Magnetic Permeability

[ramonest]

I am trying to understand the concept of magnetic permeability. Right now I'm stuck with understanding if the proper definition is:

mu = B/H
(With mu-> magnetic permeability, B->Magnetic flux density and H->Magnetic field strength)

or:

mu = dB/dH

For the simplification that magnetic permeability is constant and the BH curve crosses the (0,0) both definitions are equal (in a sense).

But if you take into account a real BH curve with the saturation it is no longer the same. If you want to calculate the inductance of an inductor working with a DC bias and some AC ripple it will have different value depending on the definition used, specially if the DC bias moves the operating point close to the "elbow" of the BH curve.

I'm using the basic formula for inductance:
L = ((N^2)*Ac*mu)/(lc)

Where N->Number of turns, Ac->Effective area of the core and lc-> effective length of the core.

No air-gap is used (which is why I was interested in the saturation or close to it case).

I would really appreciate it if someone can clarify that! Or point to some sources that properly define it.
Thanks in advance!

(I'm not sure if this should go to beginners section or is more suited for another section)

[cur8xgo]

I am trying to understand the concept of magnetic permeability. Right now I'm stuck with understanding if the proper definition is:

mu = B/H
(With mu-> magnetic permeability, B->Magnetic flux density and H->Magnetic field strength)

or:

mu = dB/dH

For the simplification that magnetic permeability is constant and the BH curve crosses the (0,0) both definitions are equal (in a sense).

But if you take into account a real BH curve with the saturation it is no longer the same. If you want to calculate the inductance of an inductor working with a DC bias and some AC ripple it will have different value depending on the definition used, specially if the DC bias moves the operating point close to the "elbow" of the BH curve.

I'm using the basic formula for inductance:
L = ((N^2)*Ac*mu)/(lc)

Where N->Number of turns, Ac->Effective area of the core and lc-> effective length of the core.

No air-gap is used (which is why I was interested in the saturation or close to it case).

I would really appreciate it if someone can clarify that! Or point to some sources that properly define it.
Thanks in advance!

(I'm not sure if this should go to beginners section or is more suited for another section)

Clarify what? You answered your own question. Permeability changes with flux in some materials. Not so much in air/vacuum/non magnetics.

[Achu]

Actually in practice the permeability is a function of H.And variation is non linear which explains the saturation of the B/H curve.
Also in the case of inductors when designing the I believe one has to consider the current being carried and the core properties.(Pardon me if I am wrong)

[Berni]

Yep you found out on your own why inductors you can buy at DigiKey or Mouser or Farnell have a rated saturation current.

Once you put enough current trough it to get you up to the top of that chart the inductance starts to rapidly change. Obviously you are not supposed to operate an inductor in that area.(Unless you need the effect of saturation as part of your circuits functionality)

[schmitt trigger]

Welcome to the exciting world of hysteresis curves and flux saturation.

Magnetic materials have permeabilities hundreds or even thousand larger than air. Which means that for a given geometry and number of turns you will have inductance values which would be impossible to achieve with an air core inductor.

But on this life nothing is free.  You have to pay a couple of penalties when using magnetic materials.

The first is that at H=0, the initial permeability is very high, but it will gradually decrease as the flux increases, until you reach saturation and the permeability drops to very low values.
The second is that those materials have remanence. In plain words it means that the magnetic flux follows two distinct paths, the hysteresis curve. And with AC-excitation, where the curve is transversed back and forth with each cycle, the area inside the curve represents wasted energy, the core loss.

[ejeffrey]

The simple definition of permeability basically assumes linear behavior which is wrong for ferromagnetic materials.

It is often easier to think about this in terms of a combination of the B = mu_0 ( H + M ) where H is the applied external field, and M is the magnetization of the material.  This is completely general -- you are just saying that the magnetic field is the sum of two terms which are defined to contain everything but it is also kind of useless.  So you want to find a way to represent M for interesting materials. For linear materials, M = chi * H.  chi can also be a function of frequency have an imaginary component.  For non-linear materials, M can be a polynomial function of H.  For hysteretic materials (like ferromagnets), M is not a function only of H, but also on the internal state (i.e., history) of the material.

You can still talk about the permeability in differential terms, that is the change in B for a change in H where it is approximately linear, but you have to recognize that is what you are looking at, not a simple globally linear relationship.

[T3sl4co1l]

Simple: which one do you want?

If you take the ratio of instantaneous B and H, you have the average permeability.

If you take the ratio of a small difference in B and in H, you have the incremental permeability (or "marginal" permeability, as accountants would call it; or in calculus, for difference-->0, the slope or derivative).

You would use the incremental version for small-signal purposes (like signal filters), and the average version for large-signal purposes (like power transformers).

Tim

[Achu]

Permeability is a ratio of magnetic flux inside of the material to the flux outside the material.
So the problem of magnetic saturation can thought like pushing a crowd into a small hall.
As more people come inside the people already inside become unhappy about it and would be reluctant to allow more to come in.As the no of people cross a limit, it becomes impossible to push any more people(or flux) inside.!!

Not my best analogy.
Correct me if I'm wrong

[bob91343]

The problem here is trying to devise a linear parameter to describe a nonlinear component.  After all, the distance you drive depends on the route and the distance measured with calipers on a map won't agree.

