Alf’s Musings #6

A bit of context

The article was originally published some months ago directly in a LinkedIn post, but I wanted to include it in its proper place in my newsletter.

The original article was a summary of how magnetic core shapes are simplified and used through what is called effective parameters. These parameters are not as accurate as modeling the whole shape in Finite Element software, but they remove so much complexity that they are the de facto method for working with magnetic cores, and even manufacturers provide them for all their products.

And because I hate to just reprint old stuff, I have gone over all texts and improved them where I thought I had something more to say, plus I have added a couple of subsections at the end.

Best shape is in the eye of the beholder

When Power Engineers talk about designing their own Magnetics there is always a point in which we all agree to disagree: the core shapes. It’s like the drinks, everybody has a favorite, which they usually go with, and it kind of works for them (until they are faced with their decisions next day/end of the project).

I have met people that always go with RMs because they are more modern, people that prefer Es, as they are cheaper, and even people that made their whole career with ETDs. But, how much of a Scientific base is behind those decisions and how much is just taste?

This Alf’s Musing is dedicated to the basics of the core shapes, those little pieces that make every magnetic component, and that we tend to neglect the time and thought they require.

Shape parameters

Anybody that has worked minimally with magnetics can easily visualize how the magnetic flux circulates through a round-section toroid: evenly distributed along the curvature, with a higher density in the interior part of the cross-section than on the outer, due to the shorter length, and therefore smaller reluctance.

But can you visualize it for other shapes, dear reader? Can you imagine how it is distributed in a simple E or P? Or how it goes around a really complex shape like a PQ or RM, where some sections are cylinders, others are truncate prisms, and others are taken out of a bad Lovecraftian dream?

Unless you are some kind of savant, or spend a lot of time doing Finite Elements simulations, you are on the sack with the rest of us, who need to simplify these esoteric shapes with some parameters that allow working and designing magnetics with them without losing sanity points.

I have divided the parameters into two kinds: effective and geometrical, and although many Engineers just look at the first ones and decide what cores to use with only them, I think that the geometrical parameters are also important, as they help us choose between shapes from different families that might have similar effective parameters.

Effective parameters

What we call the effective parameters of a shape is a little effort that is done by the designers of the shape (or by whoever is nice enough to implement the calculation) to help us use complex shapes into the relatively simple calculations we use to model the behavior of our core, like modeling its reluctance, core losses, or saturation.

Delving a bit more into them, the effective parameters of a shape, length, area, and volume; are the dimensional values as seen by the magnetic flux as it traverses our shape. They can be also seen as the parameters of the toroid equivalent to our shape.

We must always have in mind that, although they have dimensional units, they are not geometrical parameters. The effective parameters are related to the magnetic flux in our core, and should only be used for calculation where the magnetic flux is involved.

The calculation of these parameters is defined in the standard IEC 60205, although they can be found along with the main dimensions in the IEC 62317, associated with each shape in the standard.

Their calculation is not direct, involving several steps:

1. Dividing the core in small simple pieces, normally cylinders or cubes, in order to facilitate the following steps. Usually they are 5:?

• Winding column (usually the central one).
• Lateral column.
• Top/bottom plates (they are considered equal and counted as one).
• Corner connecting top/bottom plate to the winding column.
• Corner connecting top/bottom plate to the lateral column.

2. For each of these simple pieces, we calculate the length of the path of magnetic flux going through it, and the area of the section perpendicular to that path. E.g.: In the central column of an ETD, with a cylindrical shape, the length would be the cylinder height and the area that of its round base.

3. We calculate two intermediate parameters, called core constants C1 and C2, as the sum of the division of each length by its area, and each length by its area squared.

4. We calculate our effective parameters from this core constants like this:

Now we can finally use our effective parameters. But, do we know how?

Effective length

Let’s start with the simplest one, the effective length. As explained before, this is the mean length of the path traversed by the magnetic flux inside our core. In an equivalent toroid it would be a circumference going through the inside of the core (not necessarily the central circumference).

When to use it: In the calculation of the Reluctance value of our core if it were ungapped (which is physically impossible for the cores made of two pieces, as there is always a residual gap, but we will talk about this in another occasion). In non-ideal cases it will always go in series with the Reluctance of the gap(s)

When NOT to use it: In any calculation where geometrical length is expected, like in the Leakage Inductance or the Winding Losses. And no, it is not the Mean Turn Length (MLT) of the core, as I have seen some people using.

Effective area

The effective area is the area of the mean section that the magnetic flux sees as it travels through our core. It does not take into account corners and other irregularities, although in some shapes the calculation can take into account the notches on the lateral legs, like the ECs, or the central holes, like some Ps and RMs.

When to use it: As the effective length, the effective area is used in the calculation of the Reluctance of the ungapped core. Additionally, it is the relationship between the magnetic flux and the magnetic flux density, as the former does not distribute evenly along its path.

