Inductance calculation

[rakibulraju]

hi i am doing my final year project.for this i need to design a inductor and i prefer circular air core inductor.i am using the following formula in the i have attached.but it is not giving the accurate inductance.please someone suggest me a formula for calculating the inductance of a circular air core coil?

 

[MrAl]

Hi,

There are a number of formulas out there on the web you can use, but you'll only get an approximation anyway because there are variables that are hard to control.

What is this going to be used for, and how accurate does it really have to be without some sort of adjustment? For highly accurate values usually there is some sort of adjustment so the circuit can be tuned.

 

[rakibulraju]

Hi,

There are a number of formulas out there on the web you can use, but you'll only get an appoximation anyway because there are variables that are hard to control.

What is this going to be used for, and how accurate does it really have to be without some sort of adjustment? For highly accurate values usually there is some sort of adjustment so the circuit can be tuned.

'I am going to use it for resonance of LC circuit with high frequency of about 2 MHZ'.so as I need the LC circuit to resonate at a certain frequency I need a value close to accurate value so that the resonant frequency is about certain.SO please suggest me a formula for circular air core coil inductance calculation.I am using the formula I have attached below.Is it currect.please suggest me.

 

[steveB]

I agree with MrAl 100%.

I would even state it more strongly and say you are on a fools errand if you expect a formula to give the precise inductance.

This is a notoriously difficult calculation to make, especially at high frequency. There is no closed form solution for the flux over the circular loop. Hence, formulas are just fits to numerical solutions, and they are not exact answers. But even exact theoretical answers don't always give you good estimates of reality.

Let's say you forgo the simple formula and do a numerical calculation. Even there, you are going to have difficulty getting accuracy. Notice how your formula has the wire diameter "a" in it. This means that when the wire diameter is significant compared to the loop diameter, the numerical integration you do is different inside the air part of the loop, compared to the wire part of the loop. The wire diameter is 3 dimensional and there is no guarantee that the current is distributed evenly across the cross section of the wire as frequency goes up. Also, multiple loops are not going to be exactly on top of each other and you need to integrate over the slightly offset loops. So, formulating the proper numerical method of solution is hard to do.

But, even if you do a correct and proper numerical solution and consider all effects, you may still not get the exact answer, but you will be closer than using approximate formulas.

So, you use these formulas to get you in the ball park of where you need to be, and then you adjust in the real experiment. If you have many loops, then just add or subtract loops as needed. If you don't have many loops, then you need to change the loop area because adding and subtracting loops makes too big a change.

 

[MrAl]

Hi again,

I'll quote a couple of 'book' formulas here just for reference. The formula you posted does not look correct. Maybe you could quote what each variable is for, but as is it looks wrong.

'I am going to use it for resonance of LC circuit with high frequency of about 2 MHZ'.so as I need the LC circuit to resonate at a certain frequency I need a value close to accurate value so that the resonant frequency is about certain.SO please suggest me a formula for circular air core coil inductance calculation.I am using the formula I have attached below.Is it currect.please suggest me.

Single Layer Cylindrical:
I=(R^2*N^2)/(9*R+10*L)

Multi-Layer Cylindrical (one of Wheeler's Formulas):
I=0.8*(R^2*N^2)/(6*R+9*L+10*B)

where
I is the inductance in microhenries,
R is the coil mean radius,
L is the length of the coil,
B is the build of the coil (multi layer),
and all length dimensions in inches.

I checked the second formula above by interpreting Bunet's formula and deriving a new formula based on his graph and associated sub formula, and the two formulas agree very very closely. Unfortunately, i can not find a reference for Bunet on the web anywhere. Perhaps it is time to add one

As Steve and myself have noted, these formulas and others like them (some decent ones on Wikipedia too) are approximations and should be taken only as a starting point for a design of a coil especially at higher frequencies.

