This is the third item presented under the heading 'Thermoelectric Energy Conversion' (TEC). Press TEC I or TEC II to see those earlier chapters on this subject.

Transformers are used in several stages to adjust the voltage of electrical power as it is transmitted from the generators to the ultimate user. All of those power transformers generate heat owing to what are called 'magnetization losses'. Those losses cannot be avoided. The transformers are designed to keep them at a minimal level and, commercially, they are a tolerable overhead in the economics of power distribution. However, what is not general known, even by many electrical engineering professionals in academic institutions is the fact that there is a significant proportion of that loss that defies explanation. The problem was highlighted in 1936 by Dr. F. C. Dannatt (*Journal I.E.E., p. 667; 1936)* who measured the loss in various steel stampings as used in the assembly of transformers. The component of loss attributable to what are called eddy-currents, meaning the parasitic currents induced internally within each lamination forming the transformer core was found to be as much as 3.5 times greater than it should be according to electrical teaching.

Yet the teaching is not wrong. It relies on proven principles of electrical science. The electrical EMFs induced in those laminations are no different from those induced in the winding turns of the transformer and, since the resistivity of the material of those laminations is known, the electrical loss is a quantity that can be calculated with the same confidence as applies to standard electrical circuit design. Yet, for some mysterious reason, there is that threefold increase in the eddy current loss.

Indeed, Dannatt reported measurements on several sample materials of different resistivities, different mass densities and widely different thickness, ranging from 0.052 mm to 0.354 mm and found that the eddy current anomaly factor ranged from 2.8 to 3.5, that factor being the ratio of actual eddy current loss to its theoretical value as determined by the design principles we are taught as part of our professional electrical engineering education.

Onward research to explore this phenomenon and solve its secrets seems not to have attracted physicists as a worthy pursuit, with the result that six decades on from the publication of that paper by Dannatt I can survey the subject as if it were virgin territory. As you will see there is much to learn about this mystery energy loss of power that silently and unobtrusively disposes of billions of dollars worth of electricity every year, without our having understood why that happens.

Now I do have a special interest in the subject. I heard about it in 1947 and, after graduating from my Electrical Engineering studies in 1948, but during a period of onward practical training in the heavy electrical industry in U.K., I resolved to seek the chance to research that theme. In 1949, during that training period, I wrote a thesis in support of my application for a research scholarship with that anomalous loss problem in mind. My proposal in that thesis was a new method of measuring hysteresis loss by a technique which was not dissimilar from what is known today as an application of the fast Fourier transform. My aim was to analyze the waveform of the magnetic flux in the transformer core to achieve the rapid measurement of the hysteresis component of loss and track how the separate loss components varied during the cycle of a.c. magnetization.

I began my Ph.D. research on the eddy current anomaly at Cambridge in 1950 and spent three years on experiments probing all aspects of the problem that I could think of. Some of my tests revealed anomaly factors as high as 6 applicable during parts of the magnetization cycle. By this I mean that calculation of the instantaneous power dissipated as eddy current loss, as determined at different stages in the cycle of magnetization, was only one sixth of the corresponding measure of that form of power loss at that same instant.

Now I say here that I probed all aspects of the problem that I could think of at the time, but I confess, in retrospect, that there was one aspect that I should have thought of but didn't. It never occurred to me that the heat being generated as loss could regenerate itself as electricity in a way which augments the EMFs driving the eddy currents. Had I thought of that possibility, I would in all probability have dismissed it immediately from my mind, because it would have involved challenging the Second Law of Thermodynamics. However, I am, I believe, now a wiser being and I will here redeem myself by surveying the evidence which supports that contention.

I cannot claim that the resolution of the mystery of the eddy-current anomaly will save that loss of billions of dollars of electrical power in future transformer design technology. All we can do is to accept the loss, but come to know full well how that loss arises. However, inasmuch as uncovering the secret tells us how to deploy environmental heat to generate electricity, there is promise here of something even more rewarding.

Note that magnetization loss comprises two components, the hysteresis loss which transformer designers regard as increasing linearly with frequency and eddy-current loss which increases as the square of frequency. Thus, since 60 Hz is the standard power frequency used in USA and 50 Hz is the U.K. power frequency, the eddy-current anomaly is a more serious issue for U.S. power generators. Even so, the research interest in the subject seems to have been essentially a mid-century pursuit in U.K. and if there has been significant interest elsewhere or in more recent times I would appreciate being guided to references to that information.

