Coil Theory

[domesticextremist]

'Ahem.
This is my theory, what it is...'

(Name that skecth ).

Seriously, getting into rebuildables I've been thinking about coils
a lot, possibly more than is healthy.
Once you've navigated your way through the labyrinth of bewilderment
of wick, wire and mesh and got all the bits in place, the next question
everybody asks is 'How many turns of wire do I need'.

Well, this thread tries to answer that question.

I agonised for a while about posting this because it involves maths
which might put people off, even though it ends with a simple result
and also because you all might think I'm mad for even trying

Anyway, here goes. there will be diagrams and mathematical derivations
created using the open source PP* software cos it was easiest, but I can't
draw for toffee and my maths is very rusty so lower your expectations.


*PP= pen and paper

[domesticextremist]

Basics.

You can't really cover a topic like this without telling folks stuff
they already know, so bear with me. Or, if your a newbie, this is
the stuff nobody writes down 'because it's obvious'.


So, egg sucking.

Yer basic mod:



Basically, current flows through the coil to heat it.
The resistance of the coil determines how hot the coil and
vapour gets and how fast the battery drains.

Typically the coils have resistance in the range 1.5 to 3.0 ohms.

The resistance of the coil is determined by the kind of wire it is
made from and the (resistive) length of the wire. Because we twist it
into a coil it them becomes a function of the number of turns in the coil,
but it it the length of the wire that makes all the difference.

The fundamental property we are concerned with then is resistance
per unit length* (which I will label U) and is expressed in Ohms per metre,
though it is more convenient to use Ohms per millimetre for our purposes.

Your wire supplier should be able to supply U, otherwise if you know what
it is made of and the diameter, Google is your friend.

I'm only familiar with two types of wire so far - Nichrome( NiCR) and
Kanthal (Ka).

Wire	diameter (mm)	U (Ohm/m)	U (Ohm/mm)
NiCr	0.15	53.7	0.0537
Ka	0.20	44.5	0.0445

(*I haven't been able to a proper scientific name for this property,
nor an official symbol, so we will call it 'resistance per unit length' and
use the symbol U as shorthand).

[domesticextremist]

So if we know U for our wire, we can calculate how long a
bit we need to get to a certain resistance:



We simply divide our target resistance by U to get the length of wire required to generate that resistance.

Except it isn't that simple. generally, the coil making process is as follows:

• Chop off some wire
  
• Wind the coil
  
• Fix in place
  
• Trim off any leftovers
  

So even if we measured it out at first, we have no idea how
much wire is left after we've fixed it in place and trimmed off
the loose ends.

So that's not much help...

[domesticextremist]

So maybe we can calculate how long the coil is from first principles.

Suppose we have an ideal coil, which is perfectly cylindrical with a
diameter d, a pitch p(the distance between turns) and a
number of turns n:




So our resistive length is given by:



Now, in many cases the tails don't exist because the ends of the coil are
clamped directly against the terminal posts (e.g Vision, Vertigo) or they
are made of no resistance wire (e.g. Penelope, Odysseus).
But on a Genesis or DID type atty they are unavoidable.
For instance two 2mm tails from 0.15 NiCr would add 0.2 Ohms to the
total resistance, so not always inconsiderable.

If we cut it along the dotted line and flatten it out, it looks like this:



So now we can calculate the length of the coil section (L_c) as being composed of n segments of length l_c

[domesticextremist]

Phew, time for a cuppa.
More in a minute.

[domesticextremist]

That's better. Everybody paying attention?
There'll be a test at the end

Right, where was I?


So now we can calculate the length of the coil section (L_c) as being composed of n segments of length l_c

So that is to say our coil consists of n basic elements:



and we need to work out the length of the red line:



Which is a bit of a pig to work with, even with a calculator, and still
doesn't directly tell me how many turns we need. Gmph!

However, after plugging in a few numbers and generally thinking about it,
I realised that the pitch term doesn't make that much difference.
Basically, unless you have a really narrow coil with a very long pitch
(basically a slightly curled bit of straight wire) then the pitch doesn't really
add that much to the length of the coil and can be ignored and we
still have a good approximation if we consider our coil to be a stack of n
circular hoops.

[DiscoDes]

Pythagoras was half square

[domesticextremist]

Aug 29, 2012 15:18:11 GMT DiscoDes said:

Pythagoras was half square

Quiet at the back!

[DiscoDes]

here's how I do it:

1. Wind a coil 4 turns round a wick.
2. Use a multimeter, one probe on the cut end one probe roughly where you are going to cut, BEFORE cutting this end.
3. Measure resistance with multimeter.
4. if too low add another turn.
5. if too high take off a turn.
6. Cut wire when desired resistance is reached.
7. Sit back smug knowing I didn't even get a CSE pass in maths.





Seriously, nice work, though I did get a little bogged down towards the end, I get number blindness when it gets more complicated than straight addition/multiplication/division though I'm pretty good at percentages after being in Sales for many years.

[domesticextremist]

So how much of an approximation is it?
Going back to our basic element:



If the angle A is 10 degrees, then



That is to say that the circumference one circular hoop is 98.5% of our
coil segment, which isn't too bad.

Even if the pitch of the coil is the same as the diameter - which would be
a pretty loose coil:



then it is still 95.3%. Let's remember we're more doing cookery than rocket science and a five percent error is readily bearable.

So, we can ignore pitch and simplify the expression for the coil length:



Which is a lot simpler

[Gordy]

For instance two 2mm tails from 0.15 NiCr would add 0.2 Ohms to the
total resistance, so not always inconsiderable.

should that be 0.1 Ohms?

[domesticextremist]

Now, if we ignore the effect of any tails, and approximate away the
contribution the coil pitch, then the length of the coil is the total resistive
length, as in equation 4 above.

But we already have an equation for resistive length in equation 1 (way
back up there), so we can stick the two together and solve it for the
number of turns:





Wahey!

Or put into words:

[domesticextremist]

So, there you have it.



The answer, to the question

'How many turns do I need?'

is

'Turns equals res over pud'!

Ithangyou.

[DiscoDes]

Well done!

#icon_thumbsup# #icon_thumbsup# #icon_thumbsup#

[domesticextremist]

Aug 29, 2012 15:37:25 GMT Gordy said:

For instance two 2mm tails from 0.15 NiCr would add 0.2 Ohms to the
total resistance, so not always inconsiderable.

should that be 0.1 Ohms?

No there's two of them (4mm).
A bit extreme I know, but I've made sloppy coils before