To deal with the problem of inaccuracy build a coil and then provide a means for tuning. Since regular air core coils do not have a slug to slide in or out of the coil center, a common way to handle this is to design the coil so that it's inductance is too high and wind it on a coil form that is too long for the wire coil itself, then adjust the inductance by spreading the turns out slightly (which lowers the inductance) until the right value is attained. This is best done in the circuit it is going to be used for while it is running under normal operating conditions.

 

[The Electrician]

hi i am doing my final year project.for this i need to design a inductor and i prefer circular air core inductor.i am using the following formula in the i have attached.but it is not giving the accurate inductance.please someone suggest me a formula for calculating the inductance of a circular air core coil?

You haven't fully described what form of inductor you want. Is it a single layer solenoid, or a multiple layer solenoid, etc.?

You will find formulas aplenty in this document:

https://www.g3ynh.info/zdocs/refs/NBS/Sci169noerr.pdf

 

[steveB]

I believe that the OP is just talking about a simple circular loop coil. I've seen that formula before, and it was for a circular loop, as he stated.

 

[MrAl]

Hi again,

Steve:
The formula without the N^2 is for a single turn, with N^2 is approximates a multiturn coil. However, i think the constant (2) is incorrect, and possibly other errors. We'll have to look at Electricians link better.

In Electrician's link, it is very very interesting that in examples 52 and beyond a little they obtained inductance values from the formulas in units of length: centimeters <har har>. For example, L=654pi centimeters instead of L=654pi nanohenries
Funny too as this error seems to carry over several other examples. Glad we had such people working on papers like this way back then

Seriously though we should take a look at those formulas and see what gives. Are they as accurate as they claim? We should try a few and compare to other formulas and see what gives. I smell something fishy somewhere

I also believe that Maxwell made a slight error in one of his calculations way back when, but i dont think that formula appears in the link. It's a long paper though so it's a lot to read in five minutes. Looks very interesting just the same.

 

[The Electrician]

I believe that the OP is just talking about a simple circular loop coil. I've seen that formula before, and it was for a circular loop, as he stated.

Well, with all due respect, he didn't say circular loop; he said:

"hi i am doing my final year project.for this i need to design a inductor and i prefer circular air core inductor.i am using the following formula in the i have attached.but it is not giving the accurate inductance.please someone suggest me a formula for calculating the inductance of a circular air core coil?"

To me the phrase "circular air core inductor (or coil)" doesn't necessarily equate to a single turn loop, and he didn't actually say "single turn loop".

I also noticed that the formula he gave is the high frequency version of formula 119b on page 143 of Grover's book, but a single turn circular loop of reasonable size isn't going to resonate at 2 MHz with a reasonable size capacitor.

A typical tuning cap in an old broadcast band receiver had a maximum capacitance of 365 uuf (as they used to call it). To resonate 365 pF at 2 MHz requires 17 uH. To get 17 uH from a single turn loop requires a loop about 3 meters in diameter. Or he could use a much smaller single turn loop with a much larger capacitance.

Even MrAl assumed a cylindrical solenoid; I don't know what to assume.

But, maybe he really does intend a single turn loop; maybe it's going to be a loop antenna.

It's not clear to me what he wants, and as is so unfortunately typical in these posts, he doesn't give us complete information about what he's doing and where the inductor figures into his project.

At any rate, I gave him a link to the ultimate source of inductance formulas. If he comes back with more specific questions, we can offer more specific help.

 

[steveB]

Hi The Electrician,

I don't disagree with you. I was just trying to clarify the OP's intent as I saw it. Actually, I didn't mean to imply that it was a single turn loop. By "simple" I didn't mean one turn, but only that the turns would overlap, rather than extend longitudinally like a solenoid. The formula includes N^2, so i assume he is considering the possibility of multiple loops. His words were not entirely clear, but combining his words with the formula was enough for me to zero in on what is likely (but not certainly) his question.

Sorry for the confusion. Let's see what the OP says about it.

Thanks,
Steve

 

[The Electrician]

By "simple" I didn't mean one turn, but only that the turns would overlap, rather than extend longitudinally like a solenoid.

