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Old 19th Oct 2017, 6:50 pm   #1
Al (astral highway)
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Default My solution to RF/EHT cable diameter dilemma, for those interested!

Split from this thread, for those interested.

As part of the original thread, I was hedging between 2mm dia wire (14SWG) and a thicker grade. The necessary ampacity, I stated, was around 55A at HF (300KHz).

This is a working estimate based on the low-impedance power supply sourcing 1A into a tank circuit with a Q of 55, let's call it 60 for good measure.

Of course, it might be as low as 10 in practice, but I won't know that until I've actually built it.

I wondered if others would be interested in the thought experiment I completed to decide that the 2mm dia wire would be fine?

Well.. Here goes.

*The inductor in the tank circuit has a design inductance of 140uH.

*Whatever wire is used, that's the design figure. So more turns, more spaced turns, thicker or thinner wire and more strands of wire are all potential variables, but I was always only concerned with double-enamelled wire, single strand.

That's good to 200 degrees centigrade. It doesn't have a 'jacket' as Argus25 describes insulated wires (like it!) so fire is not an issue as with PVC covered wire.

*The substrate (form) is heat-resistant as coated with stove enamel, baked at 70 degrees. I think the paint is good to 400 degrees centigrade, so plenty of headroom.

As I thought around this, I considered how a single strand of very thin wire is used as a 13A fuse in domestic circuits. It reminded me that ampacity with wires is not so much to do with their electron-carrying capacity, but rather more to do with I^2R (heating) losses, otherwise the relatively tiddly size of every fuse would be the absolute determinant of electron carrying capacity in the entire circuit.

That isn't what happens at all... whatever concentration of joules/cross sectional area, the wire carries on taking what's thrown at it, heating and heating, until it fuses. A long time before then, there's a point of equilibrium where the wire is heating up but also radiating heat at the same rate. Good news for nice thick enamelled wires with spacing between turns...

The rate of heating is fiddly to determine - another part of a thread, perhaps, but I did take a look. With 14SWG, fusing eventually happens with a continuous DC current of 166A (sorry, I can't remember the source, but we see the picture...)

On the other hand, here is an RF circuit, where skin effect is quantifiable and significant. Russell_w_b helpfully pointed out that at the design frequency of 0.3MHz, skin effect means resistance is 3.7 times normal DC resistance.

But this alone wouldn't get us very far, as 3.7 times 0.00532R per metre (14SWG wire) doesn't amount to very much in a small inductor. I needed to make another comparison.

Here's what I did.

I compared two theoretical windings, winding 1 and winding 2, both with the same inductance, 140uH. I considered their winding length for the same inductance, including some extra for 'stretch' between windings.

I use this applet, which allows for different coil diameters and winding lengths and also determines Q of the inductor (not to be confused with the Q of the tank circuit) at the design frequency.

Winding 1 - 14 SWG, or 2mm OD, using the relevant dimensions, takes up 13.85metres of winding over 46.2mm winding length.

Winding 2, with a whopping 4mm OD, takes up 17.5 metres of winding over 110mm winding length.

But here's the thing... the inductances are the same. So, all we need now is the inductive reactance, X_L

Which...
=2 * pi * f * L
=6.28 * 3e3 * 140e-6 = 264 R

And so now we can get the resistance at RF per metre for comparison.

Winding 1 has impedance 264/13.85=19R per metre

Winding 2 has impedance 264/17.5 =15.1R per metre

So that gives us a a bit of a 'cost/benefit'/ effort background on which to make a decision. Both windings get 'rather hot' if actually fed with continuously with 55A at this frequency, and there are unacceptable heating losses.

On the other hand, they won't see that current continuously, or anything like it. The whole circuit is class C, so even if I ran it in CW, the heating losses would be much lower.

Further, I'm planning on operating it in a pulse mode with a low duty cycle, maybe 10 to 25%.

So, although it looks worthwhile trading up without doing any deep thinking, it actually doesn't look like a necessary plan when we do.

That's me. I'm just sharing this for those who might have any level of interest, as decision-making is something we all do on this forum the whole time!
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Old 19th Oct 2017, 8:57 pm   #2
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Default Re: My solution to RF/EHT cable diameter dilemma, for those interested!

Interesting - bet that kept you quet for five minutes!

If you wish to wind an inductor to the required value with minimum resistance (shortest length of wire for your 'L' and hence higher Q) then make it so the diameter is 2.46 x the length, although having a squat, flatter coil may not be what you want.

Wheeler's formula (below) should get you away to a reasonable starting point, although it is really based on a flat 'current sheet', not round wire. But as long as the length is greater than 0.8 of the radius, you'll get about 1% accuracy. I can vouch for this, having measured many single-layer coils of varying sizes with both bridge and tape-measure!

Inductance (uH) = ( radius^2 x No. of turns ^2) / ((9 x radius) + (10 x coil length)).
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Old 20th Oct 2017, 7:38 am   #3
Al (astral highway)
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Default Re: My solution to RF/EHT cable diameter dilemma, for those interested!

That's useful insight on the ratio of diameter to length and its effect on inductance, Russell. I'd love to wind a flat pancake indicator for the tank but I think it would just be too space-consuming.

Thank you. I am familiar with Wheeler's formula but I didn't know, and it's reassuring, that it can be quite so precise and accurate (not the same things!) to ad close as 1%. Truly astonishing!

All: please note small typo in my sum: it is of course 300e3 in the second line - distracted transcribig it from my notes , but the X_L result is correct as I did the calc based on the correct figure.
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Old 20th Oct 2017, 7:51 am   #4
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Default Re: My solution to RF/EHT cable diameter dilemma, for those interested!

I glossed over that one. I use Nagaoka's formula, which, I've found, is even more accurate - it factors in pitch and cable diameter too - and it's handy when an inductor can be measured accurately (for all practical purposes) with a tape-measure instead of carting an impedance bridge around.

Nagaoka relies on a table for values but I put the mechanics of it on a spreadsheet along with the tabulated values, and I use linear interpolation to pull out the relevant figure.
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Old 20th Oct 2017, 9:35 pm   #5
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Default Re: My solution to RF/EHT cable diameter dilemma, for those interested!

Quote:
Originally Posted by russell_w_b View Post
I use Nagaoka's formula ...

Nagaoka relies on a table for values but I put the mechanics of it on a spreadsheet along with the tabulated values, and I use linear interpolation to pull out the relevant figure.

I hadn't heard of Nagaoka, thank you. And linear interpolation from a spreadsheet! That's a true purist's route. But I love the idea that you've collected the underlying data, and it celebrates empiricism!
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Old 20th Oct 2017, 10:31 pm   #6
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Default Re: My solution to RF/EHT cable diameter dilemma, for those interested!

The spreadsheet takes the tedium out of things and it wasn't too tricky to put together. Even without the linear interpolation the results are close enough for jazz.

Various publications cite similar optimum form factors for designing inductors and it isn't too critical, with greatest loss being when the coil is long and thin and just a gentle roll-off the other way. Many coils I've been involved with are as about as long as they're wide. The 2 - 2.5 optimum ratio figure takes no account of inter-turn capacitance (and so self-resonance) and variation of a.c. resistance with frequency, but fixed coils are generally designed with this data acquired or calculated for the job in hand.
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