The widespread use of gapped ferrite cores for inductive components subject to a superposed dc current poses a challenge to the way the required properties are specified. The importance of attaining a precise definition of target values is driven by the need of miniaturization, on one hand, and the slow progress of material performance, on the other. This is because incrementing key material properties requires advancement in both material composition and processing in order to push the limits.
The conventional way to formulate a dcbias specification for a gapped core based on measurements shown in Fig. 1.

A target rolloff in the range of 10% to 30% is linked to a dc flux density as estimated from the dc current and the gapping factor β, defined by the reciprocal difference between the effective and the initial permeabilities: (Eq. 1, 2)
N stands for the number of turns, I for the current, l_{e} for the effective magnetic length of the core and g for the air gap. Note that the validity of Equation 1 improves with increasing air gaps.

It is then verified if the resulting value is distant enough, e.g. 20%, from the material saturation figure as judged by empirical data. The rolloff referred to the lower tolerance limit of the inductance minus a safety margin is then assigned to this current.
The above described semiquantitative procedure can be improved by taking into account the material behavior under dcbias. To this end, a model of the nonlinear magnetic response describing the permeability rolloff as a function of flux density is needed. One possibility is the classical model by Gans [1], which describes a universal function that depends on initial permeability and saturation. However, only a certain type of materials follows the Gans curve because features such as the squareness of the hysteresis loop (which is reflected in the inductance vs. current curve) are missing. Therefore, a closer representation has to start from hysteresis modeling.
Hysteresis Model and Reversible Permeability
The model proposed by the author [2] consists of solving the Hodgdon [3] hysteresis differential equation in the formulation H(B) with the major loop, as a particular solution. For the branches of the major loop a heuristic description has been found, which fits the lower H_{L}(B) and upper H_{U}(B) branches for soft magnetic materials: (Eq. 3a, 3b)
The fit parameters are besides the saturation B_{S}, coercive field H_{C} and permeability at the coercive field µ_{C}, the two squareness exponents a and b. The reversible permeability can be derived from the equations for asymmetric minor loops in the limit B → 0 and for a ≈ b, valid for most power ferrites as follows: (Eq. 4)
To compare the model with measured curves, a relationship between the nondirectly measurable dc flux density B_{dc} and the applied dc field H_{dc} is necessary: (Eq. 5)
These equations yield curves in excellent agreement with measured values for a variety of soft ferrite materials [4].
Rolloff at Saturation
With the help of Equation 4, the permeability vs. dc field for cores with air gap can be calculated. For the inductance, the effective reversible permeability follows from Equations 2 and 4: (Eq. 6)
and for the effective dc field: (Eq. 7)
The above equations can then be applied to answer the following question: If the shearing line described by Equation 1 is prolonged to cross the saturation line, which rolloff corresponds to the resulting value for the effective field (dubbed H_{s} for “saturation field,” Fig. 2)? The result depicted in Fig. 3 clearly shows that for sufficient large gaps, such that roughly [4] (Eq. 8)
the rolloff is 50%. This can be proofed both theoretically [4] and by measurements on a gapped toroids (Fig. 4: R9.5 in material T38, µi=10000, µe=270) for which the dc current has been rescaled by a factor (Eq. 9)
following Equation 2 and 7, yielding values in Tesla. The saturation value can then be read at the 50% saturation yielding 410 mT in excellent agreement with the directly measured value. Of course, values higher than the saturation do not have any physical meaning.
When applying the above procedure to core shapes other than rings, a distribution of crosssectional areas has to be taken into account. This means that the corresponding flux density distribution of the core has to be calculated section by section. This is shown for a highly inhomogeneous shape, such as EP13 in Fig. 5, along with a rescaling of the current by the minimum crosssectional area, which represents the bottleneck in terms of saturation. (Eq. 10)
Remarkably, the corrected EP13 curves and the ring core coincide exactly at the 50% rolloff, yielding the same material saturation figure.^{1}
Distance to Saturation (DTS)
The results presented so far show that the dc bias can be precisely defined relative to the saturation point. With Equations 4 and 6, curves for the rolloff vs. the relative distance to saturation have been calculated for typical material parameters at different temperatures, as shown in Fig. 6. With the help of these curves, the choice of rolloff can be converted via the relative distance into a corresponding current, based on the knowledge of material and core parameters.
A_{L}Tolerances
When considering the effect of A_{L}tolerances, it is important to first note that based on a statistical tolerance simulation analysis, the tolerance is proportional to the A_{L}value itself: (Eq. 11)
(Eq. 12)
The constant involves the standard deviation for the permeability σ_{µi} and the air gap σ_{g}, a factor sym for symmetric (=1) or asymmetric (=2) grinding as well as the target Cpkvalue. The above equation gives a generally valid relationship to be considered for designs. However, concrete values will change from vendor to vendor, even though an IECstandard for A_{L}values and its tolerances is in preparation.
DCBias Specification
Since the target of an application is to have a certain inductance: (Eq. 13)
the selection of A_{L}value and of the number of turns is a degree of freedom. In the frame of the present discussion, the upper tolerance limits the maximum attainable dc current and needs to be kept as small as possible. Therefore, the following specification procedure is recommended (Fig. 7):