Transformers are designed based on an optimum choice of electrical sheet steel. The choice of material determines the hysteresis loss for the standard operating condition of the core, the appropriate maximum magnetic flux range that gives high permeability, and determines the electrical resistivity, which is preferably high and usually some 5 or so times that of pure iron. Given the resistivity, the thickness of the steel laminations is the factor which determines the eddy-current loss for the chosen operating conditions. The smaller that thickness, the less the loss, at least in theory. However, there is the extra cost of the material and assembly if the sheet steel used is too thin and, considered with the hysteresis loss, there is little to be gained in designing for minimal eddy-current loss if that leaves hysteresis as too dominant a component. Accordingly one finds that these two components are much the same at 50 Hz or 60 Hz when electrical sheet steels of the order of 0.3 mm thickness are used.

Experimentally, one can distinguish between hysteresis loss and eddy current loss by measuring overall magnetization loss at two different frequencies but with the strength of the magnetizing voltage in proportion to assure that the range of flux density is kept the same. One needs to allow for losses arising from current flow through the resistance of the magnetizing winding and exclude these from the measurement. Then, by dividing the rate of loss by frequency, the plot of the increment with frequency, when extrapolated back to the origin gives the hysteresis component and the increment with frequency represents the eddy current component. See Fig. 1.

This figure has a straight line relationship corresponding to the theoretical notion that hysteresis loss per cycle of a.c. oscillation is constant for a given range of magnetization flux change in the iron of the magnetized core, whereas eddy-current loss increases with frequency squared and so the loss per cycle is a linear increase. However, in practice, one finds that the loss per cycle relationship increases as a curved relationship, the rate of rise being far more rapid at lower frequencies and then looking more linear as the frequency reaches 50 Hz, but always exceeding the theoretical eddy-current component by a quite substantial factor.

To understand this, consider now the situation where, for a given core sample, the hysteresis loss under 50 Hz operating conditions is 1 watt and, in theory, the eddy-current loss is 0.5 watt, whereas in practice the eddy-current loss is actually 1 watt. We can interpret this as meaning that 25% of the heat actually generated has found its back into regenerated electricity which augments the EMF driving the eddy currents. Here the anomaly factor is 2.Let us now work out how this would affect tests at 25 Hz, for the same range of the magnetic flux cycle. The hysteresis loss would be 0.5 watt and the theoretical eddy current loss would be 0.125 watt. However, using a little algebra you can work out that the actual eddy current loss would be 0.333 watt, giving an anomaly factor of 2.67. This gives overall heating in the core of 0.833 watt and if 25% of this is regenerated to feed it back as additional eddy current input power this adds 0.208 watt to 0.125 watt to give that 0.333 watt figure. You see, therefore, that the anomaly loss factor has increased as frequency has been reduced. Here is the first point that we can test to see if the regeneration theory holds up.

Repeat this for 10 Hz conditions, where hysteresis loss is 0.2 watt and eddy current loss is, theoretically, 0.020 watt and one obtains an eddy current anomaly factor of 4.65. To verify this, note that the heat generated in the core is 0.293 watt and 25% of this is 0.073 watt, which together with 0.020 watt is 0.093 watt or 4.65 times 0.20 watt. In reducing the frequency from 50 Hz to 10 Hz the loss anomaly factor has increased from 2.00 to 4.65.

Next, let us see what happens if we have a core sample in which at 50 Hz the hysteresis loss is as much as four times the theoretical eddy current loss. We are considering here an electrical sheet steel that is magnetized cross-wise to the rolling direction used in its fabrication. This does not take advantage of the optimum magnetic permeability which is higher for magnetization aligned with the direction in which the steel sheet has been rolled during its fabrication.

This circumstance is a laboratory test situation, rather than one of practical significance to transformer design, but it does have practical significance where steel stampings are used to form rotors and stators of electric motors. Magnetic flux in the stampings used to form such motor components can flow in all directions parallel to the plane of the steel sheet, varying as the motor operates and, of course, there can be a whole spectrum of frequency conditions as the motor speed is varied. Here, we can contemplate very high eddy current anomaly factors. Using a 2 watt figure for hysteresis loss and 0.5 watt for theoretical eddy current loss, if 25% of the heat generated is regenerated as an supplementary electrical input, then the resulting eddy current anomaly factor is 2.67. The reason is that the heat generated is 2.00 watt from hysteresis and 1.33 watts from eddy currents and 25% of 3.33 is 0.83, which together with 0.5, sums to 1.33. In short, that anomaly figure of 2.67, which is quite representative of the Dannatt measurements already mentioned is consistent with observation, even where hysteresis loss is fairly high.

