# Encyclopedia Magnetica

Encyclopedia of magnetics and electromagnetics.

flux_fringing

# Flux fringing

 Stan Zurek, Flux fringing, Encyclopedia-Magnetica.com, {accessed 2019-02-22}

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)

Fig. 1a. Fringing flux spreads from the air gap to the surrounding medium (colour map shows flux density)

by S. Zurek, E. Magnetica, CC-BY-3.0

Fig. 1b. Magnetic field strength is significantly greater where the fringing flux is present around the air gap

by S. Zurek, E. Magnetica, CC-BY-3.0

### Explanation

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)

 (1) $$F_{FF} = 1 + \frac{l_{gap}}{\sqrt{A_{core}}} \cdot ln\left( \frac{2 \cdot W}{l_{gap}} \right)$$ (unitless) where: $l_{gap}$ - length (thickness) of the gap (m), $A_{core}$ - cross-section area of the core (m2), $W$ - length of the core window (m)

### Increased copper loss

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.

by S. Zurek, E. Magnetica, CC-BY-3.0

### Increased core loss

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

by S. Zurek, E. Magnetica, CC-BY-3.0

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)

by S. Zurek, E. Magnetica, CC-BY-3.0

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.

### Reduction of flux fringing

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)

by S. Zurek, E. Magnetica, CC-BY-3.0

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

by S. Zurek, E. Magnetica, CC-BY-3.0

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)

by S. Zurek, E. Magnetica, CC-BY-3.0

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.

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.

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

by S. Zurek, E. Magnetica, CC-BY-3.0

In theory, placing the coil very close to the air gap reduced 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.