Magnetic Flux Density, Flux Linkage and Faraday–Lenz Law in Inductors

Magnetic flux density, magnetic flux and Faraday–Lenz law explain why a voltage appears across an inductor when its current changes. These concepts form the basis of inductor energy storage, transformer action, magnetic saturation and the selection of practical magnetics for power-electronic converters.

Why changing flux produces voltage

A changing magnetic field can induce a voltage in a conductor loop. In a wound component, the induced voltage depends on how much magnetic flux links the winding and how rapidly that linkage changes.

This effect is described by Faraday’s law of electromagnetic induction. Its polarity is determined by Lenz’s law: the induced voltage drives a current whose magnetic effect opposes the original change in flux linkage.

This is why an inductor resists a rapid change in current, why a transformer transfers AC energy between windings, and why uncontrolled switching transients can create substantial voltage across a coil.

Key magnetic quantities

The quantities in the table below are closely related but should not be used interchangeably. In particular, a winding terminal voltage is determined by flux linkage, whereas magnetic flux is a property of a defined surface within a magnetic field.

Quantity	Symbol	SI unit	Practical meaning
Magnetic field strength	H	A/m	Field-producing force associated with free current and winding turns
Magnetic flux density	B	T	Magnetic field through a given area; 1 T equals 1 Wb/m²
Magnetic flux	$\Phi$	Wb	Total magnetic field passing through a defined surface
Flux linkage	$\lambda$	Wb-turn or V·s	Total flux linked by a winding
Permeability	$\mu$	H/m	Measure of how a material supports magnetic flux
Inductance	L	H	Relationship between current and flux linkage in a defined operating condition
Reluctance	$\mathcal{R}$	A-turn/Wb	Opposition of a magnetic circuit to the establishment of magnetic flux

For linear magnetic materials, magnetic flux density and magnetic field strength are related by:

$B=\mu H$

In vacuum, and with good accuracy in air, the permeability is approximately the vacuum permeability $\mu_0$. In ferromagnetic materials, permeability can be much higher, but it is neither constant nor independent of temperature, frequency, DC bias and magnetic history.

Magnetic flux density

Magnetic flux density B is a vector quantity that describes the magnitude and direction of magnetic field through an area. It is measured in tesla.

For a simple uniform magnetic field passing normally through a surface, the magnetic flux density is the magnetic flux divided by the area. A larger flux through the same core cross-section therefore means a higher flux density.

In real magnetic components, the maximum usable flux density is limited by the core material, temperature, excitation waveform, switching frequency and acceptable loss. Exceeding the usable range drives the core toward saturation, reducing effective inductance and potentially causing excessive current.

Relation to magnetic field strength

The magnetic field strength and Ampère’s law describe how winding current and turns establish magnetomotive force in a magnetic circuit. In an idealised uniform magnetic path, more ampere-turns increase H, while the material permeability determines the resulting B.

The simple proportional relation between B and H is a useful first-order model only. Ferrites, powdered-iron materials, electrical steels and nanocrystalline alloys have nonlinear B – H characteristics, particularly close to saturation.

Magnetic flux

Magnetic flux $\Phi$ is the total magnetic field passing through a defined surface. It is calculated from the surface integral of magnetic flux density:

$\Phi = \int_{S} B \cdot dA$

For a homogeneous magnetic field over a flat area, this reduces to:

$\Phi=BA\cos\theta$

where A is the area normal to the field direction and $\theta$ is the angle between the magnetic flux density vector and the surface normal.

Magnetic flux is measured in weber. One weber is equivalent to one volt-second. Maximum flux in a core depends on its effective cross-sectional area and the maximum permissible flux density.

Flux linkage

Flux linkage connects magnetic field behaviour with winding terminal voltage. It is defined as the sum of the flux linked by all turns in a winding.

If a winding has N turns and each turn links substantially the same flux, flux linkage is:

$\lambda=N\Phi$

This condition is usually a good approximation in a well-coupled transformer or a closed-core inductor with low leakage flux. It becomes less accurate when leakage field, non-uniform flux distribution, winding position or complex core geometry cause individual turns to link different amounts of flux.

Flux linkage is the preferred quantity for describing real inductors because the winding voltage is directly related to its time derivative.

Faraday–Lenz law

Faraday’s law states that a time-varying flux linkage induces a winding voltage:

$v=-\frac{d\lambda}{dt}$

For a winding in which every turn links the same magnetic flux, the equation becomes:

$v=-N\frac{d\Phi}{dt}$

The negative sign represents Lenz’s law. It does not mean that an inductor “creates negative voltage” in all circumstances; it establishes the polarity reference such that the induced voltage opposes the change in flux linkage.