So the question is, do you want to know the slope of the B/H curve at a given point and then call it inductance?  Change anything, and you'll get a different number.  Why do you want the number anyway?  These components are just that, components, and not standards or anything else absolute.  The bottom line has to do with the details of the application, and how it works depends on how you use it.  That's why anything accurate requires trimming of parts, especially, say, the coils in an old fashioned organ or in an analog signal generator.

I can measure inductance accurately with a tiny signal and no bias current.  I don't know if that value offers help in circuit design.  Of course, an air core inductor hasn't got these problems.  So for very low power applications, we will stay around the origin of the B/H curve.  If there isn't enough flux to cause the core to function, you get the same thing as without the core.

I remember asking a college instructor why the curve of inductance has that first tiny bend in it, very close to the origin.  I never could get an answer but now I know.  What this has taught me is that, just because someone teaches a subject or writes a book on it doesn't mean he understands the subject.

[cur8xgo]

Whats the resistance of a thermistor
Whats the voltage across a capacitor
Whats the flux through a coil

[ivaylo]

The best 50 min you can spend on the subject - https://youtu.be/4UFKl9fULkA

[T3sl4co1l]

I remember asking a college instructor why the curve of inductance has that first tiny bend in it, very close to the origin.  I never could get an answer but now I know.  What this has taught me is that, just because someone teaches a subject or writes a book on it doesn't mean he understands the subject.

Although this isn't their fault; condensed-matter physics is notoriously hard (in fact, provably without general solution*), and ferromagnets are very hard indeed to solve.  Like superconductivity, this is one of many outstanding problems in physics to this day.

*This goes something like, we can make a Turing machine with condensed matter, therefore at some level, condensed matter necessarily leads to the Halting Problem, and is therefore provably unprovable.

Tim

[pwlps]

*This goes something like, we can make a Turing machine with condensed matter, therefore at some level, condensed matter necessarily leads to the Halting Problem, and is therefore provably unprovable.

Wow, this is an interesting philosphical point of view  . But I think it doesn't makes sense.  First you can't tell the quantum physics of condensed matter is provable or unprovable because it is not a purely logical construction.  Second the physics of the Turing machine is not at the same level as it's logic: the halting problem is just a version of the Godel theorem applied to the logic system of programming but it can't tell anything about the theory of the logic gates that are used in the machine: the theory behind the gates does not belong to the logic system that exhibits the halting problem.
Sorry for this off-topic post, it was just to challenge Tim's philophical thoughts 

[bob91343]

When someone does scientific work, it's subject to peer review.  He publishes it and others review, comment, and challenge in an attempt to arrive at the truth of the matter.

When someone is hired as an instructor, or writes a book on a scientific subject, that doesn't happen.  The qualifications are noted (sometimes all too casually) and a decision made based on the usual criterion, money.

I see technical articles all the time with gross errors.  Publishers save money on proofreading.  Some articles are so bad that they are nearly incomprehensible.  Is this a relatively new syndrome, or has it always been thus?

Sorry if this is off topic but I had to vent.

[ramonest]

Thank you all for answering.

[ramonest]

Simple: which one do you want?

If you take the ratio of instantaneous B and H, you have the average permeability.

If you take the ratio of a small difference in B and in H, you have the incremental permeability (or "marginal" permeability, as accountants would call it; or in calculus, for difference-->0, the slope or derivative).

You would use the incremental version for small-signal purposes (like signal filters), and the average version for large-signal purposes (like power transformers).

Tim

Thanks for your answer Tim.
Consider the case where the inductor has a constant field (DC component) that brings the magnetic material to a nonlinear point of the BH curve (near the "elbow" of the curve) plus a small AC component (ripple current). That could be the case of a power inductor on a switching converter in CCM with low ripple. In such case the most accurate way to predict the inductance on that operating point would be to sue the mu (permeability) corresponding to dB/dH near the operating point set by the DC component?

[schmitt trigger]

That is correct.
Many inductor manufacturers will actually provide you with a graph plotting the with inductance drop vs. DC bias.

For instance, check the graph on page 2 of the attached document

[ramonest]

The best 50 min you can spend on the subject - https://youtu.be/4UFKl9fULkA

I saw the video a while a ago. Thanks for bringing it tho, it would be interesting to replicate the experiment he does at the end of the video! I'll try to do so.

[ramonest]

That is correct.
Many inductor manufacturers will actually provide you with a graph plotting the with inductance drop vs. DC bias.

The problem I had is that regardless of the definition (mu=B/H vs mu=dB/dH) inductance drops.
I attached an example with  calculations made using the BH curve of the original post, it's vary rough specially for the dB/dH case (it's done fast with excel) but still you can see the idea. Both drop, mu=dB/dH just faster than the other case.
It's true that I haven't checked any manufacturer and their L vs I curves tho... I should have done that and maybe I would be able to see the answer there too.

[TimFox]

Looking at the “magnetization curve” (ignoring hysteresis) for a ferromagnetic material, the initial permeability is lower than the maximum permeability.  Usually, the material specification gives the initial mu (at B = 0), the maximum mu and the B where that value obtains, and the B value at saturation.