So are you describing what is sometimes called a "pancake", "disc", or "spiral" coil? Like these:

https://www.instructables.com/community/Pancake-Coil/

With a calculator for those coils given here?:

https://www.deepfriedneon.com/tesla_f_calcspiral.html

 

[steveB]

No, that's not quite what I was thinking. I'm just thinking of multiple loops with the same circular shape and all with the same radius. I guess the confusing thing is that multiple loops can't occupy the same space, so the geometry of the real coil is messy. It's a little like a spiral and a little like a solenoid, but not really. It's actually a messy group of turns. I think in practice the number of turns is usually small for this reason.

The first coil I ever made was of this type. I was about 14 years old I think. I knew almost nothing about electronics because I was just starting as a hobby. I just took magnet wire of relatively light gauge and wound a couple of hundred turns around my hand. Then I soldered a wall plug on it and taped it up. I could plug this in the wall for several seconds without it overheating and I used it to magnetize screwdrivers so the screws would stick to them. (by the way, don't recommend anyone do this. It's dangerous and I was stupid to do it, but we often do stupid things when we are young)

I can't speak to the suitability of the given formula with proper ranges for parameters, but I just know I've seen that formula a few times for this type of coil.

 

[rakibulraju]

Hi again,

I'll quote a couple of 'book' formul

Hi again,

I'll quote a couple of 'book' formulas here just for reference. The formula you posted does not look correct. Maybe you could quote what each variable is for, but as is it looks wrong.



Single Layer Cylindrical:
I=(R^2*N^2)/(9*R+10*L)

Multi-Layer Cylindrical (one of Wheeler's Formulas):
I=0.8*(R^2*N^2)/(6*R+9*L+10*B)

where
I is the inductance in microhenries,
R is the coil mean radius,
L is the length of the coil,
B is the build of the coil (multi layer),
and all length dimensions in inches.

I checked the second formula above by interpreting Bunet's formula and deriving a new formula based on his graph and associated sub formula, and the two formulas agree very very closely. Unfortunately, i can not find a reference for Bunet on the web anywhere. Perhaps it is time to add one

As Steve and myself have noted, these formulas and others like them (some decent ones on Wikipedia too) are approximations and should be taken only as a starting point for a design of a coil especially at higher frequencies.

To deal with the problem of inaccuracy build a coil and then provide a means for tuning. Since regular air core coils do not have a slug to slide in or out of the coil center, a common way to handle this is to design the coil so that it's inductance is too high and wind it on a coil form that is too long for the wire coil itself, then adjust the inductance by spreading the turns out slightly (which lowers the inductance) until the right value is attained. This is best done in the circuit it is going to be used for while it is running under normal operating conditions.

as here just for reference. The formula you posted does not look correct. Maybe you could quote what each variable is for, but as is it looks wrong.



Single Layer Cylindrical:
I=(R^2*N^2)/(9*R+10*L)

Multi-Layer Cylindrical (one of Wheeler's Formulas):
I=0.8*(R^2*N^2)/(6*R+9*L+10*B)

where
I is the inductance in microhenries,
R is the coil mean radius,
L is the length of the coil,
B is the build of the coil (multi layer),
and all length dimensions in inches.

I checked the second formula above by interpreting Bunet's formula and deriving a new formula based on his graph and associated sub formula, and the two formulas agree very very closely. Unfortunately, i can not find a reference for Bunet on the web anywhere. Perhaps it is time to add one

As Steve and myself have noted, these formulas and others like them (some decent ones on Wikipedia too) are approximations and should be taken only as a starting point for a design of a coil especially at higher frequencies.

To deal with the problem of inaccuracy build a coil and then provide a means for tuning. Since regular air core coils do not have a slug to slide in or out of the coil center, a common way to handle this is to design the coil so that it's inductance is too high and wind it on a coil form that is too long for the wire coil itself, then adjust the inductance by spreading the turns out slightly (which lowers the inductance) until the right value is attained. This is best done in the circuit it is going to be used for while it is running under normal operating conditions.

