# Interesting Math Dilemma, Elliptic Integral 2nd Kind

Discussion in 'Math' started by MrAl, Apr 19, 2017.

## Is k squared or not?

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1. ### MrAl Thread Starter Distinguished Member

Jun 17, 2014
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924
Hello there,

I was fooling around with an old formula i came up with for calculating the inductance of a coil and was comparing the results to several other formulas, including one i found on Wikipedia. The formulas involve complete elliptic integrals of the first and second kind. I then went back on the web and looked for the formula for the integral that usually just goes by the name "E" and comes in two forms, E(a,x) and just E(x). The form which i need is E(x) but it comes from E(pi/2,x) so those two are equivalent.

The real problem came in when i went to calculate a formula using E(x). I found two different results, both in fairly respectable places. The two different results are:
E(0.4)=1.399 {Wolfram using EllipticE(0.4), using k}
and
E(0.4)=1.505 {Direct calculation, Using k^2}

Those are not exact but should be within plus or minus 0.001.

I looked into this and found that the formula for this integral was:
integrate(y(x),x,0,pi/2)

where ONE FORM OF y(x) is:
y(x)=1/(1-k^2*sin(x)^2)

and that formula results in E(0.4)=1.505

Now interestingly, the OTHER FORM OF y(x) is:
y(x)=1/(1-k*sin(x)^2)

and this form results in E(0.4)=1.399.

What do you think?

2. ### SLK001 Senior Member

Nov 29, 2011
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381
You know that this site can implement TEX embedded symbology, don't you?

3. ### wayneh Expert

Sep 9, 2010
13,893
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I think those two formulas are obviously different and thus produce different results. Without following the derivations and possible assumptions that produced these formulas, it's tough to comment. Are you saying they should be equivalent?

4. ### The Electrician AAC Fanatic!

Oct 9, 2007
2,487
384
I've seen other people who are calculating inductance (besides you and me) discover that some authors use one form and others use the other form. This arises because elliptic integrals can be expressed in terms of the "parameter", or in terms of the "modulus".

See: http://mathworld.wolfram.com/EllipticModulus.html

and: https://www.encyclopediaofmath.org/index.php/Modulus_of_an_elliptic_integral

I was quite puzzled for a while when the published formulas didn't give the right numerical result when I used modern mathematical software to calculate the elliptic integrals.

5. ### MrAl Thread Starter Distinguished Member

Jun 17, 2014
4,238
924
Hi,

Yes, but i would have to look into how to post that way again, thanks.

6. ### MrAl Thread Starter Distinguished Member

Jun 17, 2014
4,238
924
Hi there,

Yes that is the heart of the matter. In short, one uses 'm' and the other uses 'k' where m=k^2.

The real problem is that when a formula is given they dont state the form they are using, they just state "E(x)" and so a formula might look like this:
y=3.2*E(x)

and so we have no idea if they are using the 'm' type or the 'k' type.

One way to tell (the only way i know of) is to calculate at least one time with some test value like say x=0.4 and see if the entire formula works out to some other values that are known or can be computed with another method. Very unfortunately, some values work out the same for both forms so we get the very same results for these test points (not the one i did which was 0.4 though).

Also interesting is that Wolfram shows TWO different results for the same function call depending on where you look on their site, so apparently even they get mixed up.

Unfortunately for the formula i am using neither form give a result that is that close so i cant really tell which form they are using. The formula i was using, in fact the two formulas i was using (as well as others) are on Wikipedia but i'll post that as another thread because that is more about calculating inductance rather than a problem with the forms of a math expression.

I think in the poll i should have included another option, "It is indeterminate if k is squared or not".

Oct 9, 2007
2,487
384
8. ### MrAl Thread Starter Distinguished Member

Jun 17, 2014
4,238
924
Hello again,

Yes, although a bit dated that might be interesting anyway so i might take a chance at the book.
I did not keep up very well on the progress in this area of physics so i dont know what new things have developed since then, if any.

This started because i ran across my old formula from back in the 1990's and decided to test it a little better. This led me to some other formulas which i had accumulated over the years that i could use to test and i also ran into a few formulas on Wikipedia which i thought i would try. Wouldnt you know it, the best formula they have does not work for every construction and give results that are off by significant factors when it is supposed to be the best formula. Investigating that formula, i ran into this little E(x) issue which was the first problem (not the only one unfortunately) but i also realized then that no matter which form i use for E(x) the inductance calculation never comes out close enough, but significantly different from many other formulas including one in an EE reference book. I tend to trust the one in the EE book because it matches other formulas within reason.

The only drawback to that book is the material is dated, but then they do mention that they give ideas for calculating other structures that dont fit the regular round or square cross section examples usually found on the web. That could be the key to creating the better formula and possibly, which would be nice, fixing the Wikipedia formula.

