Inductance Variation due to Eddy Currents

[Joe Gwinn]

I recently read (well, scanned) Bozorth's "Ferromagnetism" (Amazon.com: Ferromagnetism (9780780310322): Richard M. Bozorth: Books), the bible for ferromagnetism theory and details of all such alloys. Bozorth has a section on eddy currents, complete with partial differential equations and multiple things that vary as the square root of frequency (plus a few that vary with the square of frequency), and one of the references was particularly on point:

"Variation of the Inductance of Coils due to the Magnetic Shielding Effect of Eddy Currents in the Cores", K.L.Scott, Proceedings of the Institute of Radio Engineers (a predecessor of the IEEE), Volume 18, number 10, October 1930, pages 1750-1764. The math is laid out quite clearly. The summary is useful:

"An analysis is made of the shielding effect of eddy currents on the flux in the interior of cylindrical or flat sheet material. It is shown that the counter voltage of self inductance or an iron-cored coil is due only to the component of flux which is in phase with the flux at the surface of the core. Expressions are obtained and curves plotted showing the variations of inductance of a coil with frequency. Simple calculations and some experimental results are given. ..."

The varying magnetic field is parallel to the cylinder axis, or to the plane of the sheet.

One item would benefit from translation: "the component of flux which is in phase with the flux at the surface of the core" - The issue here is that eddy currents cause a delay, which means that eddy currents within the cylinder or sheet are not in phase with the external field. This simultaneously reduces the inductance and increases the AC resistance.

The article is half theory, and half experimental results validating the theory.

[Mike Sulzer]

I will look at that, Joe although probaby not until I post the work I have been doing over the last year and is just about done. A true frequency variable inductance (due to eddy currents) happens when there is a significant change in the current distribution within the core. However, in a situation involving mutual inductance even when there is no significant change in the current distribution, and thus no true frequency dependent inductance, the real part of the impedance increases (increasing ac resistance), and the positive imgainary part decreases (leading to the claim for decreasing inductance even n this case). However, this is just the response of a multiple component circuit which can have one value of inductance at low frequencies and another at high frequencies. For example, consider as analogy a circuit composed of C1 in parallel with the combination of C2 and R1 in series. At low frequencies where the magnitude of the capacitive reactance of both Cs is much larger than resistance, the circuit has a capacitance of the two Cs in parallel. At high frequencies, where the capacitive reactance of C1 is small compared to the resistance, the relevant value is just C1. But no one would say you have a frequency variable capacitor, just that you have a multicomponent circuit. The same should be said of the steel-cored pickup unless there really is a significant variation in current density in the core with frequency.

Is there such variation? Well, not if you believe the predictions of the simple skin effect. The current should already be out at the surface of the core even at the lowest guitar frequencies, and you just have the usual square root of frequency variation in resistance, and the circuit has no true variation of inducance with frequency. However, it appears that it is not that simple.

I just got back from a trip, and I will post as soon as I recover.

I recently read (well, scanned) Bozorth's "Ferromagnetism" (Amazon.com: Ferromagnetism (9780780310322): Richard M. Bozorth: Books), the bible for ferromagnetism theory and details of all such alloys. Bozorth has a section on eddy currents, complete with partial differential equations and multiple things that vary as the square root of frequency (plus a few that vary with the square of frequency), and one of the references was particularly on point:

"Variation of the Inductance of Coils due to the Magnetic Shielding Effect of Eddy Currents in the Cores", K.L.Scott, Proceedings of the Institute of Radio Engineers (a predecessor of the IEEE), Volume 18, number 10, October 1930, pages 1750-1764. The math is laid out quite clearly. The summary is useful:

"An analysis is made of the shielding effect of eddy currents on the flux in the interior of cylindrical or flat sheet material. It is shown that the counter voltage of self inductance or an iron-cored coil is due only to the component of flux which is in phase with the flux at the surface of the core. Expressions are obtained and curves plotted showing the variations of inductance of a coil with frequency. Simple calculations and some experimental results are given. ..."

The varying magnetic field is parallel to the cylinder axis, or to the plane of the sheet.

One item would benefit from translation: "the component of flux which is in phase with the flux at the surface of the core" - The issue here is that eddy currents cause a delay, which means that eddy currents within the cylinder or sheet are not in phase with the external field. This simultaneously reduces the inductance and increases the AC resistance.

The article is half theory, and half experimental results validating the theory.

[LtKojak]

Why is it that every time a thread like this arises, it makes my head spin and my brain hurt...?

[Tonestack]

Why is it that every time a thread like this arises, it makes my head spin and my brain hurt...?

That's how my head feels every time I try to digest Maxwell's equations. What I find to be amazing is that Oliver Heaviside took on Maxwell's work with almost no formal training in mathematics or physics.

[SpareRibs]

Why is it that every time a thread like this arises, it makes my head spin and my brain hurt...?

