Simplifying the calculation of core losses in composite inductors

Determining core losses to obtain temperature rise and ensure proper component selection

BY NICHOLAS J. SCHADE

Vishay Intertechnology

Malvern, PA

http://www.vishay.com

Accurately predicting core losses in magnetic components is difficult, and no less challenging in composite inductors. Several core loss models are available, all of which have their merits.

This article will adapt an existing model for estimating core losses in composite inductors. The goal is to simplify the process for determining core losses and to allow designers both to calculate the total losses for the inductor by combining core and copper losses and to select the proper inductor for their application based on temperature rise. This article will take into account the factors affecting core loss determination, as they are slightly different with this type of inductor.

The most notable difference is in the geometry of the component, which, as we will see later, requires different core constants for each inductor size and value. Examples will be presented to illustrate the core loss calculation and component selection.

Issues with determining core loss

Calculating core losses in inductors based on industry-standard core styles — available from numerous vendors — is a straight forward process. This is because standard core sets have a fixed geometry with associated-set core constants; all that is needed is the material characteristics.

Once the geometrical features are known, it is easy to determine core losses for an entire series of inductors based on a single core size. The only variable to account for is the number of turns.

This is not the case with composite style inductors. Due to the way the inductors are constructed, the geometric parameters of each part differ, even within the same inductor size. The composite inductor is essentially built backwards from a conventional one.

In a conventional inductor the magnet wire is wound either directly on the core, as in a toroid, or wound on a bobbin with the core halves inserted into it, as in “E” style cores.

The composite inductor is constructed by first winding a copper wire air coil. The two ends of this air coil are then welded to a leadframe, the tabs of which serve as its mounting pads. Powdered iron is then pressed around the coil assembly to form the inductor. Pressing the magnetic core around the coil is what defines a composite inductor. Since every coil is different in diameter and height, each inductor has different geometric parameters. This means that core constants must be calculated for each inductor.

The only consistent item in a series of inductors will be the iron powder; therefore, the core loss constants for this material will remain the same. There are, however, different iron powders used in a variety of product lines which cover a wide range of operating conditions. Within these product lines, the same inductance values do not use the same air coil, which means constants will be required not only for geometry, but for material as well. What it comes down to is that each inductor has its own unique parameters, even within the same family size.

Composite inductors are frequently used in non-isolated dc/dc converters. This is not an issue, but the waveforms associated with them are. Core loss characterization and the resulting data are often determined using sinusoidal excitation. Dc/dc converters, on the other hand. do not operate with sine waves, instead using a pulsed dc waveform. This means that the current waveform in the inductor that determines core loss will be a triangular wave, not a sine wave. This difference will need to be compensated for in the core loss calculations.

Increasingly dc-to-dc converters are being asked to operate at higher ambient temperatures. This in turn requires the inductor to operate at the higher temperature, in addition to its own temperature rise incurred due to power losses.

It is known that iron powder exhibits the effects of aging at higher temperatures in the form of increased core losses. These losses must be accounted for during the design process in order for a composite inductor to be used at temperatures in excess of 125C. The effects of thermal aging can be minimized by simply limiting the maximum inductor temperature to 125C or less.

Determining core loss

A good starting point is to determine the worst-case operating conditions for the circuit involved. Once these conditions are established, the core losses can be determined and the inductor will be functional for the entire operating range of the power supply. The parameters required for calculating core loss are the volt-microsecond product, duty cycle, and operating frequency of the circuit. The remaining values can be calculated from these three items.

In designing inductors, core losses are often characterized by the Steinmetz equation in mW/cm3:

Equation 1

However, this equation was developed based on sine wave excitation. What is needed is an equation that takes into account the nonsinusoidal waveforms associated with dc/dc converters. To accomplish calculating core loss with non-sinusoidal waveforms, the Modified Steinmetz Equation (MSE) was introduced:

Equation 2

This equation uses an “effective frequency” of the nonsinusoidal waveform along with the operational frequency of the circuit, while making use of the Steinmetz parameters.

The core loss equation shown previously has several constants associated with it that must be determined. They are the core constant K0, the frequency constant Kf, and the flux density constant Kb. These constants are determined by curve fitting of the experimental core loss data established from lab testing. The constants are unique to the material they are established for and do not change from inductance value to inductance value. They will of course change from material to material.

Effective frequency (fe) is a frequency other then the repeat frequency of the circuit used to correct for non-sinusoidal waveforms. The effective frequency for the waveforms experienced in dc/dc converter inductors can be characterized as follows:

Equation 3

This method equates the frequency to the sum of the changes in the slope of the flux density, divided by the changes in amplitude, divided by 2 π.

