How does one explain the loading on a pickup coil by another conductor (eddy currents

[Mike Sulzer]

Understanding how a coil is loaded by another conductor requires both theory and measurement. The measurements here use a humbucker coil with 5,000 turns of #43 wire. The six holes for the slugs are drilled out a bit so that it can use some ceramic cores that have a slightly larger diameter than the steel slugs, so that the coil can have air, steel, or ceramic cores as required for the experiments. The characteristics measured with an Extech at 120 Hz, series mode, are:
air core 979.9 mH 4.698 Kohms
steel 1739.2 mH 4.708 Kohms
ceramic 1798.3 MH 4.682 Kohms

The theory used in this post is just the description of a circuit involving mutual inductance, given here:MutualInductanceLoading.pdf The reason for using circuit analysis, when applicable, rather than solving Maxwell's equations directly is that circuit analysis is accessible to more people. Circuit analysis in effect allows the solution of complicated differential equations by the algebraic combination of pre-solved differential elements, resistors, capacitors, inductors, and mutual inductors. It is not even necessary to be aware of how this works in order to understand the solution and how it is found.

The result of the analysis referred to above is the following equation:

L_c and R_c are the inductance and resistance of the coil, as measured either by the Extech at 120 Hz or the low frequency limit of a measurement covering a wide freqeuncy range. R_se is proportional to the resistance of core, slug, or whatever, but scaled to the value that appears across the pickup coil. Omega is 2 times pi times the frequency; k is the coupling coefficient; it is a number between 0 and 1 and is determined by the amount of shared magnetic flux. The inductance of the slug, or whatever has been scaled to be the same as L_c, (which simultaneously gives the value of R_se.) and so the value L_s does not appear in the equation.

The same parameter T controls the increase in the real part of the impedance (the so-called ac resistance) and the decrease of the imaginary part (the decrease of the inductive reactance) As shown in the analysis by looking at limiting cases, this frequency variable impedance is a multiple component circuit. The inductances are fixed, but of course R_se might vary due to the skin effect.

The real and imaginary parts of this equation are each one of two independent equations, and so it is possible to solve for both R_se and k over the range of frequencies covered by the measurement. Given values for L_c and R_c, and measurements of Z_p covering some frequency range, it is only necessary to subtract R_c from both sides of the real part and L_c from the imaginary part. Then one can solve for R_se by taking the ratio of the real and imaginary parts. Then k is found by substituting into the equation for T.

This post presents one such solution, and the purpose is to verify that the analysis works. Eight turns of copper wire (about .5 mm in diameter, whatever size that might be I do not know) were wrapped tightly around the pickup coil described above, and the ends were soldered together. Then the impedance is measured across the audio range, and the analysis described above is performed. The effect of the coil capacitance is removed. The capacitance is measured; one way is to insert the ceramic cores in the unloaded coil and measure the resonant frequency. The equation for paralleling two impedances is inverted to give an equation for removing the effect of one impedance in parallel with another given a measurement of the parallel impedance.

The equation above can be thought of as containing three terms, one real, the coil resistance, one imaginary, the coil inductance, and one complex. The effect of currents in the conductor couple to the coil magnetically. The real and imaginary parts of this third term for this case are shown in this plot:. They are exactly as predicted: both are zero at zero frequency and increase slowly at first, allowing the coil inductance to be measured accurately at a low frequency, such as 120 Hz. The real term is positive and the imaginary term is negative; that is, ac resistance increases with frequency and the inductive reactance of the whole circuit decreases. This plot shows R_se: . It is very noisy at the low frequencies, and there could be some bias in this frequency range. (The third term is small at low frequencies, and the computation is therefore prone to error.) Throughout most of the frequency range, the resistance does not change all that much, although there is a significant rise near the top. The final plot () shows k. It is also nearly constant with frequency, but does show a drop at the highest frequencies. The behavior of both Rs_e and k at the high frequencies might be explained by a change in the distribution of current within the outer coil, but it is hard to tell.

All three plots show good agreement with what is expected from the theory. Next to be measured is the impedance with steel cores.

