Targeting Switcher Magnetics Core Loss Calculations

Magnetics product catalogs derive core loss vs. frequency curves by measuring the core losses that result from sinusoidal excitation at varying frequencies and voltage amplitudes. The “B” in the family of curves is the maximum flux either side of the origin of the B-H curve. Thus, the total swing in flux is twice that shown in the core loss charts. The formulas for core loss in the catalogs are empirical ones that give a best-fit to the measured values.

Most application notes when estimating the core loss of a magnetic component have a procedure similar to:

1. Calculate the total flux swing using Faraday's Law. If the voltage applied to a transformer winding is constant during a pulse, then the total flux swing is:

   ΔB=(VΔt×10⁸)/NAe (1)
   Where:
   ΔB=Total flux swing in gauss
   VΔt=Volt-seconds in the pulse
   N=Number of turns in winding
   Ae=Cross-sectional area of core in cm²

2. Assume the total flux swing from (1) is the same as that for a sinusoid with same volt-seconds. Divide the total flux swing by two and go to the core loss curves at the specified switch frequency to find the core loss per unit volume (or unit weight), either in mW/cm³ or W/lb.

3. Multiply the core loss per unit volume×cm³, or W/lb×the core's weight.

The classical procedure is easy to use. However, for pulsed operation where the duty cycle is low, the actual core loss will be higher than predicted by the classical procedure. For these pulsed applications, you'll find a better procedure some experienced magnetic designers have used.

Core Loss

Core loss is proportional to the area enclosed inside the hysteresis curve. In reality, the actual width of the hysteresis loop is influenced by the rate of change of flux, dB/dt, which has a nonlinear relationship with frequency and flux amplitude. If the frequency is doubled and flux amplitude is held constant, then the dB/dt is doubled. If the frequency is held constant and the flux swing is doubled, then the dB/dt is also increased by a factor of 2. However, when we look at either the core loss curves or the best-fit formula for any magnetic material, we see the core loss isn't directly proportional to the flux amplitude or the frequency. Instead, the actual relationships are exponential.

Consider Magnetics “P” material as a typical example. For the frequency range of 100 kHz to 500 kHz, the best-fit formula ^[1] is:

P_L=0.0434×f^1.63×B^2.64 (2)
Where:
P_L=mW/cm³
B=One-half the total flux swing in kilogauss
f=kHz

Equation (2) predicts that if the amplitude of the flux swing is held constant and the frequency is tripled, then the core loss increases not by a factor of 3, but by a factor of 3^1.63=6. Similarly, if frequency is held constant but the flux swing is tripled, then the best-fit formula predicts a core loss increase, not by a factor of 3, but by a factor of 3^2.64, or about 18.2. Obviously, with this highly nonlinear behavior, predicting the core loss in a high-frequency, low-duty cycle, pulsed operation via the classical procedure found in most application notes (assumes core loss is the same for sinusoidal excitation) will result in a gross underestimation of the total core loss.

Because of this nonlinear core loss behavior, the large exponential increase in core loss with an increase in dB/dt is due to eddy currents in the core. The amplitude of eddy currents is proportional to dB/dt and thus the voltage applied to the coil winding. This means the core loss due to eddy currents is proportional to the square of the eddy currents and proportional to the distribution of the eddy currents (core eddy current loss = I²R). Thus, the eddy current losses in the core are proportional to the square of dB/dt, which in turn is proportional to the volts per turn of the winding ^[2]. You can model core eddy current losses in a choke via a resistor across the winding and in a transformer by a resistor across the primary.

Skin Effect

The penetration depth (D_pen) of the fields in the core material is:

meters (3)
Where:
ρ = Resistivity of the material
μ_r = Relative permeability of core
f = Frequency.

Thus, there's a complex relationship between core material, frequency, and eddy currents. Magnetic materials in Permalloy tape cores are conductive, thus have a much smaller penetration depth than do ferrites that have high resistivity. In most power ferrites, the skin effect upon eddy currents losses can be ignored. However, this isn't true with tape cores whenever the tape thickness exceeds the penetration depth. Thus the exponential to dB/dt in core loss for tape cores depends greatly upon the frequency.

Obviously, an actual measurement of core loss of a magnetic component in actual circuit operation is desirable. However, any reliable measurement of core loss would require a specialized test setup, normally outside the resources of an SMPS development lab. What alternative is there? One approach is a thermal measurement of core temperature made inside a supply under actual operating conditions, which lumps the copper loss with the core loss and is affected by other variables including airflow and nearby hot components. Thus, the effects of winding loss, surface area, airflow, and surrounding hot components make it extremely difficult to deduce the core loss from a temperature rise measurement.

There's a simple, easy-to-use alternative procedure that can give the SMPS designer some confidence that his estimation of core loss under pulse operation is approximately correct. The procedure calculates the apparent frequency, finds the core loss for the apparent frequency, then multiplies that core loss by the duty cycle of the apparent frequency. We “approximate” by assuming that the one-cycle core loss for the apparent frequency is the same as that for one cycle of a sinusoid at the apparent frequency (a sinusoid with the same total flux swing).

