Eddy Currents - Skin Effect - Proximity Effect

[CharlesPotter]

Does any one have a MathCAD document to determine the optimum AC to DC resistance and losses in an inductor with copper conductor? Is it best to have AC and DC losses equal? For example: What is the AC resistance of a #12 AWG copper coil in an inductor at 100kHz at any ripple current and length? What is the optimum number of strands of smaller AWG wire would it take to equal #12 AWG? My understanding is that the AC losses are determined by the ripple current in the inductor. So keeping ripple current low will reduce the AC losses. What I am trying to determine is the best "build" of an inductor for a given application. Any discussion of this subject will be appreciated.

[RichardJ]

Charles Potter wrote:

Does any one have a MathCAD document to determine the optimum AC to DC resistance and losses in an inductor with copper conductor?

What do you mean by "optimum" losses? Do you mean minimum?

Is it best to have AC and DC losses equal?

If you mean "minimum" losses, then you would minimize both.

What is the AC resistance of a #12 AWG copper coil in an inductor at 100kHz at any ripple current and length?

That depends on how the coil is wound.

What is the optimum number of strands of smaller AWG wire would it take to equal #12 AWG?

That depends on how the smaller strands are wound compared to the larger one. And what you mean by "optimum".

My understanding is that the AC losses are determined by the ripple current in the inductor. So keeping ripple current low will reduce the AC losses.

Well, if you don't have any AC current then you can't have any AC losses.

What I am trying to determine is the best "build" of an inductor for a given application.

I know little about it, but I know that inductor design is a very complex thing. It's not just a question of the number of turns and the diameter of the coil, it is also a question of how the coil is wound and what the core material is, what shape it is, skin effects, proximity effects, etc. Especially at frequencies high enough for skin effects to make any difference (I believe that means RF). You need to get a book on the subject. Either that, or just describe your problem to a coil manufacturer and buy whatever they recommend (which is what I would do).

[CharlesPotter]

Thanks for your response, Richard. I should have been more specific with my questions. Your answers are well taken. I'm still looking for a good MathCAD document on the subject. Magnetic design requires a lot of estimating, iterating and reiterating. One good paper is: [PDF] 2003 Power Seminar - Under the Hood of Low-Voltage DC/DC Converters. An advanced text is: Wiley::High-Frequency Magnetic Components. High-Frequency Magnetic Components Solutions Manual by Marian ...

Regards,

Charles Potter

[RichardJ]

I'm still looking for a good MathCAD document on the subject.

I don't recall seeing one in the forums, but if you haven't already searched them you should try that. The old Collaboratory is still up, so the best place is to search there:

http://collab.mathsoft.com/~Mathcad2000/login

A lot of the Mathcad worksheets have not made it over to these new forums (hopefully, that is a temporary problem!)

You culd also try searching the Resource Center:

http://www.ptc.com/appserver/mkt/products/resource/list-dtav.jsp?&rccg=888&nav=5283&sec=5268&top=0&section=mathcad_resource_center

[AWesterman-disa]

As Richard says, this is a very large question. It suffers from the fact that every choice is a trade off of one power loss against another.

The exact balance of what you choose to do will depend on the balance of DC current to AC component, duty cycle, frequency (~100kHz), core material (I am assuming ferrite), ...

Overall you aren't really concerned between the AC or DC power loss, its all power that dissipates in the copper.

Some thought needs to be given to the core loss / copper loss balance & there the 50/50 rule is generally the ideal though rarely achieved.

To suffer the effect of skin depth you don't need to be running at RF

Formula for skin depth is ~76/sqrt(f) mm ( f in Hz)

So at 100kHz skin depth is 0.24mm.

At 440Hz (some 3 phase) its only 3.7mm!

That is fairly straight-forward on its own - a single layer winding &100kHz - dont exceed 0.5mm diameter wire & it should be in the correct general area.

Once you start on Proximity Effect and/or multiple layers it becomes much more complex.

Each wire affects every other wire around it, stacking layer on layer on layer the effects rapidly multiply beyond control if the skin depth : wire diameter ratio isn't considered carefully.

There have been a large number of papers written about it, I have found a lot of useful info from dartmouth college (Google : dartmouth college & proximity).

[CharlesPotter]

Here is a good resource from DARTMOUTH.EDU: search on "eddy current effects in windings" and scroll down to Introduction.

[AWesterman-disa]

This is roughly where I have started.

Most of the equations in the mathcad file attached are from the Dartmouth source.

Not sure if they are all either correct or applied correctly, but gives a start point - It is a work in progress!

Hopefully the comments as added so far will make it semi-understandable

Would appreciate any comment as to changes / improvements that can be made.

Andy

[ptc-1365791]

These two resources may help you out

1) Circuit Analysis of AC power Systems, Edith Clarke Volume II, Chapter 11

2) SAE practice paper ARP-1870, figure 32.

Jon

[jeanGiraud]

>Does any one have a MathCAD document to determine the optimum AC to DC resistance and losses in an inductor with copper conductor? Is it best to have AC and DC losses equal? For example: What is the AC resistance of a #12 AWG copper coil in an inductor at 100kHz at any ripple current and length? What is the optimum number of strands of smaller AWG wire would it take to equal #12 AWG? My understanding is that the AC losses are determined by the ripple current in the inductor. So keeping ripple current low will reduce the AC losses. What I am trying to determine is the best "build" of an inductor for a given application. Any discussion of this subject will be appreciated.<

____________________________________

Maybe you aren't doing correct from the start. Taking a static circuit RLC, the voltage will govern the current, which in turn will govern the °C and increase the resistance [R] ... and so on. Now, depending upon the frequency, equate for the effective resistance ... and so on. Is that coil for dephasing [ like a matching circuit] or is it for filtering. All those technical matters have been solved . Just purchase what you need, for your application.