Note, however, that Dannatt was not making measurements across the grain determined by the rolling direction of the steel and consider what would be the outcome of the latter calculation had two-thirds of the heat energy been regenerated as added eddy current loss. You will see that this implies an eddy current anomaly factor of 11, because a 2 watt hysteresis plus 5.5 watt of eddy current sums to 7.5 watt and if two-thirds of this heat is regenerated this adds 5 watt to the 2.5 watt input to give that 7.5 watt figure. If, therefore, we can show evidence supporting such a high eddy current anomaly factor, then we can infer regeneration of heat as electricity with a 67% efficiency.

Consider what that would mean, bearing in mind that a transformer core operates at temperatures ranging only over a few per cent as measured on the Kelvin scale. Compared with the Carnot efficiency limit set by the Second Law of Thermodynamics, we would have here a clear breach of that law. We could, if we knew how to exploit the phenomenon, convert heat into electricity and use that electricity, not to drive more eddy currents which waste energy, but rather to power a conventional heat pump working according to Carnot principles. Using only a portion of the electricity generated from heat we could pump heat to the higher temperature needed to set up the temperature gradient which powers that eddy current anomaly in the transformer. We would have the ultimate clean energy source of power. Environmental heat, which exists in abundance, consumed to generate electricity needed to run our machines and light our homes before reverting to heat shed back to the environment. It may seem impossible, but you can be sure that there are surprises ahead in the technology of the future, provided we come to understand how Nature processes energy fed into the magnetic fields which we inhabit as part of our natural environment.

Now let us suppose we have a way of testing the steel under conditions of very small hysteresis loss. How might this affect the anomaly factor? Logic says that less heat will be generated so that the eddy-current anomaly will not involve more than the feedback from the eddy current heating. That 66.7% feedback figure would indicate an anomaly factor of 3.00, which is in the range measured by Dannatt. However, we can suggest doing eddy current tests over a small range of magnetization but one confined to regions well above the knee of the B-H curve. When iron is magnetized its flux density increases rapidly and somewhat linearly as magnetizing current increases, until a point is reached at the knee of the magnetization curve. Thereafter the rate of increase reduces as it becomes progressively more difficult to increase the polarization. The reason for this is that the iron contains domain regions separated by domain boundary walls. As the walls move they increase the size of one domain and reduce the size of the adjacent domain. These domains are fully magnetized in opposite directions and all we see as net flux in the magnetic core is the difference in the flux contribution of all the domains. However, in the passage of those domain walls across inclusions (impurities) in the iron and in the flipping of the flux direction that occurs as the domains adjust to the change of magnetizing conditions, there are instabilities which we can detect as 'Barkhausen' noise. These are the source of hysteresis loss. The loss is a linear function of the number of cycles of magnetization and not a function of how fast we increase the magnetizing current.

For this reason the hysteresis loss begins to tail off once the cycle of magnetization is biased so as to be confined to regions where those instabilities cease to occur. Above the knee of the magnetization curve flux changes occur in a more controlled way as the magnetization direction in a domain is forced to orientate itself in a direction angularly displaced from the preferred axial direction in the crystal housing the domain. The result is that rotational hysteresis at very high flux density drops to zero. The eddy current anomaly factor should be much reduced, therefore, for tests well above the knee of the magnetization curve, but there is some telling evidence that appears once such tests are performed, as I shall report as we proceed.

So, we must now pay attention to those numerical factors which are a measure of the eddy current anomaly observed in electrical sheet steels. If the evidence supports what has been said above about the regeneration of heat as electricity then we can hope to move in the direction of that new energy technology.

In a book entitled *Principles of Electromagnetism*, 3rd Edition, Clarendon Press, Oxford, 1955 by E.B. Moullin, the eddy current anomaly is discussed in Appendix II at pp. 285-286. The following is a quotation from that work:

"It is as though the effective resistivity is about one-third of its true value. Now alloying and impurities always increase the resistivity; one cannot countenance any explanation which depends on the significance of decrease of resistivity. Whatever the cause of the effect may be it is certainly not due to the resistivity being much less than the value measured by a straightforward conductivity test."

Here was the verdict of the Professor heading the Department of Electrical Engineering in Cambridge in 1955. I was awarded my Cambridge Ph.D. on the eddy current anomaly in 1954. It had not entered my mind that the resistivity of the electrical sheet steel could be lowered artificially by some means peculiar to the magnetization process or from heat generation, which Professor Moullin could have declared also as something which increases resistivity. So how did Professor Moullin conclude his discourse on the subject?

