# Losses when using power transformers at higher frequency

1. Mar 16, 2013

### I_am_learning

The hysteresis iron loss is give by Ph = Kh*f*Bmax^1.6
And eddy current loss is given as Pe = Ke*f^2*Bmax^2
At first it appears both loss increase with frequency (f) but recognizing that for fixed magnitude of applied voltage, Bmax = k/f , we can see that Pe remains constant and Ph decreases with increasing frequency.
So by operating power transformer designed for 60hz, at say 2khz, losses can be minimized.
Is this all right?

2. Mar 16, 2013

### marcusl

I don't think so. A 60 Hz transformer will be nearly useless at 2 kHz. Where do you get Bmax=k/f? (and what is k?)

3. Mar 16, 2013

### jim hardy

I don't think so either.

Iron magnetizes from the outside in and that's why transformer laminations have thickness inversely proportional to frequency.

At higher frequency you are using only the outside surface of the iron so flux density is higher there. Your losses become substantial. You need thinner laminations.
If you can find an eighth edition of this old book

http://www.amazon.com/Dynamo-Electric-Machinery-Students-Electrotechnics-Volumes/dp/B005YQWUF8

chapter 6 describes the phenomenon ~ page 141. My copy is dated 1901.

here's an earlier version online but it doesn't have that paragraph in the "Magnetic Principles" chapter.

http://openlibrary.org/books/OL2046...nery_A_Manual_for_Students_of_Electrotechnics

Lest I be dismissed as a luddite for suggesting such an old reference, here's what IEEE has to say about Prof Thompson's books:
http://spectrum.ieee.org/geek-life/tools-toys/treasured-texts/0
Look for his "Calculus Made Easy" in college bookstores.

4. Mar 16, 2013

### I_am_learning

k is some constant (which depends on voltage magnitude, core cross section and no. of turns)

I thought those relations were pretty well known. In any case:
Eddy Loss Relation:
http://en.wikipedia.org/wiki/Eddy_current#Power_dissipation_of_eddy_currents

Hysteresis Loss Relation:
http://en.wikipedia.org/wiki/Magnetic_core#Core_loss

Bmax Relation:
(Look at the right sidebar "Transformer universal emf equation")
http://en.wikipedia.org/wiki/Transformer#Effect_of_frequency

I also thought 60hz transformer should be useless at 2Khz. The only problem for working at 2Khz I see, is the skin effect (and proximity effect) on the conductors.
Is skin effect that problematic at 2Khz?

Last edited: Mar 16, 2013
5. Mar 16, 2013

### jim hardy

not on the conductors but in the iron...

Lamination needs to be thin so it'll constrain eddy currents to that elongated path. Eddy currents tend toward a circular path ; since at higher frequency the circles will be smaller you need thinner laminations.

A 60hz transformer's laminations are thicker than one for higher frequency.
You can see them in old Tektronix o'scopes that had transformers suitable for up to 400 hz.

You can use a 400 hz transformer at 60 hz if it's designed for both. It''ll have laminations that are thinner than if it only had to handle 60 hz, but it'll have a lot more of them than if it only had to handle 400 hz.

Here's a recent article about just that subject.
http://www.edn.com/design/component...sformer-at-a-frequency-it-wasn-t-designed-for

and a NASA article with some burly math...http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19660001049_1966001049.pdf
its 'conclusions' paragraph surprised me.

6. Mar 16, 2013

### I_am_learning

From the article

So, it agrees that eddy loss in the IRON is independent of applied frequency.

The talk about eddy losses increasing with frequency is only when the flux in the core is assumed constant. But if the applied voltage is kept constant, then flux will decrease with increasing frequency and the result is: eddy losses remains independent of frequency.

I couldn't open the NASA article, but looking at the conclusion paragraph, its clear that they are assuming constant Phi_max not constant applied voltage. In that case, its true that the Eddy loss increases as f^2 (or f^(3/2) ). But add to that the fact that Phi_max decreases as 1/f when Voltage is kept constant, and we will get that Eddy loss remains constant.

It (eddy loss) can even decrease with increasing f, if we take P_eddy_loss proportional to f^(3/2) instead of the usual f^2.

I think you are missing the point that Phi_max (or B_max) is decreasing with increasing frequency (when V is held constant). If not, sorry for mis-thinking. :)

7. Mar 17, 2013

### jim hardy

No you're quite right , flux is in proportion to volts/hz.

I think i overestimated eddy current losses.
Sorry - I just got tired before thinking it all the way through.
Probably skin effect in the iron makes eddy current flow through less iron, hence more resistance, reducing loss as skin effect kicks in ?
Same principle as laminating. Good to get that point squared away in my alleged brain.