"I am actually trying to make circular air core inductor not cylindrical.I prefer circular coil than cylindrical because in my case I using it as a antenna.In case of cylindrical coil I know radiates omnidirectionally but circular radiates across it's axis and I need to focus to my secondary coil.I am doing wireless electricity transfer project.In this I will use two coil respectively as primary and secondary .'

 

[rakibulraju]

Hi The Electrician,

I don't disagree with you. I was just trying to clarify the OP's intent as I saw it. Actually, I didn't mean to imply that it was a single turn loop. By "simple" I didn't mean one turn, but only that the turns would overlap, rather than extend longitudinally like a solenoid. The formula includes N^2, so i assume he is considering the possibility of multiple loops. His words were not entirely clear, but combining his words with the formula was enough for me to zero in on what is likely (but not certainly) his question.

Sorry for the confusion. Let's see what the OP says about it.

Thanks,
Steve

'It is correct that in my coil the turns would overlap, rather than extend longitudinally like a solenoid.So in this case when I am using more turns the width of the coils peripheral is being thick and the diameter of the coil is incresing.So for high frequency like 2MHZ application which formula I can use.was my formula correct.I am giving the link whrer I have got it from."**broken link removed**"

 

[rakibulraju]

No, that's not quite what I was thinking. I'm just thinking of multiple loops with the same circular shape and all with the same radius. I guess the confusing thing is that multiple loops can't occupy the same space, so the geometry of the real coil is messy. It's a little like a spiral and a little like a solenoid, but not really. It's actually a messy group of turns. I think in practice the number of turns is usually small for this reason.

The first coil I ever made was of this type. I was about 14 years old I think. I knew almost nothing about electronics because I was just starting as a hobby. I just took magnet wire of relatively light gauge and wound a couple of hundred turns around my hand. Then I soldered a wall plug on it and taped it up. I could plug this in the wall for several seconds without it overheating and I used it to magnetize screwdrivers so the screws would stick to them. (by the way, don't recommend anyone do this. It's dangerous and I was stupid to do it, but we often do stupid things when we are young)

I can't speak to the suitability of the given formula with proper ranges for parameters, but I just know I've seen that formula a few times for this type of coil.

''It is correct that in my coil the turns would overlap, rather than extend longitudinally like a solenoid.So in this case when I am using more turns the width of the coils peripheral is being thick and the diameter of the coil is incresing.So for high frequency like 2MHZ application which formula I can use.was my formula correct.I am giving the link whrer I have got it from."**broken link removed**"

 

[rakibulraju]

No, that's not quite what I was thinking. I'm just thinking of multiple loops with the same circular shape and all with the same radius. I guess the confusing thing is that multiple loops can't occupy the same space, so the geometry of the real coil is messy. It's a little like a spiral and a little like a solenoid, but not really. It's actually a messy group of turns. I think in practice the number of turns is usually small for this reason.

The first coil I ever made was of this type. I was about 14 years old I think. I knew almost nothing about electronics because I was just starting as a hobby. I just took magnet wire of relatively light gauge and wound a couple of hundred turns around my hand. Then I soldered a wall plug on it and taped it up. I could plug this in the wall for several seconds without it overheating and I used it to magnetize screwdrivers so the screws would stick to them. (by the way, don't recommend anyone do this. It's dangerous and I was stupid to do it, but we often do stupid things when we are young)

I can't speak to the suitability of the given formula with proper ranges for parameters, but I just know I've seen that formula a few times for this type of coil.

''It is correct that in my coil the turns would overlap, rather than extend longitudinally like a solenoid.So in this case when I am using more turns the width of the coils peripheral is being thick and the diameter of the coil is incresing.So for high frequency like 2MHZ application which formula I can use.was my formula correct.I am giving the link whrer I have got it from."**broken link removed**"

 

[rakibulraju]

Well, with all due respect, he didn't say circular loop; he said:

"hi i am doing my final year project.for this i need to design a inductor and i prefer circular air core inductor.i am using the following formula in the i have attached.but it is not giving the accurate inductance.please someone suggest me a formula for calculating the inductance of a circular air core coil?"

