Summary: Part 3 described a method of modeling a core. In another article, hopefully to be posted before this one, is a means of creating a simple, lossless, nonlinear core model. This article is intended to discuss some core considerations and misconceptions regarding cores. It does not pretend to be exhaustive or even rigorous, but nonetheless to illustrate some important points in a discussion format. I am, of course, playing the 'expert' while my friend, the questioner, is speaking in italics.

Air gaps:

So as I was saying before coffee break, one can learn a lot from a B-H curve of an inductor or from a core model, or its material. Here is one.

Figure 1
Tape wound core material typical hysteresis loop

To a greater or lesser extent, all useful core materials display a curve somewhat similar to this (Figure 1). The curves often have less loss and are narrower. Often they saturate less abruptly, and are more 'S' shaped. This curve is for a tape wound core and is effectively 'gapless', as the core material is a continuous strip of a specially prepared magnetic material.

When this core, or any core material is driven deep into the saturation region, E.G., the 'flat' upper and lower portions of the curve, the incremental inductance approaches that of an air core inductor. Essentially all of the magnetic domains present in the material have been aligned with the magnetic field.

About the origin, in the somewhat steep and linear regions of the curve, there are large numbers of unaligned magnetic domains at any point, and an increase in H will cause a correspondingly linear increase in the number of aligned magnetic domains.

Between these two regions is a transition region. In this portion of the curve, which can be relatively large dependent on the material, the number of unaligned magnetic regions is becoming smaller, and the B-H curve flattens. This region, dependent on the core material, can be rather small or even quite large.

Let me sketch a lossless B-H curve. ..

Figure 2
Typcial Lossless Core

This drawing (in Figure 2) shows a curve of a hypothetical lossless, saturating core. Such a B-H loop approximates the midpoints of a curve for a material such as a ferrite, which can have a 'skinny' B-H loop, representing little core loss. The 'flat' top portion can represent an extended transition region, or an air-core, fully saturated region of operation, again dependent on the material. Some powdered toroid cores already have an inherent air gap distributed throughout the core, in between the many grains of material that constitute the core material. Generally such powdered cores cannot be easily modified, nor is there often a need to do so.

We are often interested in curves of un-gapped material B-H curves. Un-gapped material B-H curves are often provided for materials and core construction types where the air gap can be customer specified to the manufacturer during the ordering process, or the customer can if desired create their own air gap.

The specific cores can be of many different configurations. They could be E-I cores, where the windings are placed on bobbins, placed over the center post of the E shaped block of material, and then the magnetic core 'I' block affixed to the 'E' block which has the bobbin placed on it. They could be pot cores, in which the 'E' cross section can be thought of as being rotated through 180 degrees to form a solid 'cup' with a center post. The mating 'I' section can itself be thought of as a rotated block, or another 'E' section can be mated with the first section. One or two 'slots' are provided to bring winding ends out of the enclosure. The particular configuration is of no great concern for us at this time other than to observe that, because of mating surface irregularities, some inherent air-gap will always be present.

In the B-H curve, B is proportional to voltage, and H is proportional to current. So, to get a B-H curve of a core model, all one has to do is to plot the current through the device on the x-axis, and the voltage across the device on the y - axis, right?
Well, you are close. B is proportional to the integral of voltage across an unloaded inductance model, and H is proportional to the current through the device.

e = n * (dphi/dt)

Come to think of it, B = (µ0 * µr) * H, and H is proportional to current, so why is not B proportional to current? Well, it is a function of current, so in a linear core or in that region where the core is linear, then it is related by a constant. But of course this is a straight line and is not very interesting. In the region where the core material is nonlinear, then the core effective permeability is a function of current but not a linear one. So yes, B is in general a nonlinear function of current, but at least for me and most persons it is clearer to describe it as being proportional to the integral of the voltage across the device.

So for a transformer model, represented by a nonlinear core in parallel with the primary winding of a transformer (which may be constructed with several ideal transformers), all you have to do is to measure the integral of the voltage across the primary, and plot this against the current through the primary to get a B-H shaped curve?
That depends on what you mean by primary current. If you mean the current through the effective primary winding, which means the core model and the perfect transformer primary winding, the transformer must be unloaded, as the secondary(s) load(s) are reflected back to the primary, and appear in parallel with the core currents. Otherwise you must measure the current directly through the core model to eliminate the reflected load currents. The curve will then have the shape of a B-H curve, but to be exact some constants must modify the voltage to produce B and the current to produce H values. Bear in mind that manufacturer's B-H curves are usually provided on the basis of one turn.