I once experimented with twelve foot long solenoids where of course flux is lower than a closed transformer core.

We excited the solenoid with a triangle wave current so the e = k*di/dt relationship would be apparent.
Triangle wave is constant di/dt on each slope so we expect square wave voltage.
What we observed is the "retardation of magnetization" due to eddy currents makes this solenoid resemble inductance at low frequency only.

Traces of excitation current(upper trace) and induced voltage(lower trace) at three different frequencies.
Top traces at 3 hz
middle traces at 10 hz
bottom traces at 60 hz.

As you see voltage got bigger with increased frequency but the waveform was no longer the expected square wave.
Even at 3 hz the time for eddy currents to settle is apparent- the slow rise and fall times of the square wave is tens of milliseconds.

Those traces are from a control rod position sensor.
It has a moveable core of 400 series stainless steel which is ferromagnetic. The rod is near full out, ie magnetic core is most of the way up in the solenoid.
But with air core(rod on bottom) the waveforms looked textbook - triangle current, square wave voltage..
Clearly when we inserted the core our excitation current (triangle wave) was no longer the only current flowing.. Eddy currents delayed its magnetization.

I carried some of those prejudices with me into this topic.
I was exciting my inductance with constant current, of course you'll be exciting yours with constant voltage.

If you ever experiment with inductors remember that triangle current approach. It really made the light go on for us. With sine excitation all we saw was phase shift and that was puzzling.

I don't know why the NASA article won't open, I just tried it. It opens on every other attempt, even numbered tries give error "doesn't begin with %%pdf" (or something equally cryptic.) My Acrobat reader is only a few months old and I think I have explorer 9.

Thanks ! I am learning too.

old jim

Last edited: Mar 17, 2013
8. Mar 17, 2013

### I_am_learning

So whats our conclusion? What will happen when we operate 60Hz transformer at 2Khz?
I wish I had necessary tools to experiment.

And regarding the graphs, are you sure you did the experiment on a non-closed solenoid?
Because if it is open solenoid with much of the magnetic path in the air, then I would expect the Flux to be much lower. Then, the eddy current effect wouldn't have been so much pronounced. But maybe its because you are using solid Iron rod and not laminated one. Also, I don't know the magnitude of current and no. of turns being employed.

I really like the way jim always brings up extra materials. Thanks.

9. Mar 17, 2013

### jim hardy

Yes, that's a measurement of a control rod position detector taken from the control room.
It's an open coil surrounding the shaft that lifts the rod.
The shaft is inside a non-magnetic pressure boundary, the coil surrounds both..
The shaft is a solid rod perhaps two inches diameter.
The coil has a few thousand turns.
Excitation current was about 20 milliamps.

The device is intended to detect position of the shaft.
If you think about it in simplest terms, in a long solenoid half the magnetic circuit is outside the coil and half inside.
So as you insert a highly permeable core, you are replacing half the magnetic path with essentially a short-circuit for flux. So when core is fully inserted flux should double and we observed not far from that.

At DC the relative permeability of ferromagnetic iron is at least in the hundreds.
But with AC, eddy currents cancel magnetizing amp-turns per Lenz's law.
So with AC it's as if the relative permeability were much lower than the DC value in the catalogs.
This core's material listed a relative permeability in the range of several hundred.
But at 60 hz we observed an effective relative permeability of more like 20.
At three hz it was around 100.
At 400 hz our coil was oblivious to presence of the core - effective relative permeability of ~1.
We attributed this to its not being laminated.
And it showed a significant temperature dependence which is what we were investigating in the first place. Not surprising in hindsight, for temperature directly affects conductivity of iron which is a major term in that NASA paper.

So - to your question what happens to a power transformer at 2khz?
I never tried it so this is a guess -

Eddy currents will cancel magnetizing amp-turns to point no load current will be higher than you expect.

Leakage reactance is now at 2khz not 60hz , so is 33x as many ohms.
So the transformer will show poor regulation.

The core will run hotter than you expect because skin effect constrains flux to the outer couple thousandths of your iron laminations. Less flux, true, but it's traversing way less volume of iron so flux density is up and hysteresis is more than you expect.

A surplus automobile alternator and variable speed drill would make an interesting variable frequency power source for such an experiment. If you're a student maybe you and a professor would find it interesting to arrange a 1 credit hour lab special project.

Sorry for the tangential nature of my posts. Once again I am humbled by how much I don't know. But it builds character, I suppose, to be reminded of that and Mother Nature is always happy to oblige.

Have fun -- old jim

PS if you do experiment be advised that many good quality dmm's are accurate on their AC scale only in vicinity of power line frequency. Check against your 'scope or you'll probably repeat a lot of work.

old jim

Last edited: Mar 17, 2013