Hello,
Whats the matter, don't you want to know how to enhance the dog whistle frequencies in your pickups ?

[CRU JONES]

I recently read (well, scanned) Bozorth's "Ferromagnetism" (Amazon.com: Ferromagnetism (9780780310322): Richard M. Bozorth: Books), the bible for ferromagnetism theory and details of all such alloys. Bozorth has a section on eddy currents, complete with partial differential equations and multiple things that vary as the square root of frequency (plus a few that vary with the square of frequency), and one of the references was particularly on point:

"Variation of the Inductance of Coils due to the Magnetic Shielding Effect of Eddy Currents in the Cores", K.L.Scott, Proceedings of the Institute of Radio Engineers (a predecessor of the IEEE), Volume 18, number 10, October 1930, pages 1750-1764. The math is laid out quite clearly. The summary is useful:

"An analysis is made of the shielding effect of eddy currents on the flux in the interior of cylindrical or flat sheet material. It is shown that the counter voltage of self inductance or an iron-cored coil is due only to the component of flux which is in phase with the flux at the surface of the core. Expressions are obtained and curves plotted showing the variations of inductance of a coil with frequency. Simple calculations and some experimental results are given. ..."

The varying magnetic field is parallel to the cylinder axis, or to the plane of the sheet.

One item would benefit from translation: "the component of flux which is in phase with the flux at the surface of the core" - The issue here is that eddy currents cause a delay, which means that eddy currents within the cylinder or sheet are not in phase with the external field. This simultaneously reduces the inductance and increases the AC resistance.

The article is half theory, and half experimental results validating the theory.

That book is expensive

Very interesting though. Non-destructive testing uses eddy currents to imperfections in metals. The depth of the imperfection can be determined by this delay. So I have read. I actually did work NDT, but never got the chance to do Eddy testing.

Here is my thinking on it:
Eddy currents are induced perpendicular to the Alternating Magnetic Field. Perpendicular to that Eddy Current, is created a second magnetic field. It takes time and energy to build that magnetic field. This is speculation. I have pondered this before. I would think that a material that has High permeability compared to material with low permeability might shed some light to whether the "phase shift" is caused by a secondary magnetic field.

Cheers,
Ethan

[Joe Gwinn]

That book is expensive

Yes. Fortunately, I was able to borrow a copy.

Very interesting though. Non-destructive testing uses eddy currents to imperfections in metals. The depth of the imperfection can be determined by this delay. So I have read. I actually did work NDT, but never got the chance to do Eddy testing.

This sounds more like ultrasonic testing. My readings in the NDT field indicated that eddy currents were quantified as an effect on phase and amplitude, if memory serves.

Here is my thinking on it:
Eddy currents are induced perpendicular to the Alternating Magnetic Field. Perpendicular to that Eddy Current, is created a second magnetic field. It takes time and energy to build that magnetic field. This is speculation. I have pondered this before. I would think that a material that has High permeability compared to material with low permeability might shed some light to whether the "phase shift" is caused by a secondary magnetic field.

Almost. The eddy current loops around the changing magnetic field, and is in a plane perpendicular to the field. This is Faraday's Law of Induction.

The induced eddy currents generate a magnetic field that opposes the original change. This last part is Lenz's Law.

The extreme case is a superconductor, where the eddy current is sufficient to totally prevent entry of the magnetic field. This is called the Meissner Effect, and is a hallmark of superconductvity.

The phase shift is caused by the opposition by eddy currents slowing down the penetration of magnetic flux into the interior of the conductive cylinder or sheet.

[David King]

How does this relate to the EM force field around a power transformer?

[salvarsan]

There are less expensive sources of the book, used, like Alibris and AbeBooks.
Lowest price I saw was $35.

Alibris: Ferromagnetism - Richard M. Bozorth

The original publication was in 1951 and was never revised so it doesn't matter which one you get.
In other words, you may not see a discussion of any Alnico alloys later than isotropic A5,
so you're getting this book for its theoretical content.

[Steve Conner]

How does this relate to the EM force field around a power transformer?

To a first approximation there is none! The magnetic core acts as a waveguide, confining the power flow to the space immediately around and between the windings. Hard to explain in any more detail without handwaving and babbling about Poynting vectors.

To a second approximation, the EM field is mostly leakage flux: that is, flux that only cuts one winding, and never passes through the core.

To a third approximation, part of the magnetizing flux flows through the air around the transformer, because the permeability of the core isn't infinite.

The two last items are responsible for the buzzing in guitar pickups when they get near a transformer. They're both rich in harmonics of the AC line frequency. The leakage flux has harmonics because when a transformer drives a rectifier load, as it does in a guitar amp, the current in the windings is far from sinusoidal. The airborne component of the magnetizing flux has harmonics because the core permeability is non-linear.