[Mike Sulzer]

Let's look a the impedance of the pickup coil under test here in two different ways: with steel cores, and with ceramic cores, often called "ferrite". It is easier to make comparisons if the inductance with the two different kinds of cores are the same, or nearly so. Since these core have high permeability, it might seem that it would be necessary to match the cores closely. However, it is not: any high permeability will give nearly the same inductance.

This is something that can be understand easily using magnetic reluctance. If you have two resistors R1 and R2 of equal value in series, the total resistance changes by a significant percentage if you reduce the value of R. However, if you continue to reduce R1, its effect becomes minimal in terms of percentage of the total resistance. That is, changing the value from 1% of its original value to 0.1% does not do much to the total: the resistance cannot fall below R2 no matter how small R1 gets. The same thing applies in the coil: no matter how high you make the permeability (that is, how low the reluctance of the core), you cannot make the total reluctance of the circuit less than that of the path through the air.

So it is not necessary to match the material permeabilities because both are large; in such a case the inductance is limited by the reluctance of the flux path through the air. So the attached figure () shows that the imaginary parts are very nearly equal at low frequencies frequencies. The linearly increasing imaginary part means inductance, of course. The line for the ferrite cores increases very straight over the nearly the whole frequency range. There is a kink in the high frequencies to be discussed below. (Note that the capacitance of the pickup has been removed from the impedance by the techniques discussed in the precious post, and so it is possible to see the inductance across the whole range.) However, the line for the steel bends down some, and the impedance is really quite a bit less at the high frequencies. The cores conduct, and so they are inductors that couple to the coil inductance. From the analysis and measurement shown in the previous post, we know that the resulting circuit has multiple components that together have a a reactance that is inductive, that is positive rather than negative like a capacitive reactance, but this reactance does not increase linearly as a true inductor does. And in this case it is possible that there is more going on since current can, at least at some frequencies, flow anywhere in the core, not just in a narrow outer layer as if we had wire wrapped around a core made out of an insulator. So in analogy to the coax cable discussed in the previous post it is possible that the current density in the core varies with frequency and there is a true frequency variable inductance involved in addition to the multiple component circuit resulting from the mutual inductance.

The two real parts just have the value of the series resistance at low frequencies. The real part of the ferrite core impedance increases only slightly as frequency increases, indicating that there is little additional loss, as expected from the low conductivity cores. However, there is an exception: here is a kink in the curve at the resonance of the inductance of the coil inductance with the col capacitance. That is, even though, the effect of the capacitance in parallel with the coil has been removed, there remains an effect in the loss in the circuit. As mentioned above, there is also a kink at the same frequency in the curve for the imaginary part. If you lay a ruler on this curve so that it lines up with it at zero frequency and the high frequencies above the peak, it looks as though the impedance really does not increase perfectly linearly, but is slightly curved downward, and that this downward curve disappears above the king. It is possible that this high permeability ferrite material has some significant hysteresis loss that shows up most at the resonance because of the high circulating current in the circuit, but also has a smaller effect at lower frequencies. The effect would disappear above the resonance because most of the current flows through the capacitance.

The real part of the steel core impedance increases with frequency as expected. But it will take some additional analysis in the next post to get a better idea of what is happening with the steel cores.

[Mike Sulzer]

Now let's look at the impedance of the coil with steel cores and compare it to the impedance of the same coil with a shorted secondary, as described in the first post. There it was shown that the resistor Rse was reasonably close to constant with frequency except at very low and high frequencies, and that the coupling constant k was also mostly independent of frequency. The plot of the impedance of the coil with steel cores shown in the previous post suggests that the situation might be different with cores instead of a shorted secondary. In the plot shown there, the real part (ac resistance) increases with frequency. Another way of looking at this () shows the same information in a different way. When we look at the phase of the impedance of an air core inductor, we see the angle near zero at low frequencies, since the coil resistance dominates the impedance, but the angle increases with frequency as the inductive reactance increases, approaching 90 degrees at high frequencies. The angle of the impedance with ferrite cores almost exhibits this behavior, but the angle does not quite get to 90 degrees and there is the funny effect from the coil resonance discussed in the previous post. However, the angle of the impedance with steel cores does not even get to 80 degrees. Since the imaginary part (inductive) increases with frequency, this means that the real part (ac resistance) does as well since the angle is a function of the ratio of the two (arctangent of the ratio of imaginary to real).