To me the phrase "circular air core inductor (or coil)" doesn't necessarily equate to a single turn loop, and he didn't actually say "single turn loop".

I also noticed that the formula he gave is the high frequency version of formula 119b on page 143 of Grover's book, but a single turn circular loop of reasonable size isn't going to resonate at 2 MHz with a reasonable size capacitor.

A typical tuning cap in an old broadcast band receiver had a maximum capacitance of 365 uuf (as they used to call it). To resonate 365 pF at 2 MHz requires 17 uH. To get 17 uH from a single turn loop requires a loop about 3 meters in diameter. Or he could use a much smaller single turn loop with a much larger capacitance.

Even MrAl assumed a cylindrical solenoid; I don't know what to assume.

But, maybe he really does intend a single turn loop; maybe it's going to be a loop antenna.

It's not clear to me what he wants, and as is so unfortunately typical in these posts, he doesn't give us complete information about what he's doing and where the inductor figures into his project.

At any rate, I gave him a link to the ultimate source of inductance formulas. If he comes back with more specific questions, we can offer more specific help.

Brother I am trying to make wireless electricity transfer device in which the primary and secondary which are both LC coil should resonate at the same frequency.In that purpose i need a inductor coil I am saying about.Is there any problem with using larger value of capacitor and smaller value of inductance.Is there any problem with using larger value of capacitance?

It is correct that in my coil the turns would overlap, rather than extend longitudinally like a solenoid.So in this case when I am using more turns the width of the coils peripheral is being thick and the diameter of the coil is incresing.So for high frequency like 2MHZ application which formula I can use.was my formula correct.I am giving the link whrer I have got it from."**broken link removed**"

 

[The Electrician]

No, that's not quite what I was thinking. I'm just thinking of multiple loops with the same circular shape and all with the same radius. I guess the confusing thing is that multiple loops can't occupy the same space, so the geometry of the real coil is messy. It's a little like a spiral and a little like a solenoid, but not really. It's actually a messy group of turns. I think in practice the number of turns is usually small for this reason.

<SNIP>

I can't speak to the suitability of the given formula with proper ranges for parameters, but I just know I've seen that formula a few times for this type of coil.

This sounds like what I've always heard described as "scramble wound".

The link the OP gives shows a "coil" that appears to have multiple turns in the space normally occupied by only a single turn, a physical impossibility. The circle representing the coil is very small; I don't see how there can be multiple turns in that space, so the given formula would seem to be inapplicable because a real coil will occupy a much larger volume.

 

[MrAl]

Hi again,

Well the formula you presented was for a cylindrical coil not a flat coil, so naturally i assumed you wanted a coil like that.

Just to recap to the other readers, the formulas with "N^2" are for multi turn coils. I stated this in my post before this but apparently you missed that. If the number is 1 then of course the formula simplifies.

So now that we have been informed about the new coil shape, here is a book formula for a flat coil:
I=(R^2*N^2)/(8*R+11*B) Multilayer of length one wire diameter (flat coil)

Here again I is the inductance in uH, R is the coil mean radius, and B is the coil build height,
So for 1/2 inch diameter coil 'form' and many turns of wire bringing the outer diameter to 1 inch, the mean radius would be 0.375 inches and the build would be 0.25 inches.

Did anyone check any of the formulas in the link yet (assuming the correct units for the inductance that is) ?

 

[The Electrician]

hi i am doing my final year project.for this i need to design a inductor and i prefer circular air core inductor.i am using the following formula in the i have attached.but it is not giving the accurate inductance.please someone suggest me a formula for calculating the inductance of a circular air core coil?

Why don't you post a picture of your coil so we can see what it really looks like? Also, tell us more about its construction. What gauge wire did you use? How many turns are there? What are the dimensions of the coil?