# Addressing core loss in coupled inductors

Alexandr Ikriannikov and Di Yao, Maxim Integrated -December 20, 2016

Core loss in inductors can adversely affect system performance. Yet, predicting core loss is a complicated endeavor, especially in complex structures such as coupled inductors. This article examines core loss and the resulting effects that should be considered. The article also discusses how the core loss in coupled inductor design can be approached in order to provide a complete power delivery solution.

Magnetic components, such as inductors and transformers, are often an important part of power conversion.  Core loss in these magnetic components usually affects system performance significantly, starting with efficiency.  In that regard, magnetic components also often limit a choice for the switching frequency and greatly influence the overall solution size. Core loss is generally a complicated area of research [1-2, 12], with parametric description of how the losses depend on different parameters.  When coupled inductors were introduced and implemented in many commercial products to derive substantial system benefits [3-9], core loss estimates became even more complex.

The difficulties of core loss prediction for coupled inductors are generally associated with many different core cross sections, several different current waveforms that interact magnetically, and the different directions for many fluxes in the core: coupling, and leakage fluxes. This article provides some details of core losses in coupled inductors and necessary effects to take into account. It also illustrates that the design of the coupled inductors is more complex than discrete inductor design with a single magnetic flux and generally uniform cross section for it.  This complexity highlights the significance of the developed coupled inductor parts available from licensed suppliers, as a lot of effort and validation need to happen for each new design.

Basic core loss equation

The basic core loss is a famed Steinmetz equation (1), where B is the peak flux density, f is the frequency of applied sinusoid, Pv is the time-average power loss per unit volume, and k, α, β are material parameters.  These parameters are referred to as Steinmetz parameters, and are found by fitting the measured data for a particular material.  The original equation from Steinmetz (proposed in 1892) did not have a dependence on the frequency of the sine wave excitation, which was added later.

Equation 1

This is a basic core loss equation, which offers no physical meaning but rather a parametric fit to the measured data; therefore, a core loss can be predicted in some region of conditions around where initial measurements were conducted.  This equation is in many ways not very accurate, as it works only for sinusoidal waveforms and in particular conditions.  Many switching converters have square wave voltages applied to the magnetics, which usually results in triangular ripple waveforms for the current.  This certainly affects the magnetic flux and related core losses.  What also is a big problem is the fact that the fit parameters k, α, and β severely depend on different conditions, such as temperature, DC bias, and frequency.

A very good overview of the historical improvements for core loss modeling was presented at APEC 2012 [1]. A popular equation often used in the present industry relates to an improved generalized Steinmetz equation (iGSE) [2]. The general equation for iGSE is shown as (2), where ki is expressed as (3). Integrating over time would provide the actual (averaged) core loss (4).

Equations 2, 3 and 4

While iGSE introduces a big improvement for core loss estimation for non-sinusoidal waveforms, other effects would still have to be considered on top of it, such as dependence of the fit parameters on temperature, DC bias, and frequency. Practically, as magnetic flux density relates to the current in the winding of the inductor, it is easy to see in (4) that changes in current waveforms would be a good indication of the core loss change. For particular core and winding geometry, as well as particular switching circuit, current ripple can be calculated and translated into magnetic flux density in the core.

Typical discrete inductor has a single winding. For high-current, low-voltage applications it is often single turn or staple. The related core often has a simple shape and a single magnetic flux path, wrapping around the single turn winding. It is, therefore, relatively straightforward to define the flux density in that single flux path and relate it to the current in the winding.  Then the core loss can be estimated for that single magnetic flux.