So the coil with steel cores behaves very much unlike a "normal" coil, that is, one in
which the inductive reactance increases with frequency while the resistive part does not. Instead, both parts of the impedance increase more or less together.

So, continuing the comparison with the coil with the shorted secondary, we look at the impedance of the so called third term, the impedance in addition to the inductance and resistance of the coil. For the coil with the shorted secondary, the real part of this impedance increased with frequency at first but then leveled out. A similar plot for the coil with steel cores () has a real part that increases with frequency over the entire audio range, a very different behavior.

Still continuing with the comparison, the parameter Rse for the shorted secondary case increased slowly with frequency, almost flat in the middle of the range, but with the steel cores (), it increases nearly linearly over the whole audio range.. The coupling coefficient k in the case of the shorted secondary was nearly constant with frequency, but with the steel core (), it increases over the whole range, more slowly at higher frequencies.

To summarize so far. A simple "run of the mill" situation involving mutual inductance, such as the shorted secondary, exhibits the behavior shown by Extech measurement of a pickup coil. That is, the inductive reactance decreases, and the ac resistance increases between 120 Hz and 1 KHz. However, the pickup coil with steel cores has additional interesting behavior that is apparent from measurements over the whole audio frequency range, properties that one might mistakenly assume are obvious from the Extech measurements alone.

[Mike Sulzer]

Before beginning to attempt an explanation of why the impedance os the steel-cored pickup coil is as observed, let's look at it in another way. So far, we have been using the pickup impedance in the form R + jX, that is, as the sum of a real and imaginary part. However, from a physical point of view, the effect of mutual impedance is to load the coil, that is, put a complex impedance in parallel with the coil impedance (that is, in parallel with the inductance of the coil with the coil reistance in series with that combiation).

So what is the nature of the impedance loading the coil? We know the coil impedance alone, that is its R and L, and we have measured the impedance of the coil with the loading due to mutual coupling in parallel, and so it is just a matter of doing some computation to find the impedance that is in parallel with the coil alone.
The result is shown here ().

Let's examine the real part first. It looks very much like Rse, plotted in the previous post, but the resistance is several times higher. This is as it should be, given the measured value of the coupling constant k. Remember from the first post that Rse is scaled to the value it would have if the total inductance of the cores were the same as that of the coil. If there were perfect coupling, this would be the value appearing in parallel with the coil. However, the coupling is less than perfect, and as is well known, at least from the analysis of imperfectly coupled transformers, the loading is reduced by the square of k. So the measurements agree quite well! (As the coupling goes to zero, the loading would have to go to zero, that is, infinite resistance since the coil adn cores could not affect each other.)

Now consider the imaginary part; it increases with frequency like an inductor, but it is not a straight line. That is, although it increases, it does not increases as a straight line at some angle, and therefore, as expected we do not have a simple inductor. We expect there to be more circuit elements involved, as shown earlier, but also the inductance itself could decrease due to changes in the current distribution, similar to the effect in a coax cable, as described in the first post. The implied value of the inductance at 10 KHz is greater than the coil inductance. This is expected from the imperfect coupling, but it is not so easy to understand the details of this impedance.

[Mike Sulzer]

In this post we explain why the measurements shown in the previous post are not what might be expected, that is, the resistance does not increase with the square root of frequency, and the inductance changes little.

The skin effect in mild steel

Let's look at the skin depth of mild steel at 5 KHz, the top of the electric guitar frequency range. Consider 1018 steel; others will vary somewhat.
Skin effect - Wikipedia, the free encyclopedia gives a practical equation for computing the skin depth:

skin depth = 503 time sqrt(resistivity/(relative permeability times frequency))

where
the skin depth is in metres
the relative permeability is a number one or greater
the resistivity of the medium in Ω·m
the frequency of the current in Hz

www.public.iastate.edu/~nbowler/.../QNDE2005Bowler.pdf gives a low frequency permeability for 1018 steel of about 280. The resistivity is 1.59e-7 ohm meters*, and at 5 KZ the skin depth is 0.17 mm. Since this is a small fraction of the diameter of a slug, about 5 mm, we might expect the above equation to apply with good accuracy, implying that the resistance measured in the previous post should ivry as the square root of frequency from well below 5 KHz to the top of the measured range. It does not. Why not?