Forward Converter Example

Consider a typical 100 kHz, two-switch forward converter with 20 ms of hold-up time at low input line. Assume that at high line the steady-state voltage, seen on the primary of the power transformer, is that shown in Fig. 1. Also, assume the total flux swing in the P-material core is 1600 gauss.

In Fig. 1, one complete cycle of flux swing is completed in 5 μsec. Although the switch frequency is 100 kHz, the apparent frequency seen by the transformer during a combined period of PWM pulse time and flyback time is 200 kHz. The duty cycle for the apparent frequency is 5 μs/10 μs=½. Using the classical method to estimate core loss, we divide 1600 gauss by 2 and go to the P-material loss curve at 800 gauss and 100 kHz, the switch frequency. The core loss from the P-material loss curve is 45mW/cm^3[1]. However, using the apparent frequency method, we go into the loss curves at 800 gauss and 200 kHz to get about 130mW/cm³, which we then multiply by the one-half duty cycle of the switch frequency to get 65mW/cm³. The 65mW value is 44% more than the 45mW value obtained by the classical method.

Note that the above analysis uses data from core loss curves at 80°C. At temperatures much lower or higher than 80°C, the core loss is higher. Also note that the flux vs. time transition (Fig. 1b) is a triangular wave, not a sinusoid. In most PWM supplies, the flux swings vs. time are triangular since flux is proportional to the integral of voltage.

Magamp Example

Assume a 2714A amorphous-material saturable reactor in a magamp post regulator connected to the 100 kHz forward converter above. Assume that the turns and cross-sectional area of the saturable reactor (SR) are such that during the blocking time and reset time that there's a total flux swing of 6000 gauss and that the voltage waveforms are those shown in Fig. 2.

Consider region (a) in Fig. 2 as one-half cycle at an apparent frequency of 1/(0.8 μs+0.8 μs) or an apparent frequency of 625 kHz. The duty cycle of this apparent frequency while the SR is blocking is 0.8 μs/10 μs=0.08. Consider region (b), the reset voltage, as one-half cycle at a second apparent frequency of 1/(2×2.7 μs)=185 kHz. The duty cycle of the apparent frequency during reset is (5 μs-2.3 μs)/10 μs=0.27. Now we go to the loss curves for 2714A material in^[3] at both 625 kHz and 185 kHz for a 6000 gauss/2 flux swing, then multiply each core loss by its corresponding duty cycle and sum the two results together. The calculation based upon the loss curve figures and best-fit formula is:

Core loss/lb=587W/lb×0.08 duty cycle+100W/lb×0.27 duty cycle =74W/lb (4)

The core loss for a frequency of 625 kHz and 3000 gauss is outside the range of the core loss chart. Thus, the core loss (P_L) for 625 kHz was predicted from the “best-fit” formula:

P_L ??? 0.0458×10^-4×f^1.55×B^1.67 (5)
Where:
B=Flux density in Teslas

(Note: This is a Magnetics Inc. formula whose accuracy at 625 kHz isn't known. However, it's the best available data).

Using the classical procedure given for magamp design, the predicted core loss for a 3000 gauss flux swing at 100 kHz on either side of the origin of the B-H curve, the core loss would have been only 42W/lb. Compare this to the 74W/lb obtained using the apparent frequency procedure. This discrepancy explains why saturable reactors in magamp post regulators seem to run hotter than first predicted. Another factor is the added area inside the hysteresis loop resulting from driving the core deep into and out of saturation. Driving the core in and out of deep saturation adds a significant amount of stored energy. However, much of this energy transfers back to the primary.

To find the total core loss via the apparent frequency procedure, multiply the watts per pound times the weight of the core. For example, assume a 50B10-1E core weighing 3.5 g. The estimated core loss for the saturable reactor example above is 74W/lb×3.5 g×1.0 lb/454 g=0.57W.

Full-Bridge Example

A good example of the usefulness of the apparent frequency approach is to estimate the core loss in the resonant choke in series with the primary of the transformer in a phase-shifted full-bridge supply^[4]. The resonant choke Lr is shown in Fig. 3.

Fig. 4, on page 22, shows a typical transformer primary current waveform for the 200 kHz phase-shifted full bridge. In a choke, the B field is directly proportional to the current, and the voltage across the choke is proportional to the derivative of the current (V=Ldi/dt). Since we know that the core loss increases exponentially with dB/dt, then for a quick approximation of core loss, we will consider only the high dB/dt portion of the flux curve and its duty cycle. At the leading edge of the pulse, the bulk voltage is divided between the resonant choke and the leakage inductance of the transformer. For the transition we can assume that Cc is a short circuit. Assume that the flux transition from (a) to (b) in Fig. 4 is 1200 gauss and that the transition time is 0.5 μs and the switch frequency is 200 kHz. The apparent frequency from combining the 0.5 μs positive transition with the 0.5 μs negative transition is 1/(0.5 μs+0.5 μs)=1 MHz and the duty cycle is 1 μs/5 μs=20%. Assume P-material ferrite. Enter the loss curve for 1 MHz at 1200/2=600 gauss to get 5.5W/cm³. Multiply 5.5W/cm³ by 20% duty cycle to get 1.1W/cm³.

