Under what conditions does the inductance vary with frequency?

[Steve Conner]

So something like this:

Transformer - Wikipedia, the free encyclopedia

with the secondary terminals shorted?

[Mike Sulzer]

So something like this:

Transformer - Wikipedia, the free encyclopedia

with the secondary terminals shorted?

Yes, although I think that model might be a bit more complicated than necessary.

In the discussion "A new model for...", the approach was a bit different. A simpler model (Hagen, RF Electronics) consisting of an ideal transformer with a leakage inductor in series with the secondary (no core loss) was used to derive the core loss model. The currents circulating around the cores were represented as a shorted secondary with some resistance. This was shown to provide the necessary loss in the low KHz range while partially unloading the coil at the higher frequencies so that the resonance occurs. This was not sufficient; it was necessary to make the "secondary" resistance frequency variable (from the skin effect) in order to get a reasonable match with the measurements.

[RedHouse]

Maybe we need a group-buy ....to get them a room!

Appears it might need to be a ménage à trois!

[Joe Gwinn]

Is it available on line? I have not found it, and finding a paper copy in Puerto Rico seems unlikely.

I assumed that the observatory library is a university library and so could get anything. I don't think that it's online anywhere.

Anyway, here it is: Well, no it isn't. It exceeds the Forum's ~2 MByte file size limit by at least a megabyte, being a scan. So I need a direct email address. I will be offline from ~10 AM tomorrow (Friday) until Sunday the 30th, but I will send copies to whoever gets an email to me in time before I drop off the screen.

But in any case is it really necessary? Is a resistor in series with the leakage inductance really a poor model?

Yes and no. Did you mean in parallel?

Since you have a Wien bridge, you could make measurements at a series of frequencies on a humbucker coil (cores in and out) and see if they disagree with the ones I presented above.

I built the bridge from small parts and made such measurements many years ago, when I was qualifying the Extech LCR meter. The bridge and Extech agreed to a fraction of one percent. I can dig up the data in a week.

The trick with the bridge is to use ordinary 5% and 10% components but measure the resistance and capacitance values (which will be fairly pure) to within 1% using a digital multimeter, and use a calculator to solve for inductance and series resistance.

The only other thing that must be good is the oscillator - it must have low harmonic distortion. If it does not, the harmonics will not be nulled, and will obscure the null.

[LtKojak]

Appears it might need to be a ménage à trois!

Yeah, the more, the merrier.

[Mike Sulzer]

I built the bridge from small parts and made such measurements many years ago, when I was qualifying the Extech LCR meter. The bridge and Extech agreed to a fraction of one percent. I can dig up the data in a week.

I am not disputing the accuracy of the Wien bridge and the Extech as regards to their ability to measure complex impedance (and in the case of the Extech, to interpret the measurement in terms of two simple models). My concern is with the interpretation of the measurements of complex impedance when the model is not so simple.

[Mike Sulzer]

I have read the Legg paper, which you kindly supplied.

The Legg paper represents the various losses in an inductor as a series resistor that is the sum of a resistor accounting for the the copper loss in the coil and and one accounting for the losses in the the core (equation 7). Let us consider eddy current loss. This resistance is a function of frequency squared (equations 11 through 19).

The frequency variation of this resistor suggests that it is an equivalent resistance, and that the physical process is more simply represented by a resistance in parallel. The question that arises then is whether or not this use of equivalence resistance affects the value of the inductance that is interpreted from the Wien bridge measurements.

Consider the equivalence relationship between an inductor with a series resistor (Rs) and a parallel resistance (Rp). That is, suppose we have a parallel resistor but we interpret the measurements of complex impedance as if it is in series (so that we can, for example, subtract off the copper loss resistor). How must we transform the values?

The impedance of the series combination is 2*pi*f*Ls*j + Rs. The parallel network consisting of Rp and Lp can be put in the same form (imaginary part plus real part) with some algebra. We equate the two expressions to look for the transformation relationships.

This is not simple. But if we make the assumption Rp is much larger than 2*pi*f*Lp* over the frequency range of interest (high Q), then we see that Lp = Ls and Rp = ((2*pi*f*Ls)^2)/Rs. The latter shows that the parallel resistance has a simple frequency variation, given the squared dependence for Rs from the paper, as the physics suggests. The former says that the error in inductance goes to zero at very high Q because Ls and Lp are the same. (Measuring one is the same as measuring the other.)