9. ### MrAl Thread Starter Distinguished Member

Jun 17, 2014
4,238
924
Hello again,

The book arrived yesterday afternoon and i took a look at it today. I see that yes it is very very dated. In fact, it comes from a time before the real computer age where everyone has a computer in their home. That's a bit of a problem because they go to great lengths to avoid things we would do in passing today, like for example the elliptic integrals, which they avoid like the plague, and instead give partial series' solutions which are not nearly as accurate.

The base price of the book however was only \$12 USD, and with today's book prices that's pretty good and with the information they do include even though dated i believe it is worth the price.

The book is written a little strange too, where for example in chapter 2 they outline a procedure for a given calculation, then state that if the calculation is done a certain way the result is the formula below. When they say "below" they actually mean 3 chapters later in chapter 5, where the actual formula appears for the first time. This threw me at first, because i thought there was an editing error where they forgot to include the formula.

Add to that the illustrations are pretty poor, and not well drawn at all. Maybe a sign of the period. A few of them are ok though.

Additionally, there is a lot of inference required in order to actually read some of the formulas, because certain things are not explicitly stated. For example, a formula in the book uses variables a and b, explains explicitly what 'b' is, but omits any explanation for 'a'. Reading on, we find that they give another formula whcih shows 2*a/b and says that is diameter over length, so we can 'deduce' that 'a' is a radius. That's the only way we could ever be sure 'a' was a radius because they dont state that explicitly.
Conversely, they often explicitly state the units being used, which is often omitted in a lot of books.

However, again, twelve dollars for a book like this isnt bad because it does give a lot of information which is still interesting today. If this book was 50 dollars though i would have to send it back for sure

10. ### The Electrician AAC Fanatic!

Oct 9, 2007
2,487
384
What book are you referring to? The F W Grover book?

What inductance geometries are you referring to in this thread? Which Wikipedia page has the formulas for said geometries?

When testing formulas it's good to have a known exact result to compare. Have you got a source for some known results? The Grover book is a good source for some known results.

11. ### MrAl Thread Starter Distinguished Member

Jun 17, 2014
4,238
924
Hi,

Yes that's the book. It's not bad, but i had to go to references outside the book to get answers to problems i should have been able to use the book alone for. I attribute this to the age of the writing which back then tried to meet certain simplified goals that are no longer really needed today with modern computers and computer programs. In other words, a modern version of that book would probably be an ideal book on inductor calculations from physical constructions.

The main geometry is the solenoid coil, single layer. There are also mutual inductance calculations of interest but i hold off on that for now because my main ooncern was to test an old formula, and i happened to run into some formulas on wikipedia:
https://en.wikipedia.org/wiki/Inductance#Self-inductance_of_simple_electrical_circuits_in_air

especially the first two for the solenoidal single layer coil. I may have been able to confirm that first one now though as it looks like the integration they are using involves the squared parameter not the non-squared parameter which we talked about previously.

I have other formulas from other books, most notably a formula from W.B. Boast from an EE handbook that came in graphical form but i was able to extract and curve fit to turn it into a straightforward formula. The results from this calculation match my own formula as well as another regular formula which we've all seen:
L3=(r^2*N^2)/(9*r+10*L)*1e-6 --single layer cylindrical {inches, Henries}

These three match within reason so if i see a formula that is not close to those results, which come from vastly different sources, i have to conclude that it is not right (for reasonable constructions).
Sorry to say that since they match so well (as the second one on Wikipedia) that they must all be derived from the same root source, namely the current sheet approximation. This doesnt bother me too much right now though, but later i will probably want to update that, depending on available time. So the current sheet version is considered acceptable for now.

What was interesting so far is that the simple formula above matched the others to a decent degree of accuracy. Considering the mathematical operations that would have to go into a brute force field calculation, that formula almost mocks the physical science behind it

Back to the Grover book:
I dont want to make it sound like the book is 'bad', it's not bad, it's just a little outdated that's all. It's still very useful even today. A modern book would probably have programs in it for the calculations, as well as not be afraid of using integrals in the calculations. Back then integrations of some kinds were harder to do on a regular basis, so the idea was to look for simplifications. The simplifications make the calculation seem easier, but it's always at the cost of accuracy. This is the very problem i ran into almost immediately. The calculation for Nagaoka's constant was given as three term series. Maybe i didnt do it right? I'll double check that. Maybe i was just looking for higher accuracy than what was considered adequate for the time period of the book. [UPDATE: I did it right but was looking for better than three or four digit accuracy for better theoretical comparison to other formulas even though in real life we would not be able to get that accuracy with these kinds of formulas].

I'll provide some numerical results later if you want to try to calculate some of these too.
For the formulas i tried now, i get a result of 1.26uH for a given construction for example and very little variation from that with any formula.

Last edited: May 5, 2017