Changing the subject. air gaps are of course bad ….. or are they? Why would we want to buy a high permeability core, and then add an air gap?
In general, air gaps can be very useful and desirable. Look at the B-H loop I sketched before (Figure 2). Curve 1 represents a hypothetical B-H curve for an ungapped ferrite core. Its relative permeability in the unsaturated region is rather large. Curve 2 represents the same core material, but with an air gap. We know that

B = (µ0 * µr) * H

Looking at the two curves, the maximum flux density value at the onset of saturation is unchanged. However, for the gapped core in curve 2, at what was the original saturation magnetic field intensity value of H for curve 1, the corresponding flux density is noticeably lower. The gapped core is somewhat removed from saturation for this same value of H. In fact, the new value of magnetic field intensity H required to saturate the magnetic core material of curve 2 has greatly increased.

The corresponding effective permeability of the device in curve 2 has decreased. At the same time, its inductance in the linear region has also decreased. What has happened?
Without going into too many details, the reluctance, or resistance to the magnetomotive force or 'H' has increased. Using the reluctance model, the magnetic path the flux flows through presents a 'resistance' that is inversely proportional to the permeability. However, in the air gap the flux encounters in a serial manner a very low permeability air gap, which has a high 'resistance'. And just as resistances in a series DC circuit, the reluctances in a serial magnetic path add. The air gap has a much higher reluctance than that of the magnetic material, so the reluctance of the air gap predominates. Flux is proportional to the magnetomotive force divided by the reluctance. The flux density decreases for the same stimulus, and does the effective permeability. Inductance is a function of permeability so it decreases also.

This is bad news, right?
Not really. For one thing, the allowable current which the inductor will pass without saturating the device is greatly increased. But its inductance has at the same time decreased, L being a function of permeability.

So there is no net gain, right?
Wrong. There is often a very decided advantage. And that is in the energy storage capability of the inductor. The energy stored in an inductor is ½ LI2. So while the inductance has decreased, the maximum current it can pass without saturating the core has increased, and the energy storage capacity has increased by the square of the current. Also, the number of turns required to gain a given inductance has increased. With a small number of turns getting the desired inductance, or transformation ratio for a transformer may be difficult.

Now the energy stored in the gapped inductor is really stored in the air gap, but this is of little concern. Energy storage is of particular importance in flyback transformer operation.

Well wait a doggone moment!!! If energy is stored in an air gap, why not just eliminate the core altogether and be done with it?
Good question. Here is what is hoped is a good answer. First, the magnetic material guides the path of the magnetic flux, and prevents this flux largely from coupling with other conductors and inducing currents in those devices. But second and more importantly, consider a fairly typical inductance of perhaps 100, or even 10 mH. How many turns of wire do you suppose are required to get an air core inductance of this value? Now how much wire resistance would there be? The answers are lots, and lots.

Okay, but I am not building a flyback transformer, one where energy is stored in the inductor during a 'charge' cycle, but I am using one where energy is being transferred by means of transformer action. In this case, where the inductor (core) is effectively in parallel with the primary winding of an ideal transformer, the transformed load(s) is(are) effectively in parallel with this magnetizing inductance (of the primary winding). So why use a gapped core here?
Another good question. This is more complex. The magnetizing inductance does indeed appear in parallel with the transformed load impedance(s). In general one would desire the largest primary or magnetizing inductance possible. If one were driving the transformer with an AC waveform, it would be essentially as you say.

I am driving a transformer primary with a unipolar drive waveform. One end of the primary is connected to +30V. The other end is be periodically switched to ground, and then opened . The switch has a 50% duty cycle. That would present an average 15V DC to the primary winding!!!! Why does the transformer not saturate with a net DC voltage across the primary?
Glad you mentioned that. You see, the transformer primary does NOT see a net DC applied to it. It would seem so, but that is not the case. When the switch opens, the voltage across the primary will reverse. Remember the switch opens and closes, and while it is open or off, its impedance is very large.