When the Q is not high, the assumption that Ls = Lp fails, and there is error in the inductance measurement. (Rs and Rp are functions of both Rp and Lp.) Thus there is an assumption in Legg's work that the Q of the inductance is high. (I do not see where he has stated this, but I might have missed it.)

The same thing applies to measurements with the Extech. If the Q is high, then even if we have both a series and a parallel resistor, the measurement of the inductance is accurate if we assume just a series resistance.

But if the Q is low (humbucker pickup), the conditions are violated; the additional terms that were ignored for high Q are large in this case and there is error in the inductance measurement. Note that this is true even if the Extech measures the complex impedance with perfect accuracy. The problem is in the interpretation, not in the measurement itself.

[Joe Gwinn]

I have read the Legg paper, which you kindly supplied.

Quite welcome. Made me reread it, too. I think I'll put a copy up on my website too.

The Legg paper represents the various losses in an inductor as a series resistor that is the sum of a resistor accounting for the the copper loss in the coil and and one accounting for the losses in the the core (equation 7). Let us consider eddy current loss. This resistance is a function of frequency squared (equations 11 through 19).

The frequency variation of this resistor suggests that it is an equivalent resistance, and that the physical process is more simply represented by a resistance in parallel. The question that arises then is whether or not this use of equivalence resistance affects the value of the inductance that is interpreted from the Wien bridge measurements.

While it is clear that this frequency-dependent resistor is representing some deeper physical phenomena, largely eddy currents in the present case, how does it follow that the effect is better represented by a shunt resistor? While this has been your assertion all along, Legg says no such thing. Legg (on page 51) uses shunt capacitance and conductance only to model coil self-capacitance and DC leakage, and uses the series-resistor model for everything else.

Consider the equivalence relationship between an inductor with a series resistor (Rs) and a parallel resistance (Rp). That is, suppose we have a parallel resistor but we interpret the measurements of complex impedance as if it is in series (so that we can, for example, subtract off the copper loss resistor). How must we transform the values?

The impedance of the series combination is 2*pi*f*Ls*j + Rs. The parallel network consisting of Rp and Lp can be put in the same form (imaginary part plus real part) with some algebra. We equate the two expressions to look for the transformation relationships.

As discussed above, Legg does no such thing.

This is not simple. But if we make the assumption Rp is much larger than 2*pi*f*Lp* over the frequency range of interest (high Q), then we see that Lp = Ls and Rp = ((2*pi*f*Ls)^2)/Rs. The latter shows that the parallel resistance has a simple frequency variation, given the squared dependence for Rs from the paper, as the physics suggests. The former says that the error in inductance goes to zero at very high Q because Ls and Lp are the same. (Measuring one is the same as measuring the other.)

This assumes the shunt resistor model, which Legg does not use to model eddy current effects.

However, returning to the original question, Legg says in multiple places that eddy current shielding causes the observed inductance to decrease with increasing frequency. A clear statement appears above equation 30 on page 45:

"Also, eddy currents set up magnetizing forces within the magnetic material which more or less neutralize that applied by the coil winding, and thus effectively shield the inner parts of magnetic laminations of wires. Such eddy current shielding reduces the total flux in the core, thus decreasing the inductance and loss resistance observed at higher frequencies."

Equation 30 is a partial differential equation (PDE). As discussed in previous postings, PDEs are required to capture phenomena such as eddy currents, although for equation 30 the object is magnetic hysteresis. Probably Steinmetz (footnote 4 on page 41) is where this is done for eddy currents.

A supporting statement appears on page 46, near the bottom:

"The apparent permeability (mu sub fm), which is calculated from the measured inductance, decreases as the measuring frequency is increased."

And, at the top of page 52:

"Thus the observed inductance tends to increase at higher frequencies on account of distributed capacitance, in contrast to its tendency to decrease on account of magnetic shielding in the core according to equation (33)."

Equation 33 is one of the results of the PDE equation 30.

When the Q is not high, the assumption that Ls = Lp fails, and there is error in the inductance measurement. (Rs and Rp are functions of both Rp and Lp.) Thus there is an assumption in Legg's work that the Q of the inductance is high. (I do not see where he has stated this, but I might have missed it.)

Legg does not mention any limitation to high Q, and in fact his equations give the full story, including magnetic hysteresis and viscosity effects in addition to eddy current effects.

The closest approach to a Q limit in Legg is on page 55, where it is observed that for accurate measurement of magnetic material properties, it's helpful if the Q is high, and the two test coil examples Legg includes have inductances in the low millihenries and Q=20.

