.

The mitigations follow directly from the mechanism. Size conductors against the skin depth rather than against the DC current. Interleave primary and secondary sections so the peak magnetomotive force between windings is halved and, in a well-interleaved build, the proximity loss falls by roughly a factor of four. Use foil or flat wire for high-current, low-turn windings where a single layer can span the bobbin. Use Litz wire—many individually insulated strands transposed so that each occupies every radial position along the length—where the frequency is high enough to justify the cost, remembering that Litz stops helping once the strand diameter approaches the skin depth or the strand count inflates the winding's parasitic capacitance.

All of this competes for a fixed window area. The window utilization factor, the fraction of the available window occupied by conducting copper, is typically around 0.4 for round magnet wire, and lower still where safety margins, layer insulation, or triple-insulated wire consume space. Every improvement to AC resistance—more strands, thicker insulation, interleaving with insulation between sections—spends some of that budget, which is why winding design is an optimization rather than a checklist.

Coupled Magnetics

When two windings share a core, the flux one produces links the other, and the pair is described by a mutual inductance M = k√(L₁L₂), where the coupling coefficient k runs from zero for entirely independent windings to one for perfect coupling. The dot convention records the relative winding sense: currents entering the dotted terminals produce flux that adds. Getting the dots wrong reverses the sign of M and, in a converter, usually destroys something.

The same physical component can be modeled as a magnetizing inductance in parallel with an ideal transformer, with leakage inductances in series with each winding representing the flux that fails to couple. Which view is useful depends on the role. A forward-mode transformer wants k as close to one as construction allows, so magnetizing current stays small and leakage-driven voltage spikes stay manageable. A flyback component wants a deliberate gap and behaves as a coupled inductor: it stores energy during the on-time and delivers it during the off-time, and calling it a transformer obscures the fact that it must be sized by energy storage.

Two coupled structures deserve separate mention because they are neither inductors nor transformers in the ordinary sense. Coupled inductors in multiphase converters share flux deliberately between phases; inverse coupling reduces the ripple current seen by the output capacitor while allowing a smaller effective inductance during transients, improving step response without sacrificing steady-state ripple. Common-mode chokes wind two or more conductors on a shared core so that the fluxes from the intended differential current cancel while common-mode current in the same direction on both conductors produces additive flux. The result is high impedance to common-mode noise and, ideally, none to the power or signal current. The cancellation is imperfect, and the residual leakage inductance—typically a fraction of a percent to a few percent of the common-mode value in a well-wound toroid—is often used deliberately as differential-mode filter inductance. Because the differential flux largely cancels, a common-mode choke can use a high-permeability material, frequently nanocrystalline or a high-permeability manganese-zinc ferrite, that would saturate immediately if the full load current produced net flux.

Parasitics and the Impedance Curve

Every wound component carries capacitance between adjacent turns, between layers, and between windings. Lumped as a single equivalent parallel capacitance, it resonates with the inductance at the self-resonant frequency, above which the component is capacitive and no longer performs the function it was chosen for. The impedance-versus-frequency curve, rather than a single inductance value, is the honest description of a real part: inductive with a slope of plus one on a log-log plot, peaking at self-resonance, then falling capacitively.

Winding style dominates this capacitance. A single-layer solenoid has little, because adjacent turns differ by only one turn's worth of voltage. A multilayer winding wound back and forth places the start of one layer next to the end of the next, putting the full layer voltage across the interlayer capacitance. Sectioned or bank winding, which keeps the turns adjacent in space also adjacent in potential, can lower self-capacitance by an order of magnitude and is standard practice in high-impedance RF chokes.

Interwinding capacitance in a two-winding component is a different concern: it couples high-frequency common-mode noise straight across an isolation barrier, defeating the isolation at exactly the frequencies where noise is worst. It also trades against leakage inductance, since the interleaving that reduces leakage brings more conductor area into proximity across the insulation. A grounded electrostatic screen between windings intercepts that displacement current, at the cost of build height and an additional insulated layer.

Suppression components invert the usual priorities. A ferrite bead is chosen for the resistive part of its complex impedance, not its inductance: the goal is to dissipate noise energy as heat rather than reflect it. Datasheets accordingly specify impedance magnitude at a stated frequency, conventionally 100 megahertz, and the useful information is the full curve showing where the reactive and resistive components cross over. Quality factor Q, the ratio of reactance to resistance, works the same way in reverse: a resonant-circuit inductor is selected for the highest achievable Q, and a bead is deliberately the lowest-Q component in the design.

Measurement and Characterization

Wound components are the least predictable parts in most designs, so measurement is not optional. A handful of techniques cover most needs.