Method for calculating core loss

To make use of the core loss equation, the effective frequency (fe) and maximum flux density (Bpk) need to be established. These items are then plugged into the MSE to determine core loss per unit volume.

Since the volt-microsecond product of the circuit is known, or can be calculated, we will make use of it to determine Bpk. Composite inductors can be characterized in terms of their ability to handle a certain volt-microsecond product corresponding to 100 Gauss. This parameter is referred to as the ET100 constant. As before, this constant will be different for each inductor size and value. This constant can be used to determine Bpk as follows:

Equation 4

where ETckt is the volt-microsecond product of the circuit and the units are in Gauss. The volt-microsecond product of the circuit can be taken from the IC manufacturer’s datasheets and application notes or calculated from Vout, frequency, and duty cycle.

It can be shown that the effective frequency depends on the duty cycle of the circuit and the operational frequency:

Equation 5

Changes in the flux density in the magnetic circuit are proportional to changes in the inductor current, assuming that the inductance remains constant. Core losses are caused by this ripple according to the flux density levels. The inductor is subjected to the input voltage during the switch-on time and a voltage of the opposite polarity during the switch-off time. The resulting plot of the flux density is composed of two straight-line portions that correspond to the integration of these two voltages over time.

Once the effective frequency and the maximum flux density are known, core losses can be estimated using the MSE. The result from this equation will be measured in mW/cm3, but to simplify calculations the core volume is rolled into the K0 constant, giving us the final version of the MSE for core loss (Pfe) in watts:

Equation 6

This results in a different core constant for each inductor that can be found on the application data sheet. The core loss constants and associated parameters for a partial list of IHLP composite inductor products are listed in Table 1, separated by inductance value and material type.

Table 1. Core loss constants and associated parameters for a partial list of IHLP composite inductor products

Selecting the proper composite inductor

Start the inductor selection process by establishing the selection criteria for the part. Composite inductors have a recommended maximum component temperature of 125C. Subtracting the ambient temperature will give us the maximum allowed temperature rise for the part, although if this number should exceed 40C it is recommended that 40C be used for the allowed temperature rise. Core losses should be limited to 1/3 of the total losses to mitigate any aging effects associated with the powdered iron in the core at elevated temperatures.

Data sheets list a heat rated current (Iheat) as a parameter, which represents the current needed to produce a certain temperature rise indicated on the data sheet. The problem is that this temperature rise is typically measured using dc current and is due to copper losses only, and does not take into account core loss. However, this information is useful since it can be used to determine maximum power losses allowed in the inductor by multiplying the temperature corrected resistance of the inductor by the heating current squared. This will be the power loss (Pheat) to produce the temperature rise associated with the Iheat parameter.

To determine core losses, a designer can calculate the peak flux density using the volt-microsecond product of the voltage waveform across the inductor during operation. Using this core loss estimate, the designer can balance the combination of core and copper losses to keep the total losses and associated temperature rise less than the maximum 125C operating temperature of the composite inductor. Care must be taken to insure accurate copper losses by accounting for the resistance increase of copper due to the change in temperature from room temperature (where DCR is specified) to the maximum operating temperature and losses due to ac components such as skin effect. Lastly, check that Ipeak is Isat (found on the data sheet). Due to the soft saturation feature of IHLP composite inductors, Ipeak can exceed Isat without a serious reduction in inductance, as observed in ferrite style inductors.

Some examples

Consider an inductor for the following “buck” circuit: Vin = 12 V, Vout = 3.3 V, Iout = 5 A, frequency = 500 kHz, ambient temperature of 50C, and a ripple ratio of approximately 0.4, where ripple ratio is defined as the ratio of ac to dc components. A ripple ratio of 0.4 is a good compromise between inductor size and output capacitor rms current. Allowing for an estimated switching voltage drop of 0.5 V and a diode drop of 0.5 V, the above conditions produce design requirements of: duty cycle = 0.317, volt-microsecond product of 5.19, and a required inductance of 2.60 µH. Since we wish to keep the ripple ratio around 0.4, and an inductance of 2.60 µH is not a standard value, we will choose a standard value above and below the calculated inductance.