"What is the AC resistance of a #12 AWG copper coil in an inductor at 100kHz at any ripple current and length ? "

A long useless piece of uncertain results, just test it. BTW, AC resistance does not exist ... rather "Impedance". If you say 100 kHz, you implicitly decide "sinusoidal wave". The attached work sheet is all red, no idea what it's doing, and mois not defined. What do you mean by "ripple current" ? Ripple means the wavy part above the max constant current. In one hand you let figure a filtering coil, on the other hand not clear because you haven't abstracted the project. Click on the image for a start. In the Mathcad collab, there are two superb work sheets: one is from Lou about the waveform analysis , and the other one from "jmG" about matching a transmission line. But none got involved with the skin effect ...

jmG

[AWesterman-disa]

Hi Jean,

unfortunate that your version of mathcad does not support the fundamental constants, µ₀ is just the vacuum permeability constant µ₀ = 4π×10⁻⁷H·m⁻¹

AC resistance is a terminology that has been used in many papaer from a number of differing sources.

It has to exist - What we are trying to express is the lossy (pure resistive) part of the total impedance of the wire (length, diameter etc to be specified later).

The AC impedance that you refer to could be a pure inductive element giving rise to zero power loss, but that is not our concern.

In my field (switch mode power supplies) the current flowing in a power inductor will typically be a repeating arbitrary waveform with a repetition frequency of (say) 100kHz and composed of a number of elements including square/rectangular , triangular and sinusoidal etc pulses.

All of these components (sinusoidal especially) may have a fundamental frequency much higher than the base repetition frequency.

The sample current waveform in the mathcad sheet is just a 100kHz triangular ramp superimposed on a DC (output) level as is typical for an output choke in either a "Buck" or "Forward" converter. It could be a much more complex waveform, but for the purposes of discussion simple is as good as anything.

It then allows us to split the given waveform into its harmonic components - for a waveshape as described this is relatively straightforward,

Losses for each frequency component can analysed and summed to give a total loss for the composite waveform

The difficulty with skin effect is that for any given **sinusoidal** waveform, the frequency of the sinusoid will determine how much of the conductor is effectively used.

If the diameter of the wire is too large for the frequency component being analysed then only an annulus can be considered for the conduction area of the wire and so the effective "AC resistance" of the wire will increase with increasing frequency giving rise to additional power losses compared to those calculated from Irms and Rdc.

Additional to that is the "Proximity effect" which has been analysed in many papers, some referenced in earlier postings and most of the mathematics of it is far too complex to explain here.

The simplest viewpoint for a partial understanding of the effect is to consider a spiral wound copper foil of thickness much greater than the calculated skin depth at the frequency of interest and a current flowing in it.

To a simplistic analysis:

The current flow in the first layer will only flow in 2 strips (top & bottom).

The current in the top layer of the first turn will then induce some current flow in the bottom strip of the second layer (~equal and opposite in polarity)

To keep the current continuous from the first layer to the second, the current in the top strip of the second layer must be the sum of the total current in the first layer + the opposing current induced in the bottom strip of the second layer.

Continuing this argument to the third layer the current flow in the bottom strip is induced from the current in the top strip of the second layer , etc etc ...

In the extreme the power losses incurred in sucessive layers increases :

1^2, 1^2 + 2^2, 2^2 + 3^2 , 3^2 +4^2 ...

1, 5, 13, 25, ...

Obviously this is an extreme condition that could never apply in a real situation.

As a start point I posted a segment of a work sheet, which I did say was a work in progress, with some comments added to try to explain what I was trying to do.

The area "various equations" is where I have included the information that I have extracted to begin to demonstrate the effect in a practical,and calculable manner using what I have managed to understand from the squiggly equations that are generated by the people with rather more active brain cells than me.

Short term it seem to relate to a couple of examples that I have working on a testbench at the moment - whether it is accurate or not still remains to be seen.

If I can get a few more viewpoint on the subject then it may shed more light & understanding on the overall topic.

Once I can understand it better then (maybe I can write the abstract)

Best Regards

Andy

[jeanGiraud]

Andy,

In simple words, you have :

1. A system producing wave forms, waveforms of different shapes,

2. The period of the system is fixed ... equivalent to 100 kHz,

3. The system is the exciting source

4. The combined transport and load is unknown

The project consist in:

1. Knowing/measuring the static of the transport+load

...... neglecting the source .

2. Plot the wave shapes,

Check if all/none or whichever ones have Laplace transform

3. If Laplace ==> apply Laplace circuit analysis

4. if NOT Laplace ==> apply Fourier analysis.

5. The anlysis is to reveal the current at the load ,

6. ... from 5) apply more conclusive as applicable maths.

Mathcad tools available:

1. Laplace analysis [jmG]

2. Fourier analysis [Lou] ... adapted jmG style

Note: Fourier will do it all ... if Laplace applies to all = simpler

jmG