Impedance sweeps. An LCR meter gives inductance and Q at spot frequencies; an impedance analyzer sweeps the whole curve and reveals the self-resonant frequency, the AC resistance rise, and any secondary resonances from winding structure. Use a four-terminal-pair fixture and compensate for the open and short conditions; at the milliohm and nanohenry level, lead inductance and contact resistance dominate an uncompensated measurement. Always record the excitation level and any DC bias, since both change the answer.

Bias sweeps. Inductance under DC bias is the number that matters for a power choke, and it is not the number on the front of the datasheet. A bias current source in series with the analyzer produces the inductance-versus-current curve directly, and the manufacturer's saturation rating—defined variously as the current causing a 10, 20, or 30 percent inductance drop—should be checked against the definition in use before it is compared across suppliers.

Coupling measurements. Leakage inductance is measured by shorting the secondary and reading the primary inductance. Mutual inductance follows from two series measurements: connect the windings series-aiding and series-opposing, and M = (L_aiding − L_opposing) / 4. The coupling coefficient then follows from M and the two open-circuit inductances.

Core loss. Datasheet loss curves are measured under sinusoidal excitation on ring samples; IEC 62044-2 covers magnetic properties at low excitation level and IEC 62044-3 covers power loss and amplitude permeability at high excitation. Because converter waveforms are not sinusoidal, the defensible verification is a B-H loop measurement under the actual excitation, taken with a sense winding for voltage and a current probe for magnetomotive force, with careful attention to phase error—a fraction of a degree of phase shift between the two channels produces a large error in a quantity derived from their product.

Integrity tests. A surge or impulse test detects shorted turns that a DC resistance measurement cannot see, since one shorted turn out of a hundred changes resistance imperceptibly but rings the applied impulse very differently. IEC 61007:2020, Transformers and inductors for use in electronic and telecommunication equipment—Measuring methods and test procedures, collects the standard test methods for electric strength, resistance, power loss, inductance, impedance, and transformation ratio, and is the usual reference when a component specification must state how a parameter is verified.

Selection Trade-offs Across the Family

The choices above interlock, and a change made for one reason propagates. Raising the switching frequency reduces the volt-second product and therefore the turns and the core size, but it increases core loss at a given flux density and increases AC winding resistance, so beyond some frequency the component grows again. Raising flux density reduces turns and copper loss but increases core loss; for a transformer the optimum often falls near the point where the two are comparable, while an energy-storage inductor is usually limited by saturation instead. Adding gap linearizes and stabilizes inductance but adds fringing loss and demands more turns for the same inductance. Interleaving cuts leakage inductance and proximity loss but raises interwinding capacitance.

A useful first question is what the component is for, because the three roles impose different priorities. A component that stores energy is sized by gap volume and saturation, and its figure of merit is energy stored per unit volume at an acceptable temperature rise. A component that transfers power is sized by the volt-second product and window area, and it wants tight coupling, low magnetizing current, and no more gap than the topology demands. A component that presents impedance—a bead, a common-mode choke, a filter choke—is specified by an impedance curve across a frequency band, and its material is chosen for loss rather than against it.

The last trade-off is whether to specify a catalog part or a custom build. A catalog inductor arrives characterized, qualified, and second-sourced, and for the great majority of designs it is the correct answer. A custom wound component becomes worthwhile when the requirement is unusual—a specific leakage inductance for a resonant tank, a particular isolation rating, an awkward mechanical envelope, or a volume large enough that a few points of efficiency justify the engineering. Choosing between them well requires understanding the physics either way, which is the reason for reading this material before opening a catalog.

Summary

Every wound component reduces to the same model: ampere-turns drive flux through a reluctance, the flux induces voltage in the turns that link it, and the shortfall appears as heat. Reluctance, the effective core parameters standardized in IEC 60205, and the inductance factor A_L turn that model into arithmetic. Permeability is a family of measurements rather than one number, and the useful one is always the one measured under the intended excitation, bias, and temperature.

An air gap, not the core, stores the energy in an inductor, which is why energy storage demands gap volume and why gapped designs hold their inductance tolerance. Losses divide into core loss, described empirically by the Steinmetz relation and its non-sinusoidal extensions, and winding loss, dominated above audio frequencies by the skin and proximity effects and estimated first by Dowell's analysis. Coupling between windings, described by mutual inductance and the coupling coefficient, distinguishes a transformer from a coupled inductor from a common-mode choke, though all three are the same physics with different priorities. Parasitic capacitance sets the self-resonant frequency that bounds every component's useful range.

These principles are what remain constant while materials, frequencies, and topologies change. With them established, the component-level detail in the articles on inductors and transformers, the material properties in the article on magnetic materials, and the converter-level design loop in the power electronics section all fit into a single coherent picture.

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