A linear circuit model is derived which will show complex proximity effect losses for arbitrary waveforms.

Introduction

In this special article, Dr. Ridley reports on work resulting from the magnetics presentations at APEC 2016. It is shown how a linear circuit model can be used inside any version of Spice (or other circuit simulator) to get accurate results for the ac proximity loss resistance of magnetics windings. The equivalent circuit is derived directly from the physical structure of the windings and supported by measurements. This important work makes the findings of Dowell’s equations easily accessible to any engineer.

Inductor Circuit Model

Figure 1 shows the traditional, and quite simple, circuit model for a high-frequency inductor. This equivalent circuit has been used for over 100 years to characterize the important constituent components of an inductor.

Figure 1: Traditional Inductor Equivalent Model

For quite some time, power supply engineers have been using simple circuit models like these to simulate their power converter circuits. Unfortunately, the simulations fall short of observed reality, especially in the area of power dissipation. Simulated inductors simply do not have the same temperature rise as the real inductors in the circuit. This can lead to unexpected failures or greatly-shortened product lifetimes. This problem has become more exaggerated in the last decade as many off-the-shelf inductors are now available, and designers push to higher frequencies with high ripple currents in the inductors. This is being done in an effort to minimize the inductor size. Not paying attention to ac resistance can be a fatal error.

More Complete Inductor Circuit Model

I have told design engineers many times that a much more complex circuit model is needed if you want to be able to get better simulation results for inductors. Generally, I have thought that the magnetics models are simply too complex for a circuit simulator to be able to handle them effectively. At the recent APEC conference in Long Beach, California, I presented what I thought was the minimum-complexity model needed to approach reality in simulation. This circuit is shown in Figure 2, although even this circuit is a little simplified for clarity in this article.

Figure 2: More Complex Model Needed to Capture Most of the Important Behavior of Inductors

There are several modifications in this circuit compared to the standard circuit model used in the past. Firstly, the fixed inductor value is replaced with two discrete and different values – if the current through the main inductance exceeds the saturation level, a second and much lower value is used to represent how much inductance is left after the core is saturated. In reality, there would need to be a continuously-variable value of inductance with current, but two discrete values can often be sufficient to model many of the important phenomena that can occur with inductor saturation.

Secondly the core loss resistor, usually just a fixed value, is replaced with a ladder R-L network to represent the frequency-dependent nature of the core loss. The resistances of this network would also need to be modulated with the amplitude of the drive to the core, but that is beyond the scope of this article. Details of the derivation of the frequency dependent core loss model are given in [1].

I don’t know of anyone using such a model to simulate core loss properly – they usually just calculate the core loss separately from the circuit waveforms and the flux swing in the core on each cycle. This can provide reasonable accurate figures for the core dissipation although the calculations are usually time-consuming.

The third major modification to the traditional inductor circuit is the inclusion of a network to represent the frequency-dependent nature of the winding resistance referred to as proximity loss. This is shown as another ladder network to the left of the circuit.