A new user-friendly computational model - a kind of nomogram - for various ferrite materials and converter types helps developers of power supplies to select the right materials for specific applications. The nomogram developed by EPCOS allows the user to design a converter whose actual core loss is both known and can be restricted to a preset level. This optimizes the energy efficiency under the given operating conditions and reduces both costs and environmental impact.

The magnetic power dissipation of power ferrites is determined essentially by the hysteresis of the material and thus by the magnetic flux density. Another important role is played by the operating temperature. As a rule, most manufacturers of power ferrites provide extensive data on core losses in ferrite cores. Such loss measurements are usually based on a sinusoidal magnetic flux. However, sinusoidal signals occur very rarely, if at all, in practical applications such as switch-mode power supplies.

Core losses are largely determined by the amplitude and type of the temporal path, i.e. the waveform, of the magnetic flux density that affects the material at the operating temperature. Thus the values specified by the manufacturers, which are based on ideal sine curves, are of little use in practice.

A mathematical model developed by EPCOS allow a user-friendly Excel-based nomogram for the calculation of core losses that consider the signal form of the magnetic flux actually dominating in the application. This nomogram may be used within the scope of the design process to select the power ferrite whose behavior best satisfies the requirements of the real situation.

The equivalent sinusoidal frequency

As a rule, the loss density p_vsin in a ferrite core for a magnetic flux density amplitude B with a sinusoidal waveform is calculated from the Steinmetz equation as follows:

Where f is the frequency, T the operating temperature and C_m, C_t1, C_t2 and C_t0 are material constants. The terms x and y thus represent the Steinmetz frequency and Steinmetz induction exponents respectively, defined for the selected operating condition.

For general non-sinusoidal waveforms of the magnetic flux density, Albach, Durbaum and Brockmeyer have extended the equation as follows:

The special feature of this equation is the introduction of the equivalent sinusoidal frequency f_sineq resulting from the periodic magnetic flux B(t), its waveform and its two extremes (B_max and B_min):

For segmental linear flux waveforms with k-1 linear segments in a period, the integral (iii) is replaced by the summation:

Where the variable r is defined as follows:

From (i), (ii) and (v) we obtain:

As a result, r^x-1 is the ratio of the actual core loss of the converter to its value with a sinusoidal signal having the same frequency and flux density amplitude at the same operating temperature.

It is important that all the considered waveforms of the magnetic flux have a single maximum and minumum per cycle, so that several hysteresis loops do not occur within a cycle.

Figures 1 to 5 show typical switch-mode power supply topologies as well as their flux waveforms together with an approximated equivalent sinusoidal waveform is compared. The ratio r is determined for the three specific cases with the aid of the duty cycle δ, the extinction cycle ξ and the resonance frequency f_r (for the resonant converters).

	FIGURE 1: PUSH-PULL CONVERTER

Duty cycle: 0 < δ <= 1, and in both cases from (iv):

	FIGURE 2: FLYBACK CONVERTER WITH DISCONTINUOUS CURRENT PATH

Frequency ratio: 0 < δ < ζ <= 1, with the extinction cycle ζ and from (iv):

	FIGURE 3: FLYBACK CONVERTER WITH CONTINUOUS CURRENT PATH

Frequency ratio: 0 < δ <= 1 and from (iv):

	FIGURE 4: RESONANT CONVERTER WITH ZERO CURRENT SWITCH

Frequency ratio: 0 < δ <= 1, from (iii) for a resonant frequency f_r:

which in turn yields:  r = 1 / δ

	FIGURE 5: RESONANT CONVERTER WITH ZERO VOLTAGE SWITCH

The value r results from the resonant frequency f_r, the charging recovery time of the inductance t_r and (iii):

where ζ = t_r * f_r. ist.

Ratio of actual to sinusoidal core loss

The dependence of the value r on the circuit topology is shown in Figs. 6 and 7 in the supplementary PDF file. Thus it increases as the duty cycle decreases for a push-pull converter and for resonant converters, both with a zero current switch and a zero voltage switch. With a flyback converter, in contrast, r increases with asymmetrical duty cycles.

The ratio r^x-1 between the actual converter losses to those with a sinusoidal signal increases with the magnetic flux as the duty cycle approaches unity for pushpull converters (approaching symmetrical for a flyback converters), but decreases with the magnetic flux at low duty cycles (asymmetrical for a flyback converter). This is shown in Figs. 8 and 9 in the supplementary PDF file. However, r^x-1 reduces with temperature at duty cycles approaching unity for a pushpull converter (symmetrical for flyback converters) and rises at low duty cycles for a pushpull converter. (asymmetrical for flyback converters). This is shown in Figs. 10 and 11 in the supplementary PDF file.

It follows that for anticipated cases of extreme low (asymmetrical for flyback converters) duty cycle operation the converter should be designed to operated at relatively lower temperatures and moderately higher amplitude of flux density.

For duty cycles where r is less than unity, the ratio r^x-1<1 and the actual converter core loss will be less than sinusoidal core loss.

The nomogram as a development aid

EPCOS has developed a user-friendly nomogram for ferrite materials N49, N87, N92, N95 and N97 as well as for various converter types on the basis of the considerations and calculations given above. Further variable parameters are the operating frequency (between 25 and 1000 kHz), the magnetic flux density (between 25 and 300 mT), the operating temperature (between 25 °C and 120 °C) and the frequency ratios.

The following data can be derived from the nomogram: r, r^x-1 as a function of the duty cycle, p_vgen as a function of temperature, magnetic flux density and frequency. The nomogram thus helps not only to select the right material but also to design the converter with the actual core loss figure and output curves to optimize core loss. The nomogram is available on request.

Author: Probal Mukherjee, Ferrites Development, EPCOS India