The heating current of composite inductors is specified based on a 40 C temperature rise due to the current only; therefore we will need to derate the dc current to allow for the heat rise due to core losses. We want to select an inductor with a higher heating current than the 5 A the circuit demands. Reviewing the data sheets, we find the IHLP-2525CZ-01 2.2-uH inductor will handle 8 A with a maximum DCR of 20 mΩ, and the IHLP-2525CZ-01 3.3-µH part will handle 6 A with a maximum DCR of 30 mΩ. The question is which one will be the better choice for the operating conditions?

We will start by comparing core losses of the two inductors. The data sheet specifications for the two inductors are summarized in Table 2.

Table 2. Datasheet specifications for the IHLP- 2525CZ-01 2.2-µH and the IHLP-2525CZ-01.3-µH

Start by determining Bpk using equation (4). Plugging in the circuit ET product and the ET100 of the inductor gives us a Bpk of 519.3 G for the 2.2-µH part and 339.4 G for the 3.3-µH part. The effective frequency from (5) will be the same for both inductors at 367.752 kHz. It is the same because the duty cycle and operational frequency are the same for both parts. The resulting core losses using (6) are:

and

This result might lead you to believe that the 3.3-µH part is the better choice, but as we will see, it isn’t in this case.

To continue with the selection process we need to determine total losses, and for that we will need the copper losses. To calculate copper losses we must first determine operational resistance (Roper), circuit ripple current (∆I), and the inductor dc current (Idc). Roper is the temperature-corrected resistance of the inductor in the circuit and can be found, assuming a 40C rise in temperature, with

Equation 7

Delta I (∆I) for a buck inductor is related to the output voltage, inductance in uH, frequency in Hz, and duty cycle according to

Equation 8

The resulting ∆I for the circuit is 2.36 A for the 2.2-µH inductor and 1.57 A for the 3.3-µH inductor. The power losses associated with ac effects in the winding can be determined using the following equation, where K1is the ac loss constant based on each components construction:

Equation 9

The dc power losses are simply the losses, where Idc is equal to the dc output current in a buck circuit. This means the copper losses will be 0.743 W for the 2.2-µH part and 1.008 W for the 3.3-µH part. Adding these results to the core losses produces total losses of 0.919 W and 1.113 W, respectively. The temperature rise can be found by multiplying the total losses by the thermal resistance (Rth). The temperature rise for each inductor will then be

and

It can be seen by adding the temperature rise to the ambient temperature that neither inductor will exceed the maximum component temperature of 125C. Based on this, the HLP-2525CZ-01 2.2-µH inductor is the better choice for this circuit.

The last item to check is to see if the peak inductor current (Ipeak) is less then the saturation current of the inductor. The peak current can be determined by

Equation 10

Ipeak therefore is equal to 6.18 A, which is less than the 14.0-A specification for this inductor.

By simplifying the calculation of core losses in composite inductors and applying this information as part of total losses in determining temperature rise and proper component selection for a given set of design parameters, the following conclusions can be drawn concerning composite inductors:

The core geometry of each inductor value is different, even within each family of inductors. This is due to the manner in which they are constructed, which is essentially backwards from traditional inductors.

The core loss constants associated with each inductor value are different as well. This comes about because of the differing geometries.

Composite inductors are manufactured with different blends of iron powder. These different blends have their own unique constants also.

In the past core losses were often estimated using sinusoidal excitation. Composite inductors are used in circuits operating with non-sinusoidal waveforms that require the core loss equations to account for this.

Composite inductors are susceptible to the effects of thermal aging and should be operated below the recommended 125C component temperature unless the increased core losses are accounted for.

Appendix

Calculating required inductance

To calculate required inductance for a buck circuit, start with the relationship

During the converter ON time, V will equal Vin – Vsw - Vout, giving us the expression:

where D is the duty cycle and T is the period of the waveform.

Solving for L and substituting 1/f for T, we get:

Plugging in the circuit parameters, the required inductance is

References

1) C.P. Steinmetz, “On the law of hysteresis”, AIEE Transactions, Vol. 9, pp 3-64, 1892, Reprinted under the title “ A Steinmetz contribution to the ac power revolution”, introduction by J.E.Brittain, in Proceedings of the IEEE 72(2), 1984, pp. 196-221.

2) J. Reinart, A. Brockmeyer, and R.W. De Doncker, “Calculation of losses in ferro- and ferromagnetic materials based on the modified Steinmetz equation”, Proceedings of Annual Meeting of the IEEE Transactions Industry Applications Society, 1999, pp. 2087-2092, vol. 3.

3) Jieli Li, T. Abdallah, and C.R. Sullivan, “Improved calculation of core loss with non-sinusoidal waveforms”,IEEE Industry Application Society annual meeting, 2001, pp. 2203-2210.