Inductor coil parasitic capacitance, using the cosh function - diyAudio
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Old 11th January 2017, 05:53 PM   #1
AndrewT is offline AndrewT  Scotland
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Default Inductor coil parasitic capacitance, using the cosh function

I have a formula for predicting the capacitance of an air core inductor.

Cparallel (pF per turn) = B * Er / {11.45*cosh^-1(S/D)}
where B = coil diameter mm
Er = dielectric constant (of air) = 1
S = gap between the coil turns mm
D wire diameter mm
I think cosh^-1 is the inverse cosh function.
Can I get a scientific calculator to give that answer, when S/D = 1?
If I press
2ndF, then hyp, then cos (1), I get 0
or
hyp, then 2dF then cos (1), I get 0
or
hyp, then cos (1), I get 1.54

Any idea if any of these answers could be correct?
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Last edited by AndrewT; 11th January 2017 at 05:58 PM.
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Old 11th January 2017, 06:44 PM   #2
Waly is offline Waly  Umm al-Qaiwan
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Quote:
Originally Posted by AndrewT View Post
I have a formula for predicting the capacitance of an air core inductor.

Cparallel (pF per turn) = B * Er / {11.45*cosh^-1(S/D)}
where B = coil diameter mm
Er = dielectric constant (of air) = 1
S = gap between the coil turns mm
D wire diameter mm
I think cosh^-1 is the inverse cosh function.
Can I get a scientific calculator to give that answer, when S/D = 1?
If I press
2ndF, then hyp, then cos (1), I get 0
or
hyp, then 2dF then cos (1), I get 0
or
hyp, then cos (1), I get 1.54

Any idea if any of these answers could be correct?
cosh(x)=(e^x+e^(-x))/2=(e^x+1/e^x)/2

If cosh(x)=1, then the only option is e^x=e^(-x)=1 (since y+1/y=2 has y1=y2=1 as solution), which in turn means x=0.
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Old 12th January 2017, 05:28 AM   #3
AndrewT is offline AndrewT  Scotland
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Unfortunately that makes the C/turn to be infinity since the cosh^-1 is in the divisor.
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Old 12th January 2017, 09:38 AM   #4
DF96 is offline DF96  England
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I guess that means the formula is wrong.

Is it possible that you have misunderstood the formula, and perhaps S is not the space between turns but the distance from wire centre to wire centre? That means that turns which are touching will have S=D, at which point you might expect quite high capacitance. Not infinite capacitance, but the formula probably only works when there is some space.
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Old 12th January 2017, 09:47 AM   #5
AndrewT is offline AndrewT  Scotland
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I copied the formula from here and although they say that S = distance between , the drawing does show what you suggested. Centre to centre rather than gap/space.
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File Type: png capacitance of coil.png (74.1 KB, 86 views)
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Old 12th January 2017, 09:50 AM   #6
AndrewT is offline AndrewT  Scotland
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Back to the question: how can I use a scientific calculator to find cosh^-1?
so that I can see comparisons of close spaced coils with wider spaced coils.
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Old 12th January 2017, 10:29 AM   #7
DF96 is offline DF96  England
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On my calculator (Casio fx-570) it appears that 'hyp' and 'inv' can be pressed in either order to get inverse cosh. I suggest you play around with a few known values (cosh itself is easy to calculate, as it is the average of two exponentials).
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Old 12th January 2017, 10:46 AM   #8
AndrewT is offline AndrewT  Scotland
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My Sharp does the same:
hyp > 2ndF > cos (number) gives the same result as 2ndF > hyp > cos (number).

And for S/D = 1.005 (very thin insulation on touching coils) cosh^-1 (1.005) results in 0.09996
and for big gaps equal to wire diameter cosh^-1 (2) = 1.317
i.e. the capacitance has gone down by 92%. That's an enormous reduction in parasitic capacitance. But the inductance will also have gone down so I would need more coils pushing the capacitance back up a bit.

Thanks Waly & DF.

However there are a few discussions on calculating parasitic capacitance of single layer coils and there seems to be much disagreement on what formulae can be relied on to give sensible answers.
One formula gives Cparasitic = coil diameter * X
irrespective of the number of turns nor of the wire diameter.
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Last edited by AndrewT; 12th January 2017 at 10:54 AM.
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Old 12th January 2017, 10:47 AM   #9
Elvee is offline Elvee  Belgium
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Quote:
Originally Posted by AndrewT View Post
Back to the question: how can I use a scientific calculator to find cosh^-1?.
If it isn't on your physical calculator, you can use the windows scientific calculator:
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Old 12th January 2017, 11:10 AM   #10
AndrewT is offline AndrewT  Scotland
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Again, thanks.

I use the scientific Windows calculator and it loads up that way why I open it.
But I forgot to go and look !

And it gives the same answer, provided I enter the number first and then inv > cosh.
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