Flux fringing - a phenomenon in which the magnetic flux flowing in a magnetic core spreads out (or fringes out) into the surrounding medium, for example in the vicinity of an air gap.¹⁾²⁾

Flux crowding is closely related to flux fringing, but flux leakage is usually treated as a different phenomenon.³⁾

Flux fringing is especially pertinent to magnetic cores with an air gap, for instance in flyback transformers or PFC inductors. The magnetic core is designed so that a well defined gap is placed in the magnetic circuit for storing energy in magnetic field.

An air gap constitutes a magnetic discontinuity in the magnetic core. Relative permeability of air gap is $\mu_{r,gap} \approx 1$ whereas for the core material is usually much much greater $\mu_{r,core} >> 1$. The magnetic flux is forced to flow through the gap which represents significantly greater reluctance than a comparable length of the core.

However, the reluctance of the given local part of the air gap depends not only on the length of the gap, but also on its cross-sectional area. The volume of medium outside of the gap, but immediately next to it represents similar cross-sectional area with the same lower permeability, so its effective reluctance is similar to that of the gap. Therefore, the magnetic flux is shared between the air gap in the core and the neighbouring volume outside of the core.⁴⁾

As a result, fringing effect lowers reluctance of the magnetic path and thus increases inductance of the winding made on such a gapped magnetic core.⁵⁾

The fringing flux factor $F_{FF}$ by which the inductance increases depends on the geometry of the magnetic core. For simple cores it can be approximated by the following equation:⁶⁾

The magnitude of fringing flux is relatively large, because of the concentration of the flux in the magnetic core. Hence, there can be significant eddy currents generated in any conductive material placed in the volume of the fringing flux. Such additional losses are exacerbated at higher frequencies - following the same principle as induction heating.⁷⁾

This also applies to windings which are made out of highly conductive materials like copper or aluminium. As shown in Fig. 2, the part of the winding placed directly next to the air gap can be subjected to excessive heating caused only by the eddy currents in the copper wire. The additional power loss might be small as compared to the total loss of the transformer or inductor, but it can locally create a high-temperature hot spot. In Fig. 2 the part of the winding subjected to the fringing flux operates at a temperature greater by 50°C than the rest of the winding.

In practice, for a uniform air gap the high-intensity magnetic field due to the fringing flux is produced over a distance roughly equal to the length of the air gap.⁸⁾ This is visible in the simulation results in Fig. 2.

Therefore, with the part of the winding placed away by approximately such distance the high-intensity magnetic field was not penetrating the windings any more, and the local copper loss was reduced drastically, so that the hot spot temperature decreased to almost the same as the rest of the winding.

Fig. 2. Fringing flux (red curve) spreads from the air gap to the surrounding medium. Top images show thermal imaging, middle images show FEM simulations and at the bottom are the photographs of actual prototypes. The hot spot temperature was reduced by over 50°C by moving a part of the winding away from the air gap with fringing flux. The cross-hair pointer is set at the hot-spot, whose temperature is shown in the upper left corner. The images show a PFC inductor with a significant air gap.

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Fringing flux is especially important for AC devices with laminated cores (Fig. 3). For a uniform air gap the flux will fringe equally on all sides.

Fig. 3. Fringing flux causes additional loss in laminated cores, due to significant planar eddy currents

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On the edges of laminations the eddy currents will not exceed their normal amplitude, because the involved dimensions are the same and so is the flux density. The effective thickness of the laminations is usually chosen to produce manageable magnitude of eddy currents, to keep the total losses at acceptable level.

However, the fringing flux flows through all the sides, including those with large surface area of the laminations, as shown in Fig. 3.

The amplitude of flux density is of the same order of magnitude as in the rest of the core, but for the perpendicular component of flux (normal to the surface) the active “thickness” of the lamination is equal to the full width of the strip. Eddy current loss is roughly proportional to the square of the thickness, so excessive losses can be produced by such high-amplitude planar eddy currents.

Similar losses will be developed in any other conductive parts, e.g. metal cases or supporting bars if placed within the space affected by the fringing flux.

Fig. 4. Radially laminated core eliminates large planar eddy currents (only a part of a cross-section through the core is shown)⁹⁾

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It is possible to make the gapped part of the core as laminated radially (Fig. 4).¹⁰⁾¹¹⁾

With such construction the all the sides expose only edges of the laminations, so there are no large surfaces in which high-magnitude planar eddy current could be induced.

However, manufacturing of such cores is more labour intensive, and thus the costs are proportionally higher. Also, due to the way the laminations are stacked in wedge-shaped packets the stacking factor is reduced as compared with normally laminated cores.