In this case the transformer will, if all goes well, operate on a minor loop. Using the hypothetical device of Figure 2, one can envision a 'DC' operating point perhaps half way between the origin and the core saturation point. When the switch is 'ON', the primary voltage is 30V and the operating point 'walks' up the curve to the saturation point. When the switch is off, the load voltage reverses, and hopefully the load removes the energy stored in the core and returns the operating point to the origin in this example, or in general to as far below the operating point as the on switch cycle moved it above the operating point. Typically one would wish to stay well away from the saturation point, however. What a gapped core does in this case, is both confine the magnetic field AND provide a wider range of allowable operating points for variable loads, input voltage variations and other circuit parameters which change over time, temperature and so on.

Actually, this situation is not that much different than that of a filter inductor at the output of a switching regulator, that is continuously conducting. The inductor is charged during a switch on time, and the current increases. The switch is opened, and the inductor current still supplies the load through another switch of diode, but the voltage across the inductor reverses. The inductor current decays. The inductor does pass a net DC current, but it does not see a net DC voltage across itself.

In the transformer situation, the primary voltage will indeed reverse, just as in the case of the inductor. But in this case one has to rely on the secondary loading to 'reset' the magnetics. (Now of course the leakage inductance present at the primary will also store energy, and it in general is desirable to provide a 'snubber' circuit to remove this energy and prevent dangerous spikes from occurring when the switching device turns off. There are ways to recover much of the stored energy, but this is too complex to go into during this break.)

Well, gapped core or not, it is always best to have some core then, to confine the magnetic field and minimize the leakage inductance, correct?
Well, you are right on the first part. Amazingly and counter-intuitively, the leakage inductance is pretty much independent of the core material, and hence it is essentially a constant for a given inductor/transformer topology and winding physical arrangement. This has been shown several times in the literature. Magnetics Incorporated in one of their application notes provides an emperical formula for determining leakage inductance for specific toroid cores which is independent of the core material, being a function of how the winding is distributed and its dimensions. It is not very exact, as varying winding methods can cause it to vary by 50% or more, but the important point is that it is independent of the core material. Hence it is unchanged whether the core is saturated or not.

There is something I am confused about. You mentioned a 'minor loop'. But the B-H curve shows no such thing. How can this occur?
A typical B-H curve is of course the result of specific currents or voltages applied to a core which has no DC component. The curve is the response of the 'inductor' in response to those specific stimuli, in accordance to the defining equations or characteristics of the core material. When DC is present, the trajectory, or the path in which the B-H values of the device will trace in accordance with the AC variations, will be about a DC operating point, in accordance with the defining equations of the device. As a first approximation it will be an 'S' shaped curve about the DC value of H. but it can vary quite a bit in shape dependent on where the DC component places the operating point.

So a air-gap is useful then?
It can be. The shape of the air-gap can be important as well. Some devices are built with a 'stepped air-gap'. This causes the magnetic field in the vicinity of the gap to concentrate near the narrower portion of the gap (which has a lower reluctance). This portion of the device will saturate before that near the longer gap. The device will exhibit an initial higher inductance until a portion of the material saturates, when the device will exhibit a 'step' to a lower value. Thus for low values of DC excitation the device will exhibit an inductance higher than for larger values of DC, creating what is called a 'swinging choke', useful in some filter applications. One can use a 'wedge' shaped air gap to create a more smoothly varying inductance with DC excitation. But of course these changes will also occur with instantaneous excitation as well.

Why is there a gap or 'hole' in the middle of a typical B-H curve? What causes this?
The specific cause of the 'middle' portion of the curve can be complex. It represents a loss component, however. Most magnetic materials will retain some magnetization after being 'magnetized' and the excitation is removed. There will be some residual flux present. This must be overcome to return to a benign, zero flux state.

So a 'fat' curve is bad?
Often this is the case, however to get a very sharply saturating transition, one has to either have a core which has a large remnant magnetization, resulting in a 'square' loop, or one has to operate the device over a very large range of H to see this behavior. Abrupt changes are often useful in certain applications.