He wrote:

"For some obscure reason the currents must be larger than we have calculated and, in some obscure manner, this must be brought about by the distortion of the wave-form caused by the hysteresis loop. Dr. Brailsford has made an important contribution towards the solution of the problem (seeJournal I.E.E., vol. 95, Part II, p. 38; 1948), but the answer is not yet known. We do not propose to discuss here the possible mechanisms for the effect. It will suffice to report its existence to the reader and to impress on him that there is a loss in iron stampings which varies as frequency squared and whose magnitude is some two or three times the legitimate eddy current loss, appropriate to conditions when the effect of imperfect penetration is certainly negligible."

I may add here that one of the aspects I had considered in my research was the possibility that the hysteresis loss could increase with frequency squared if the domain switching involved retardation. A time-lag effect in the magnetization process would show itself with much the same signature as an eddy current loss. Also, there was the prospect that the inhomogeneities in the steel attributable to domains having sizes commensurate with the thickness of the laminations could explain some measure of increased eddy current loss. For example, in the extreme case of a single domain boundary wall parallel with the plane of the lamination and located midway across its thickness, the full measure of flux change produced by magnetization would be effective from the central plane and that could even explain that anomaly factor of 3.00. However, such ideal domain structures do not exist in reality, so one must rule out that possibility.

The only other textbook discussion of the anomaly that has come to my attention is that in a 1966 book by Brailsford who by then had become Professor of Electrical Engineering in the University of London. These were the days, incidentally, when a professor at a British university was normally 'the professor', meaning the leading person in the relevant academic discipline. Brailsford's book was entitled *The Principles of Magnetism*, published by Van Nostrand, (London, Toronto, New York and Princeton). Professor Brailsford had been one of the External Examiners of my Ph.D. thesis.

His verdict on the anomaly, as stated in his book, was that " its origin is a matter of speculation", a statement made by reference to his paper co-authored with R. Fogg and entitled *'Anomalous iron losses in cold-reduced, grain-oriented transformer steel'*, which appeared in *Proc. I.E.E., v. 111, p. 1463 (1964).*

Brailsford's book, together with my own thesis record, provides the essential evidence I need here to support the case developed above. He refers to an interpretation by E.W. Lee entitled *'Eddy-current losses in thin ferromagnetic sheets'* which appeared in *Proc. I.E.E., v. 105C, p. 337 (1958)*. Lee had calculated the anomaly factor on the assumption that there were large domains spreading between both faces of the sheet lamination and that the planes of domain walls were at right angles to the sheet surface. Then, according to the ratio of domain width to lamination thickness it could be argued that the eddy current anomaly factor could increase, possibly to value of 10 or so. However, Brailsford dismisses this on page 240 of his book with the words: "It seems difficult to explain the observed results on this basis, for the domain widths would need to be two or more times the sheet thickness to obtain value of anomaly factor greater than 3, say, and the domains do not in fact appear to be as large as this."

He then goes on to say: "It is also necessary to explain the increase in anomaly factor with decreasing frequency, particularly for frequencies below about 10 Hz and also the large variation of the factor with direction in the sheet. The loss mechanism operating does not, indeed, appear to be well understood."

So one sees that, some 32 years ago, which was then some 11 years on from the date of Professor Moullin's book, the verdict remained the same. Indeed, the mystery still prevails and awaits solution. However, compare Brailsford's words with what I deduced above. I offered an explanation of the increased anomaly factor at lower frequency and, using the same argument, I said that magnetization across the grain of the steel could lead to higher hysteresis loss which would produce very much greater anomaly factors. On page 239 of his book Brailsford presented data showing how the anomaly factor in a grain-oriented material varied with direction of magnetization related to the direction in which the steel had been rolled. It was as high as 10 over a full range of flux amplitudes when the angle was 90^{o}. It was between 6 and 8 for angles between 25^{o} and 54^{o} and it was between 3 and 3.5 or so for magnetization along the intended 0^{o} direction.

So here is the evidence supporting my case that the eddy current anomaly is attributable to an effective reduction of resistivity brought about by heat being regenerated to produce EMFs which augment the current flow. This is effectively an action that reduces resistivity, though in reality the resistivity remains the same and we have heating in full measure along current paths parallel to the lamination surface but very substantial cooling at the seat of the regenerative action, which I see as being traversal flow paths across the thickness of the lamination. I will explain this in detail in the next Chapter of this pursuit TEC IV

I note that I have, in the above commentary, deliberately not brought to bear the further evidence I can offer from a new analysis of my thesis findings based on the thermoelectric regeneration proposition. I will hope to introduce that evidence as we develop this theme further in these Web pages, but I feel the reader will find more immediate interest in the development of the discussion the TEC IV chapter.