The target rolloff RO with respect to the nominal inductance is defined as the lower inductance limit: (Eq. 14)

The current corresponding to the target rolloff referred to the upper tolerance (of µ_{e}) is the setting parameter: (Eq. 15)
The above procedure provides a valid specification for all possible A_{L}values within the limits, provided the tolerance range is smaller than the rolloff: (Eq. 16)
Otherwise, the inductance vs. dccurrent curve corresponding to the lower A_{L}tolerance limit would lie outside the minimum inductance value defined by Equation 14.
Temperature Effects
The material parameters of ferrite materials are known to be strong temperaturedependent. To adapt the above results from a given temperature, i.e. 25°C, the following parameters need to be corrected in their temperature dependence:
B_{s}: Look up value in material data sheet; the slope of the temperature curve is in the range of 0.8 … 1.5 mT/°C for power grades and 1.6…2.3 mT/°C for highperm grades
µ_{e}: Since the air gap and hence µ remain constant throughout the temperature, the value at a different temperature is given by (Eq. 17)
DTS: cf. Fig. 6
The adaptation of these parameters mainly affects the setting current given by Equation 15. Because of the comparably large gaps considered here (Eq. 8), the minimum inductance may be considered as temperatureindependent. This is desirable from an application point of view and correct as long as T_{2}>T_{1}. In most cases, µ_{i}(T_{2})>µ_{i}(T_{1}).
The specification procedure can be therefore complemented with a final step:
3) Setting current at the temperature T2. (Eq. 18)
The advantage of this equation as opposed to a direct calculation for every temperature is that a known specification that has been defined at, say, room temperature can easily be transferred to a different temperature. Even though the ratio of saturation values is the dominating factor (see example below), the DTS term is necessary to have correct setting currents for all temperatures. The specification procedure can be best illustrated by the following example (Fig. 7):
Core: RM8
A_{e}=64 mm^{2} l_{e}=38 mm A_{min}=55 mm^{2}
Material: N87
µ_{i}(25°C)=2200 µ_{i}(100°C)=4000
^{2}B_{s}(25°C)=465 mT ^{2}B_{s}(100°C)=370 mT
a(25°C)=2.9 a(100°C)=5.1
µ^{c}(25°C)=5500 µ_{c}(100°C)=4300
H_{c}(25°C)=21 A/m H_{c}(100°C)=13 A/m
Inductance: L=1.3 mH A_{L}=160 nH +/ 3% (µ_{e}=75.6) and N=90
Rolloff: RO=20%
Result:
1) From Eq. 14 follows
2) From Fig. 6:
3) For T_{2}=100°C with DTS (100°C)=8%:
Conclusion
A welldefined dcbias specification for ferrite cores with a sufficiently large gap starts from the saturation properties of the core and the material via the 50% rolloff point. The result is a stable specification with higher values in both setting current and minimum inductance as compared to the conventional method.
The equations shown are not limited to the case of large gaps, which represents the most common design case and which can be handled in a simple way. As shown in [4] the model also can be applied for ungapped cores.
Acknowledgments
The author thanks D. Lange for suggesting the publication of this work and for fruitful discussions with him and with Dr. H. Meuche and Dr. J. Wrba. The author also thanks to F. Fontes and I. Kopecka for performing the precise measurements needed to assess the validity of the model.
References

1. R.M. Bozorth, Ferromagnetism, D.van Norstrand Co. Inc., New York: 1951.

M. Esguerra, “Modelling Hysteresis Loops of Soft Ferrite Materials,” International Conference on Ferrites ICF8, Kyoto: 2000, 220222.

M. L. Hodgdon, IEEE Trans. Magn. 24 (1988) 3120.

M. Esguerra, to be submitted to IEEE Trans. Magn.
_{1} A patent for the usage of the method presented to certify the saturation value of any core shape has been filed.
_{2} The minimum saturation values correspond to the typical data sheet values minus 5%. This recommended correction may vary for technological reasons depending on core shape and size, material grade and vendor.
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