This is an unacceptably high power loss. A core volume of only 10 cm³ results in an estimated 11W of core loss. If we had used the traditional approach, considering 1600 gauss at 200 kHz, (entered curve at 800 gauss) the estimated core loss would have been only 0.13W/cm³. This “traditional calculation result” would have seemed “marginally acceptable” until the supply is tested and choke winding begins to smoke. A larger core and larger air gap are needed in this example to reduce the overall flux swing and core loss.

Snubber Example

A good example of how to use the apparent frequency method is one in which a saturable reactor (SR) core is in series with an output rectifier diode, as shown in Fig. 5.

Assume the output filter is in the 100A output of a 100 kHz supply. Also assume that the SR cores are Magnetics 50B12-1D^[3] and tests indicate that when rectifiers are hot, the flux swing in the SR2 core is 4000 gauss in 200 ns, both for the set and reset times.

Use the apparent frequency method to estimate the core loss. The apparent frequency is 1/(200 ns+200 ns)=2.5 MHz. Duty cycle is 400 ns/10 μsec=0.04. Obviously 5 MHz is outside the parameters of the published core loss curves, so we will use the best-fit formula:

P_L ??? 0.351×10^-4×f^1.5×B^1.8 (5)
Here:
B=0.2 Teslas.
f=2.3 MHz
P_L=7530 W/lb

Multiply by the 0.04 duty cycle to get 301W per lb. Multiply by 1.2 g core weight to get:

Core loss=301W/lb×1 lb/454 g×1.2g=0.8W

Compare this estimate to one derived by the traditional method. Enter 100 kHz loss curve at 4000/2=2000 gauss to get 60W/lb. Multiply 60W/lb by 1.2 g core weight to get 0.16W. This value is only 20% of the core loss predicted by use of the apparent frequency and duty cycle method.

PFC Circuit

One difficult core loss problem is found in the resonant PFC circuit, shown in Fig. 6. Assume current in L1 is continuous and all components are ideal. At the beginning of a switch cycle, L1 is flying back into Cb through D1. S2 turns on. Current ramps up through Lr, as shown in Fig. 7, on page 24, until voltage across Cr goes to zero. S1 turns on, S2 turns off, and current through Lr ramps down as Lr flies back into Cb through D2 and D3. SR1 prevents high-frequency ringing due to reverse recovery of boost rectifiers D2 and D3^[4].

Core loss of Lr choke is difficult to calculate because not only is the peak current and thus the flux swing modulated at a 120 Hz rate, the apparent frequency and duty cycle are also varying at a 120 Hz rate. There are several approaches for estimating the choke's core loss, including:

1. Pick the worst-case steady state, input, ac low-line condition. Estimate the core loss for the maximum (peak current) via the apparent frequency and duty cycle method, then multiply result by some “J factor” between 0.5 and 0.7.
2. Pick a condition on the skirt of the ac waveform for which the current is perhaps 0.8 that of the peak current at worst-case, steady state low line, then apply the method.
3. Do simulation using assumed estimated circuit conditions. Repeat at regularly spaced intervals of the rectified 60 Hz input sinusoidal current waveform, then average the results.

In any case, there are many factors (and some unknowns) at work that no matter what method is used, one should do a verification test in the lab. The bottom line is the temperature rise at full load and low input line after supply has warmed-up. Some factors that effect temperature rise are: winding loss (which will be dominated by the proximity effect), the 120 Hz current changes in the resonant circuit, which in turn effect the function of many other variables. Other factors at work are the value of the components, the surface area of choke, airflow, ambient temperature, the core material, the delay time after voltage across Cr reaches zero and S2 is turned on, the current in L1, and even the snap-off characteristics of D1. Don't rely solely upon calculated values to predict temperature rise. There are simply too many variables at work.

The core loss in SR1 is also very difficult to estimate accurately, for many of the same reasons as is Lr. In a typical circuit, SR1 may be five turns on an amorphous tape core with Ae of 0.1 cm³ and le of 5 cm. The core will saturate at less than 0.5 Oe, which occurs at about 0.4A. Before it saturates, the SR blocks hundreds of volts. With only five turns, the dB/dt in core is extremely high. The apparent frequency seen by SR1 is in the megahertz range. It will run hot. For a PFC switch frequency of 100 kHz, a good approach is to mount the SR up off the p. c. board and into the airflow for better cooling. SR1 will require heat sinking at 200 kHz and up.

Acknowledgment

Warren Martin, formally of Fair-Rite and Magnetics (now retired), introduced the author to the method of using apparent frequency and duty cycle to estimate core loss of a forward converter transformer.

References

1. Magnetics Ferrite Core Catalog, 1997.
2. Unitrode, “Magnetics Design Handbook”, Mag 100A.
3. Magnetics, “TWC-300T, Design Manual Featuring TAPE WOUND CORES.”
4. Unitrode, “Power Supply Design Seminar,” SEM-900, Topic 3, pp 4-1 to 4-18, 1993.

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