A magnetic field can change in several ways:

• The magnitude of magnetic flux density changes with time
• The loop or winding moves relative to the magnetic field
• The effective enclosed area changes
• The angle between the field and loop changes
• Another coupled winding changes the flux in a shared magnetic core

Transformer action, inductive sensing, generators and wireless-power transfer all follow the same physical law, although the magnetic circuit and coupling arrangement differ.

From flux linkage to inductance

For an ideal linear inductor, flux linkage is proportional to current:

$\lambda=Li$

Combining this relation with Faraday’s law gives the familiar inductor voltage-current relationship:

$v=L\frac{di}{dt}$

This equation shows that inductance limits the rate at which current changes for a given applied voltage. A higher inductance produces a lower current ramp for the same voltage and time interval.

However, practical power inductors are not perfectly linear. A more general expression is:

$v=\frac{d\lambda}{di}\frac{di}{dt}$

The differential inductance is:

$L_{\mathrm{diff}}=\frac{d\lambda}{di}$

This differential inductance can decline significantly at high DC current as the core approaches saturation. Nominal inductance stated in a datasheet should therefore always be considered together with its test frequency, test voltage or current, temperature and DC-bias condition.

For a broader explanation of this practical behaviour, see Current-Dependent Inductors: Using Non-Linear Inductance in Buck Converters and PFC Stages.

Magnetic circuits and reluctance

A magnetic circuit provides a low-reluctance path for magnetic flux. In a simplified, linear magnetic circuit:

$\Phi=\frac{\mathcal{F}}{\mathcal{R}}$

where the magnetomotive force is:

$\mathcal{F}=NI$

For a uniform magnetic path, reluctance is approximated by:

$\mathcal{R}=\frac{l_e}{\mu A_e}$

where l_e is effective magnetic path length and A_e is effective cross-sectional area.

Under these idealised conditions, inductance can be expressed as:

$L=\frac{N^2}{\mathcal{R}}$

This relationship is useful because it shows the main levers available to a magnetic designer: increasing turns increases inductance approximately with the square of turns, increasing effective core area reduces reluctance, and a longer magnetic path increases reluctance.

The model assumes uniform flux, negligible leakage and approximately constant permeability. It is valuable for initial estimation, but detailed component design also requires core-loss, temperature, fringing, winding-loss and parasitic-capacitance analysis.

Energy storage and air gaps

The energy stored by a linear inductor is:

$W=\frac{1}{2}LI^2$

A more general magnetic-field expression is:

$W=\int_V \left(\int_0^B H\,dB\right)dV$

For a linear medium, magnetic energy density can be written as:

$w=\frac{1}{2}BH$

A gapped power inductor uses a low-permeability region—usually an air gap or distributed gap—to control inductance, improve DC-bias capability and store magnetic energy. Because the field strength is much higher in the low-permeability gap than in the core for comparable flux density, a substantial share of the stored magnetic energy is associated with the gap region.

The phrase “energy is stored in the air gap” is useful engineering shorthand. In practical components, the energy is distributed throughout the magnetic field, including fringing field around the gap and regions outside the core.

Gap fringing

Flux spreads outward as it crosses an air gap rather than remaining perfectly confined to the core cross-section. This effect, known as fringing, increases the effective gap area and may increase local eddy-current loss in nearby windings, copper foil, shields or conductive hardware.

Spacing windings away from a concentrated air gap, using distributed-gap materials where appropriate, and considering field-sensitive components near the inductor can improve thermal and EMC performance.

For an application-oriented explanation, see Why Power Inductors Use a Ferrite Core With an Air Gap.

Saturation and DC bias

Core saturation occurs when further increase in magnetic field strength produces a much smaller increase in magnetic flux density. As a result, the effective or differential inductance falls.

In a switching converter, this reduction can produce a steeper current ramp, increased peak current, higher MOSFET and diode stress, excess winding loss and potential loss of current-loop control. Saturation therefore must be assessed at maximum input voltage, peak current, worst-case temperature, tolerance stack-up and the intended switching waveform.

A gapped core increases the current that an inductor can support before a large inductance drop occurs. It does not eliminate saturation; it redistributes magnetomotive-force drop and makes inductance more predictable over a useful current range.

Core and winding losses

Inductance alone does not determine whether a magnetic component is suitable for a converter. Losses and temperature rise determine the usable operating range.

Core loss

Core loss is influenced by:

• Flux-density swing and DC operating point
• Switching frequency and waveform
• Core material and geometry
• Temperature
• AC field distribution, including gap fringing

A high flux-density swing can reduce core volume for a given energy-transfer requirement, but it usually increases loss. Design therefore requires a compromise among component size, efficiency, temperature rise and cost.