As mentioned above, the distance over which the high-amplitude flux fringes out is roughly proportional to the length of the air gap (Fig. 5). Hence, one large gap can be divided into several smaller gaps, with similar total volume and in this way the effective permeability and the energy storing capabilities are preserved, but the flux fringing is significantly reduced (Fig. 6). Such solutions are commonly applied especially in large reactors.¹²⁾¹³⁾

Fig. 5. Fringing flux (red curve) spreads from the air gap to the surrounding medium¹⁴⁾

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Fig. 6. Flux fringing can be reduced by used using multiple distributed air gaps

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Powder cores have air gap distributed within the whole volume of the core. Effective permeability is substantially lower (typically between 14-160), but it is uniform throughout the whole volume.

Fig. 7. Flux fringing is practically eliminated in powder cores (FEM simulation for core with $μ_r$ = 26)

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Without the concentrated air gap flux fringing is practically eliminated (Fig. 7). Namely, there is no specific region in the immediate vicinity of the magnetic core which is exposed to high magnitude of magnetic field.

Fig. 8. Flux fringing occurs even if the coil is wound tightly - in this idealised FEM simulation the coil occupies 100% of the window area

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Powder cores can be made from several materials, but in all cases the core offer similar performance from the viewpoint of elimination of flux fringing.

Nevertheless, flux leakage is still present (Fig. 7) and for this reason such cores are used usually as toroids, because uniform distribution of windings around the toroid reduces the flux leakage effects. As an additional effect, because the drop of magnetomotive force is distributed more uniformly around the core, the proximity loss in the winding is reduced because the number of effective layers of the winding is reduced (see also Dowell's curves).

In theory, placing the coil very close to the air gap reduces the flux fringing effect, so that flux is better confined within the air gap.¹⁵⁾¹⁶⁾

However, as evident from Fig. 8, the fringing is not eliminated in this way, and the drawback of increased copper loss may outweigh any advantages of the reduced fringing, especially at higher operating frequencies.

Fig. 9. Planar coils for wireless charging (top and bottom sides) for low power application such as mobile phone; the black backing is made out of bonded ferrite

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Wireless charging can be accomplished by using magnetic coupling between the primary coil of the transmitter, and the secondary coil of the receiver. In other words it can be said that the energy is transmitted in the near field (inductive coupling, rather than electromagnetic radiation in far field).¹⁷⁾

By the requirement of being “wireless” the coils have to be separated by some non-magnetic material (non-metallic casing of the device and the spacing through air) which contributes to significantly reduced magnetic coupling (e.g. k ≈ 0.2)¹⁸⁾, large magnetic reluctance, and large flux fringing between the coils.

The magnetic coupling can be improved (but only by a small amount) by employing a “magnetic reflector” in the form of magnetic core made typically out of soft ferrite or other suitable magnetic materials such as amorphous or nanocrystalline ribbon. Multiple layers of magnetic and non-magnetic materials can be used to optimise the efficiency of the energy conversion as well as improve electromagnetic shielding, so that less magnetic field “leaks” outside of the charging structure.¹⁹⁾²⁰⁾ The thickness and length/width of the layers can be optimised for avoiding magnetic saturation, weight of the portable/moving part, efficiency, cost, and tolerance for misalignment or difference in positions between the transmitter and receiver. Different widths of different layers can be employed, as necessary for a given design.²¹⁾

However, in such devices the length of the air gap is the main parameter which determines the flux leakage, and reduction of flux fringing is very difficult in a general case. If the external magnetic field has to be reduced then “active shields” (energised coils counteracting the main field) might need to be employed.²²⁾

Fig. 10. Magnetic flux lines around a wireless charger, with the transmitter coil (bottom) and the receiver coil (top) with layered magnetic material

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Because of large air gaps, materials with lower magnetic permeability (such as Magment) can be employed in such systems, especially for transmission of high power.

Because of the large amount of magnetic flux in the air gap, large amount of energy is stored in the magnetic field. For wireless chargers, this energy has to be re-delivered for every cycle of AC current. For this reason, resonating approaches are typically used, so that the reactive energy is stored in local capacitors (at least one in primary, and one in secondary), and operation near a resonating point allows achieving higher efficiency of energy transmission, even though the circuit can be fed with a rectangular voltage waveform (on-off).²³⁾

The presence of large air gaps in such systems is a general drawback (it impedes efficient energy transmission) but on the other hand the inductance of the primary and secondary windings change by only a small amount so that operation near the resonant point is affected by a smaller amount.

• Flux crowding
• Flux leakage
• Air gap
• Magnetic circuit