Note that while magnetic hysteresis and viscosity effects may possibly have some effect on tone in guitar pickups, they have limited effect on inductance and AC resistance. Hysteresis and viscosity are bulk properties of the magnetic material used for slugs, screws, magnets, and perhaps baseplates.

The same thing applies to measurements with the Extech. If the Q is high, then even if we have both a series and a parallel resistor, the measurement of the inductance is accurate if we assume just a series resistance.

But if the Q is low (humbucker pickup), the conditions are violated; the additional terms that were ignored for high Q are large in this case and there is error in the inductance measurement. Note that this is true even if the Extech measures the complex impedance with perfect accuracy. The problem is in the interpretation, not in the measurement itself.

Even if one accepts the necessity of a shunt-resistor model, all this depends on the definition of low and high Q.

The Extech users manual says that full accuracy (~1%) is achieved if the Q is not less than two, and my experience is that the Extech will work accurately well below that. One can test this by putting a potentiometer in series with an inductor and observing the effect on measured inductance as the series resistance is increased.

With the Maxwell-Wien Bridge, one can work down to very low Q values, but bridge balancing using a digital voltmeter as the detector becomes very slow because the null is so shallow. If one instead uses a phase-sensitive detector, one can easily balance the bridge despite the very shallow null.

To put some numbers on it, at 1000 Hz, a 2 H inductance has a reactance of 12,566 ohms, so if the AC resistance is 10,000 ohms, the Q is ~1.3. I recall making bridge and Extech measurements down to Q=0.5 or so.

[Joe Gwinn]

Legg article has been posted

I just updated my website to include the Legg article:

V.E. Legg, "Magnetic measurements at low flux densities using the alternating current bridge", BSTJ, v.15, page 39, January 1939.

This is the real deal. Some math and physics background is advised, although it's possible to glean much of the gist by skimming the text while ignoring the equations.

Legg was interested in the design of RF and audio frequency inductors and transformers for use in telephone systems. Vacuum-tube amplifiers were well established and widely used in 1939, while the invention of the transistor (also at Bell Labs) was nine or ten years in the future.

[Mike Sulzer]

As discussed above, Legg does no such thing.

(Concerning the analysis that I presented) I never wrote that he did. It is my analysis, not Legg's, that shows that he needs the high Q approximation in order to get an accurate measurement of L. I think it is right. I can fill in the details if you have doubts.

The Extech users manual says that full accuracy (~1%) is achieved if the Q is not less than two, and my experience is that the Extech will work accurately well below that. One can test this by putting a potentiometer in series with an inductor and observing the effect on measured inductance as the series resistance is increased.

I have never disputed the accuracy with a series resistor. Suppose you have both a series and a parallel resistor. That is, with a series resistor of large enough value to make the Q low, and the parallel resistor of small enough value to also make the Q low. That is the issue. The coil core loads in parallel; that is the natural physical model. Using an equivalent series model requires checking with the analysis I presented above.

[Joe Gwinn]

(Concerning the analysis that I presented) I never wrote that he [Legg] did. It is my analysis, not Legg's, that shows that he needs the high Q approximation in order to get an accurate measurement of L. I think it is right. I can fill in the details if you have doubts.

The original claim was that the apparent variation of inductance with frequency was entirely due to imperfection of the instruments, the true inductance not changing. However, Legg amply demonstrates that the inductance does in fact vary with frequency due to eddy current shielding.

For the record, Wien 1898 arrived. It's quite the tome at 95 pages, and is also on the measurement of magnetic properties of ferromagnetic materials. I am able to read the German, albeit quite slowly, and I've been looking around. Fig 1 is the circuit diagram of the bridge Wien is using, a Maxwell bridge (balances inductor against inductor), and it assumes a single resistor in series with the inductor.

The effect of low Q is independent, and as discussed before one would see the eddy current shielding effect even with a superconducting core and thus high Q.

That said, I don't doubt that low Q can cause measurement errors, but we will have to hang a number on it to know if it will matter in guitar pickups. I recall an equation relating Q, actual inductance, and measured inductance, but I don't recall the analysis that led to that equation, so I don't know if this was due to the parallel-versus-series issue, or simply due to the total energy loss rate per sinewave cycle. Legg sort of dances around the issue in the unnumbered equation at the bottom of page 51, where his parameter G is the inverse of your shunt resistance.

I have never disputed the accuracy with a series resistor. Suppose you have both a series and a parallel resistor. That is, with a series resistor of large enough value to make the Q low, and the parallel resistor of small enough value to also make the Q low. That is the issue. The coil core loads in parallel; that is the natural physical model. Using an equivalent series model requires checking with the analysis I presented above.