How does the skin effect work?

Imagine looking at a vertical plane edge on. The plane is a boundary between two types of material. To the left is a conducting material; to the right just air. Now let there be an alternating current flowing up and down in the conductor; it makes a time varying magnetic field that points horizontally. By Faraday's law of induction, the time varying B field makes a time varying electric field around vertical loops, oriented so that it reinforces the original current towards the boundary, but tends to cancel it away from the boundary. There is a figure near the top of the Wikipedia link above; it is for a wire, but it works the same. Thus current tends to be shoved out towards the boundary. This also reduces the magnetic field away from the boundary. Remember also that Faraday's law is frequency dependent: the higher the frequency, the stronger the electric field that causes the current to be pushed to the boundary. So the key steps here are: 1. ac current makes a time varying field magnetic field; 2. which makes (by the law of induction) a time varying electric field, 3., which drives current towards the boundary.

We can now make this one step more complicated: the current could be the result of an applied time varying magnetic field. This is what happens when you measure the impedance of a pickup: the current through the coil causes a magnetic field that opposes the buildup of the current and results in some current flowing in the conducting cores. The equation above for the skin depth is derived using infinite dimensions. The skin depth describes exponential decay, which extends forever in theory. This is a good approximation for many real situations, but is it good enough for the short open cores of a pickup? To answer this, we look at a toroidal transformer core, and change it into something like a pickup core.

The skin effect in a toroidal core and a pickup core

Consider a toroidal core make of a high permeability material, non conducting. If we wrap some turns around this core, we have an inductor. It does not matter where we put the turns since the core fully encloses the flux resulting from current in the wire, so let's put the wire all in one small region. This coil has some inductance L resulting from the ac magnetic field that is excited throughout the core in a direction pointing around the core resulting from an ac current flowing in the coil. In this situation, the B field in the core is simply determined from the current, the number of turns and the permeability of the core. When the flux is fully contained, it is simple.

If we let the core become conductive and excite the coil with a high enough frequency, the current flowing in the core will be concentrated near the surface of the core, the magnetic field inside the core is reduced, and the inductance is lowered. The fall off of the current is approximately exponential, with a skin depth computed from the above equation. The smaller the skin depth, the closer the decay is to a true exponential, and the more accurate is the simple description implied by the term "skin depth".

Now let the core be non-conducting again, and remove a section of it, but not where the wire is wound. The magnetic field in the remaining part of the core is reduced, and as a consequence, so is the inductance. As we remove more of the core, the magnetic field and the inductance are further reduced. When only a short open core remains, the field and the inductance are both only a few times higher than with an air core. That is what the measurements shown in the first post verify.

So what happens if the short open core is made conducting? The mechanism explained above for the skin effect still operates, but the magnetic field is much less than when the core is closed. Since the time varying magnetic field makes the electric field (throughout the law of induction) that results in the current shifting to the boundary, the skin depth is larger than predicted. In the case of the pickup core, the effective permeability is on the order of 1% of the material permeability, and so the skin depth at 5 KHz is more like 1 mm than 0.1 mm. 1 mm is too large for the exponential decay model to apply accurately, and so we do expect the resistance to increase with the square root of frequency, but rather with some other function that might be difficult to compute.

So the results of the measurements for the resistance in the previous post are not surprising, but what about the inductance? The measurement shows little change with frequency (the imaginary part of the impedance increases nearly linearly with frequency), while the fact that the resistance changes shows that the current distribution in the core is changing. The question, then, is this: should the inductance decrease with frequency with changes in the current distribution (and thus magnetic field) as it does, for example, in a coax cable 9as described above)? The answer is no, the inductance should not change very much, and the reason is the same as we have seen before: the cores are short and open. Just as changes in permeability of the material have little effect on the inductance, the current distribution does not either. After all, the magnetic circuit has no way of knowing whether the permeability or the current distribution is changing. The high reluctance of air where the field lines must complete is the same in either case.

Conclusions

So in conclusion:
1. The skin depth in short open cores with high magnetic permeability is not what the usual equation implies.
2. The resistance due to the cores should not increase with the square root of frequency, but with some other function.
3. The inductance should remain close to constant even if the current distribution within the core changes.

* (MatWeb - The Online Materials Information Resource)