Winding loss

Winding loss includes DC copper loss and AC loss. At higher frequency, skin effect and proximity effect can raise resistance substantially above its DC value.

The physical placement of windings, conductor diameter, copper thickness, layer count, gap position and nearby conductive hardware all influence AC winding loss. These effects become particularly important in high-current converters and wide-bandgap switching designs.

For material-specific context, see Core Materials, Permeability and Their Losses.

Application fit

Application	Magnetic function	Main design priorities
Buck and boost converters	Stores and releases energy during each switching cycle	Inductance under DC bias, saturation current, ripple current, DCR, core loss and temperature rise
PFC inductors	Stores energy while shaping input current	High peak current, low core loss, thermal margin, audible-noise behaviour and EMI
Flyback converters	Uses a gapped coupled inductor to store and transfer energy	Magnetising inductance, peak flux density, leakage inductance, gap design, isolation and winding loss
Forward, half-bridge and LLC transformers	Transfers energy with limited intentional energy storage	Magnetising current, volt-seconds, core loss, leakage inductance, insulation system and interwinding capacitance
EMI filters and common-mode chokes	Presents impedance to unwanted noise currents	Impedance versus frequency, saturation under DC imbalance, insertion loss, leakage and safety requirements
Current-sense transformers	Converts changing primary current into a secondary signal	Core material, frequency response, burden resistance, reset behaviour and saturation margin

Design-in notes for engineers

• Use the correct inductance value: Compare nominal inductance with the value at the actual DC current, temperature and switching frequency; a nominal LCR-meter value may not represent operating behaviour.
• Check saturation margin: Use the manufacturer’s inductance-versus-current data, saturation-current definition and temperature conditions. Do not compare saturation-current figures across suppliers without checking the stated inductance-drop criterion.
• Calculate ripple current from operating extremes: Inductor current ripple is influenced by applied voltage, switching period and effective inductance. Evaluate minimum and maximum inductance across tolerance, temperature and DC bias.
• Separate core and copper loss: Low DCR does not guarantee low total loss. High-frequency winding loss, core loss and gap-fringing loss can dominate depending on topology and frequency.
• Treat gaps as an EMC and thermal feature: A concentrated gap can create fringing fields that couple into nearby copper, low-level sensing circuits and magnetically sensitive components.
• Use realistic permeability: Initial permeability, effective permeability and differential permeability are not interchangeable. Effective permeability in a deliberately gapped inductor is strongly determined by magnetic-circuit geometry.
• Verify component measurements: Confirm measurement frequency, AC test level, DC bias, fixture method and temperature before comparing inductance or Q values between suppliers.
• Consider parasitics at fast switching edges: Leakage inductance and winding capacitance can generate ringing, voltage overshoot and common-mode noise. These effects are increasingly important with SiC and GaN switching devices.
• Validate with waveform data: For critical designs, verify current, winding voltage, temperature and—where practical—core flux-density margin under transient, overload and start-up conditions.

Conclusion

Magnetic flux density, magnetic flux and flux linkage describe different aspects of the same electromagnetic behaviour, but they should not be used interchangeably. For practical inductors, the most important design consequence is that a winding voltage is created by the rate of change of flux linkage, while real component performance is shaped by core material, air gap, saturation, loss mechanisms and operating current.

For engineers working with switching converters, magnetics should therefore be evaluated not only by nominal inductance, but also by DC-bias behaviour, thermal performance, core loss, winding loss and measurement conditions. In practice, Faraday–Lenz law is the link between field theory and everyday inductor selection, transformer design and converter troubleshooting.

FAQ

How to use this knowledge in practice

• Check whether the datasheet inductance is specified at zero bias or under operating current.
• Compare nominal inductance with inductance-versus-current behaviour over the expected DC bias range.
• Review saturation current together with the stated test criterion, temperature and allowed inductance drop.
• Consider both core loss and winding loss at the intended switching frequency and ripple current.
• Pay attention to air-gap fringing, leakage inductance and layout-sensitive EMI effects in compact power designs.

Further reading

• Magnetism, Ampère’s Law and Magnetic Fields Strength
• Why Power Inductors Use a Ferrite Core With an Air Gap
• Core Materials, Permeability and Their Losses
• Inductance, AC Inductors and DC Inductors Explained in Video

Source

This article is an editorially expanded revision of the existing Passive Components Blog knowledge article on magnetic induction, magnetic flux and Faraday’s law. The content should be used as educational guidance; final magnetic-component selection and design release should be based on the applicable manufacturer datasheet, magnetic-material data and measured operating conditions.

References

1. Magnetic Induction, Magnetic Flux and Faraday’s Law
2. Inductance, AC Inductors and DC Inductors Explained in Video