When NOT to use it: In any calculation where a geometrical area is expected. The most common misuse of this parameter is in the calculation of the Reluctance of the gap, being used instead of the geometrical cross-sectional area of the column where the gap is. Another common misconception is to use the effective area for the calculation of the saturation magnetic flux density, which is wrong as it will be explained in the section for the minimum area.

The geometrical dimensions must be used in the calculation of the gap reluctance

Effective volume

The effective volume is the product of the effective area by the effective length, and it can be interpreted as the volume “filled” with magnetic flux. It is a way for us Engineers to estimate the useful volume of a core, the volume that will generate heat.

When to use it: Probably the most common use of this parameter is the calculation of the total losses from the volumetric losses, given by material loss estimation models (Steinmetz, Roshen, iGSE)

When NOT to use it: In any calculation where a geometrical volume is expected. It must not be confused with the real volume, or with the volume of the cube containing our core, and most definitely must not be used to calculate the total weight from the ferrite density. The intuitive reason for this is that there is volume of our core where none or little of the magnetic flux circulates (e.g.: the exterior corners of an E shape), but still has material and weight.

Important geometrical parameters

As I said in the introduction, there are other shape parameters that many people tend to ignore, and that might be the difference between a mediocre and a good design. How would you choose between an E and an RM core with equivalent effective parameters? Does it matter? In my honest opinion, yes, it matters. Let me show you why.

Minimum area

The minimum area, as it name says, is the minimum cross sectional area traversed by the magnetic flux. It is used to calculate the saturation magnetic flux density, and not the effective area, as it is through this minimum area where the saturation of our core will begin, equivalent to a bottleneck.

Gap perimeter

The gap perimeter is the perimeter of the area of the core where our gap is placed. It is used in the calculation of the Reluctance of the gap if we want to take into account the effect of the fringing field, although many formulas directly use the area enclosed by this perimeter.

The reason why the perimeter must be used, is that this fringing effect is produced along the fringe of the gap, and the longer this distance, the larger the effect.

Using methods that use the area instead of the perimeter is fine as long as the shape of the gap is square or round, because the models were developed for those, but this is not true for some shapes like the EFD or the lateral legs of complex shapes, like PQ or RM.

Lateral leg area/perimeter

And now that we have mentioned the lateral legs, I would like to highlight a common misconception when gapping all three legs: Many designers assume that the Reluctance of a lateral leg is the double of the Reluctance of the central leg, which is true for simple shapes like Es or Us.

If we are working with complex shapes, like PQ, P, PM or RM, we must consider that the perimeter of these legs are much longer than the perimeter of the central round leg, and their Reluctance, with or without considering Fringing Effect, is going to be very different of the double of the central one.

This effect is especially large in the case of P and PM shapes with gaps in the three legs, leading to a worst-case error of near 50% in the inductance calculation.

Winding window area/shape

This area is the one between the winding column, the lateral column, and the top and bottom plates. It is the part of the core where we place the turns of our inductor/transformer.?

Some people might argue that this space does not have any effect on the magnetic core that is not defined by the aforementioned parameters, but I might argue back that it constrains one of the most important parameters of our magnetic component: the Current density.

For example, all the energy of our inductor, including the part that we are transforming and storing as magnetic flux, has to go through the winding window, and twice that energy (once as input and once as output) in the case of a transformer.

And because we don’t want to burn our wires, we have to control the current density through our winding window (and the heat that it produces) to be balanced with the heat dissipation of our system. This dependency contradicts the extended belief that we can have smaller magnetics at higher frequencies. But this is a topic for another time.

Shape of the winding column

This might parameter effects or design in three ways. The first one is that simple square shapes like Es and Us tend to be easier to manufacture and therefore cheaper, so if we are working on a design with a tight budget, going directly for these shapes might be a good initial filter.

The second way the shape affects is in the winding losses, as winding turns around a square column produces longer wire lengths and higher DC resistance, but also a longer distance where the skin and proximity effect will generate losses.

The last one is for leakage inductance. A longer turn length means a longer distance where the turns have to couple, so for an equivalent interleaving a square column will have more leakage inductance than a round column.

Angle covered by lateral legs

This parameter is the total angle that a given turn around the winding is covered by ferrite material due to the lateral legs used for the return of the magnetic flux. It affects our design in several ways: the magnetic field around the wires is shaped by the ferromagnetic material of the core, so the proximity losses and the leakage inductance won’t be the same in the part of the winding that is covered by ferrite material than in the part that is open.

The amount of wire exposed to the ambient also increases the thermal management, as the natural convection in this proportion of the winding will be much higher than in the space around the legs, where the heat has to be evacuated through the ferrite material. A magnetic with high core losses will benefit from a large covering angle, as the core will have more area to dissipate its heat, while in a magnetic with high winding losses will be better to have the maximum amount of wire exposed to the exterior, so the wires can evacuate their heat.

Taking this into account will also help us choose the orientation of our magnetic inside converter in order to maximize the effect when we have forced air cooling.

Economic parameters

I wanted to talk a bit about the origin of the shape names and about the non technical reasons for choosing cores, such as availability and price, but it will have to wait until the next chapter of Alf’s Musings

See you then!