I found a relevant analysis just today in the "Instrumentation Reference Book", 3rd ed, Boyes, Butterworth 2003. Figure 20.57 on page 469 has the two inductor models, one with a collection of shunt resistors and the other with the equivalent series resistor. One can read the text at Amazon.com. The text says that this table summarizes data from "Electronic Measurements and Instrumentation", B.M.Oliver and J.M.Cage, McGraw-Hill 1971. I have ordered it from the library. With luck it will tell the story, and the limitations of the equivalency.

[Mike Sulzer]

The original claim was that the apparent variation of inductance with frequency was entirely due to imperfection of the instruments, the true inductance not changing.

The original claim was that a single measurement of complex impedance cannot be expected to measure inductance accurately when both series and parallel resistance are present. But the inductance still might change with frequency. Indeed, the model you do not believe suggests that as the frequency increases and the magnitude of the impedance of the leakage inductance exceeds the value of the resistance in series with it, this series combination becomes inductive. This inductance is in parallel with the inductance of the coil, resulting in a lowering of the overall inductance. (This was the point of the title of this discussion.)

[Joe Gwinn]

The original claim was that a single measurement of complex impedance cannot be expected to measure inductance accurately when both series and parallel resistance are present. But the inductance still might change with frequency. Indeed, the model you do not believe suggests that as the frequency increases and the magnitude of the impedance of the leakage inductance exceeds the value of the resistance in series with it, this series combination becomes inductive. This inductance is in parallel with the inductance of the coil, resulting in a lowering of the overall inductance. (This was the point of the title of this discussion.)

Well, this dispute has been going on for at least three threads. The original claim was that the inductance did not vary with frequency, or alternately that the handling of high and low tones was the same. When it was counter-asserted that the inductance and tone do in fact vary, the counter claim was that the variation of inductance with frequency seen with impedance bridges and LCR meters was entirely a measurement artifact. This was ultimately completely refuted by Legg -- eddy current shielding is real, and it does cause inductance reduction. Which isn't to say that practical instruments necessarily are perfect. Specifically, measured inductance does appear to vary with Q, but this is independent of the inductance variation does to eddy currents. Practical coils will show both effects.

Oliver and Cage arrived, and I have been reading it. The chapter on impedance measurement (written by an expert from the standards world) is right on point, and at least partially supports your claims on the imperfections of instruments that measure complex impedance at a single frequency. I have not yet had the time to figure out the underlying mechanism.

[Mike Sulzer]

This was ultimately completely refuted by Legg -- eddy current shielding is real, and it does cause inductance reduction.

Then it is funny that transformer models including core effects do not have frequency variable inductors. Or is it possible that a leakage inductance in series with a loss resistor includes what you are calling eddy current shielding? When you connect a resistor to the secondary of a transformer, the current in the primary goes up, but the flux through the core remains constant because of the current in the secondary. This can be thought of as a shielding effect.

So I would claim that my model has the shielding effects in it without resorting to a frequency dependent inductance.

[Joe Gwinn]

Then it is funny that transformer models including core effects do not have frequency variable inductors. Or is it possible that a leakage inductance in series with a loss resistor includes what you are calling eddy current shielding? When you connect a resistor to the secondary of a transformer, the current in the primary goes up, but the flux through the core remains constant because of the current in the secondary. This can be thought of as a shielding effect.

So I would claim that my model has the shielding effects in it without resorting to a frequency dependent inductance.

The transformer guys don't care about the precise inductance, as transformers work the same so long as the inductance of primary and secondary coils is adequate.

A toroidal transformer can be built with essentially zero leakage inductance, and yet shows eddy current variation if the frequency is too high for the thickness of the laminations. Also note that Legg and his colleagues used toroidal samples (also called a Rowland Ring) for precisely that reason, and to eliminate air gaps. Legg defines the core shape on page 40 as a "thin annular core of mean diameter d".

So far, every article and book I've read on this subject, these having been published over the last century starting with the foundations and founders of electromagnetic theory, all say that eddy current shielding is real and causes inductance reduction, and the equations they give cannot be derived from transformer theory except as some kind of approximation. I would submit that the issue is settled, in fact was settled before any of us were born.

As for the exact mechanism, I have ordered two books by Steinmetz from the library. I gather that Steinmetz published the standard analysis of the effect of eddy currents. There will be partial differential equations for sure.