Lemme ask you something. I have had problems in some of my transformers matching the core losses to the winding losses. The core loss varies with frequency, while the winding losses vary dependent on wire diameter and also frequency, due to skin effects. What is the best way to do this?
A design rule often stated is that the copper losses should equal the core losses. This rule was derived for low frequency AC power transformers. It has NO bearing on the design of modern electronic transformers.

Why is this?
For one thing, anything that would add heat to a transformer is to be avoided. Thus one tends to use the maximum (manufacturable) winding wire sizes and types one can to minimize heating. One usually minimizes core losses for the same reason. .

Okay then, I guess I can buy that. But one thing has always bothered me. For maximum power transfer, the load impedance should be equal to the source impedance. How do you match the output loading on a transformer to the winding resistance, leakage inductance and transformed source impedances to the secondary side to get the maximum power transfer?
Well, the problem is you are incorrectly stating the requirement. For a given source with a given source impedance, for maximum power transfer the load impedance should equal the source impedance. But with a given load impedance, the maximum power transfer occurs when the source impedance is zero. All of the power is present at the load under those conditions. We generally wish to have the lowest possible source impedance, which means holding the winding resistances, leakage inductances and all other impedances which could appear in series with the load to as small values as is possible.

To change the subject, I have several coupled inductor transformer models. How can I add leakage inductance to them?
I hesitate to tell you, but leakage inductance is already present in those models IF the coefficient of coupling is less than unity. One can transform this model into a 'T' model and see this. However this model assumes that the equivalent secondary leakage inductance, transformed to the primary, is equal to the primary leakage inductance, which often is not the case. It may or may not be a good approximation for any specific transformer, depending on how the windings are constructed. As one example, a 1:1 transformer, with the primary winding and secondary windings wound 'two in hand' would have very similar leakage inductances. Other configurations in general would not.

I guess you are telling me that all of many of the things I learned in school about magnetics were wrong?
No, I am saying some of the things you thought you learned in school are wrong.

I have a few questions though, that I would like to ask Namely, ….
Forgive me for interrupting, but the coffee break has been over for a few moments. I need to get back to work, and I am sure you do too. The boss is sorta grumpy over long breaks, and I am getting paid by the hour, working on a contract basis.

I will mention to the boss we were working during the break if he grumbles. Can we do this again sometime?
Well perhaps. Let me write a little blurb about what we discussed and perhaps some other topics in my off time and send it to you. Perhaps some others might wish to read it, so if you wish you can share it with them.

Thanks. I need to go to the floor so I will see you later at the cubicle. See you later.

Conclusions:

The intent was not to belittle schooling, but often what is taught, while correct, is easily subject to misinterpretation. Every topic herein has been come up more than a few times for me. I hope that a few useful ideas have been presented and possibly a few erroneous ones corrected.

Magnetics, electromagnetism, is not something most engineers work with regularly, and often appears to be a black art. Yet consider, what other device type is there which a rather typical engineer is so deeply involved in the details of creating? Surely not semiconductor devices nor IC's. Most devices are selected and the engineer is not involved with their design and manufacture.

When I went to school transformer courses were something to be endured. This of course was during the dark ages. But if you stand out in a hot sun long enough without a hat it will start to make some sense.

General references:

1. Article Title: "SPICE Models For Power Electronics"
Author: L.G. Meares and Charles E. Hymowitz
http://www.eettaiwan.com/ARTICLES/2002APR/PDF/2002APR22_POW_EDA_DA_AN215.PDF

2. Intusoft Power Specialist's App Notebook. http://www.intusoft.com/psbook.htm

3. TWC-S2, How to Select the Proper Core for Saturating Transformers, Magnetics Inc.
http://www.mag-inc.com/pdf/twc-s3.pdf

4. TWC-600, Tape Wound Cores Design Manual.
Magnetics Incorporated
http://www.mag-inc.com/pdf/TWC-600.pdf

5. Non-linear Saturable Kool Mu Core Model
Scott Frankel, Analytic Engineering
http://www.aeng.com/pdf/core2.pdf

6. PCB Café book exerpt
SMPS Simulation with SPICE3
Stephen M. Sandler
http://www.pcbcafe.com/BOOKS/SMPS/222coresb.php