In the majority of conventional inductor designs, calculating the total power loss was limited to core and copper losses (I^2*Rdc). In today's SMT inductor designs, which use gapped ferrite cores, the fringing effects induce circulating current in the winding, which effectively redistribute the current in the winding. Calculating dc copper losses without the fringing analysis can cause a significant error in winding loss calculation.
Consider the current carrying conductor in Fig. 1. Any current carrying conductor will generate two fields: one internal to the conductor and one external to the conductor. Fig. 1 shows the external fields at different distances (A, B and C) relative to the center of the conductor.
Among the flux lines in Fig 1., magnitude for flux at point A (Ba) is greatest because it's closest to the conductor and decreases as the distance from the conductor increases.
Now, let's place a nonmagnetic, non-carrying current conductor, as shown in Fig. 2, into the stray fields so that the flux lines (Ba, Bb and Bc) intersect the conductor with a 90-degree angle.
By examining the non-current conductor alone, we can see the creation of eddy currents with directions (Fig. 3) at points A, B and C caused by the stray fields Ba, Bb and Bc. The direction of the eddy currents is opposite to the field that generated them.
Fig. 4 shows the distribution of the current through the non-current carrying conductor with a higher current distribution on the left side. This causes an increase in loss over what the dc resistance alone would suggest. The net current is the sum of the main current through the winding and the eddy currents caused by the fringing fields.
We can see that the stray fields not only create circulating current in a non-carrying conductor, but also cause a non-uniform current distribution in the conductor.
Now that we've seen the effect of external fields in generating eddy current, let's examine Fig. 5. The new bead inductors are very small with current capabilities over 30 A and sizes ranging from 0.25 × 0.25, with not much surface area for heat dissipation. Calculating the eddy currents into the copper loss will give a better estimation of the total loss and allows the designer to predict the temperature rise more accurately.
To minimize the effect of fringing field, we can increase the distance r1. However, keep in mind that increasing r1 will reduce the much-needed copper width for high current applications. Also, changing the angle will change the normal field to a tangential field and reduces the effect of the net flux.
By integration from r1 to r2, we can obtain total flux Where flux density is normal to the whole surface, where B = µo *I /(2*π*r).
Note: integral of du/u = ln[u]+constant.
µo = permeability, Io= current in Amps, r = distance in (cm) and cos(õ) is the angle in degrees.
The above formula shows two ways to minimize the fringing field on the copper loss calculation:
By increasing r1 or the distance from the edge of the copper to the area of fringing (Fig. 7).
By increasing the angle in order to reduce the total flux (Fig. 8). This effect also can be achieved by bringing down the copper, away from the fringing and replacing the high influence of the normal field by low influence of tangential field.
The gap-fringing field can cause a significant amount of eddy current loss, which needs to be added to the main copper loss calculation. This increase in loss would cause a temperature rise in the part and will affect the saturation profile of the inductor. As shown in Fig. 9, by proper combination of separation distances (r1 and X), the copper width can be maximized to reduce the dc copper loss and also minimize the fringing field.
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