# Calculating mean path lengths in a magnetic circuit. Can it be this simple?

#### Noodler

Joined Dec 29, 2015
5
I am studying electronic principles and currently electromagnetic induction.

I have been having some trouble establishing mean path lengths and cant seem to find any information in my course notes.

I have looked on this forum and found what I think is some good information, but it looks a bit simple.
I have applied this to my question with the result showing as 73mm, please see attachment for question and my working out.
I think the hole or partial penetration in the plate at the bottom of the loop may be relevant to later questions regarding flux density
and MMF, but I am not sure.
Does this look right? it seems too simple.

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#### Kermit2

Joined Feb 5, 2010
4,162
I don't think that simple formula works if all 4 sides are not the same thickness. One side of your square has a 1mm thick side.
I got a slightly larger answer by deducing the length of your vertical path to be 17.5 mm and the horizontal path to be 21 mm. Do you see where I got the 17.5 mm?

#### Noodler

Joined Dec 29, 2015
5
I don't think that simple formula works if all 4 sides are not the same thickness. One side of your square has a 1mm thick side.
I got a slightly larger answer by deducing the length of your vertical path to be 17.5 mm and the horizontal path to be 21 mm. Do you see where I got the 17.5 mm?
I think I may becoming a bit number blind, I am attempting to resolve issues surrounding flux density and MMF for the circuit also, but with very little success at all. It may well be time for a break.

I cant see where the 17.5mm is from on the vertical path.

I am assuming that other than the 1mm section on the bottom, the other three sides are all 6mm.
As such the long vertical path looks like it has an overall length of 20mm plus 1mm. Giving 21mm.
Whilst the short path would be 21 minus the 6 at the top and the 1 at the bottom. Giving 14mm.
Regarding the horizontal path, the long path appears to be 25mm, whilst the short is this figure minus 2 x 6mm. Giving 13.

#### Kermit2

Joined Feb 5, 2010
4,162
You have a vertical distance of 20mm plus the 1mm steel plate. The mean path will start at the mid point of the upper section. This point is 3mm inside the upper leg. The coresponding halfway point on the bottom leg is 1/2 mm inside. So we have 20 minus 3 and an additional 1/2 mm. This adds to 17.5 mm of vertical distance on each side.
And DAMN IT.
The calculator gave me 77 mm as an answer the first time and gives me 73 mm now.

19+19+17.5+17.5=73

I think I have data entry dyslexia

#### Noodler

Joined Dec 29, 2015
5
You have a vertical distance of 20mm plus the 1mm steel plate. The mean path will start at the mid point of the upper section. This point is 3mm inside the upper leg. The coresponding halfway point on the bottom leg is 1/2 mm inside. So we have 20 minus 3 and an additional 1/2 mm. This adds to 17.5 mm of vertical distance on each side.
And DAMN IT.
The calculator gave me 77 mm as an answer the first time and gives me 73 mm now.

19+19+17.5+17.5=73

I think I have data entry dyslexia
Many thanks Kermit.
I have worked it both ways now and got the same answer.
The main thing is that it appears that I am actually finding the mean path length.
I was a little concerned, as it looked too simple in the end.

Thanks again, I can now continue to work the remainder of the problem.
That said I may well be back with more queries regarding my understanding or lack of it.

#### Jony130

Joined Feb 17, 2009
5,351
But for this circuit you have two mean path lengths. One length for Fe-Si and the second one for carbon steel plate.

#### MrAl

Joined Jun 17, 2014
9,179
Hi,

The mean magnetic path is just a measurement of length of the magnetic path, no matter what it goes through. There are other properties that are considered separately when all the items are not of the same material, such as the normal air gap (not this particular air gap however). The other properties would definitely include the width of the air gap (if it was a simple air gap) and for this problem the length of the steel plate. Those lengths would also need to be considered, but the mean path is still just the total distance a magnetic field line takes if it was located at the very center of the three dimensional sides. This single line is at the very center so for a side that has a square cross section 6mm by 6mm and any length the line is 3mm in from all surfaces of that one side. If the cross section was 6mm by 8mm then the line would be 3mm in from one surface and 4mm in from the other surface. But usually these problems only use 2d drawings so we only see 6mm or 8mm and then we assume that the other dimension is also uniform, so we just divide by 2 and get either 3mm or 4mm respectively.

So the first thing to do is calculate the mean magnetic path, then later we would have to know the length of the steel plate and perhaps the length of the core that is not the steel plate, but only if the permeability for each material was different, which they usually are. So it does make sense to note that there could be more than one length involved, but this problem seems to be asking for that one mean length.

The mean length that has already been noted follows a line at the very center of each arm of the construct, and runs the length of an arm from the center of the butt at one end to the butt of the other end. This has already been drawn out nicely in the attached pdf.

For a circular core the mean path is the sum of the outside circumference and the inside circumference, then divided by 2. For a square or rectangular construct it would be the same i would think, although we could prove this algebraically.

In any case though, the mean path here is 73 because no matter how you calculate it, the result is the same.

I'll see if i can work up a proof if i have time. For example, if the width of one side is A and outside length L, and the width of one of the butts is B and the other C, then the mean length of the side A is:
Lm=L-B/2-C/2

and we would just need an expression for the whole thing and another expression for the other method and then equate the two. If they equate algebraically, then they are the same. For that one side though we would also know the inside length K is:
K=L-B-C

so the mean this second way would be:
Lm=(L+K)/2
so:
Lm=(L+L-B-C)/2=(2*L-B-C)/2
so:
Lm=L-B/2-C/2

so it does seem to prove that both methods are the same because every side would be calculated using the same procedure. We might have to look at angled sides too, but that's probably the same since the circular shape works the same.

Last edited:

#### Noodler

Joined Dec 29, 2015
5
But for this circuit you have two mean path lengths. One length for Fe-Si and the second one for carbon steel plate.
Thank you.
I believe that the two different material types are related to another part of the question regarding calculation of MMF.
That said I could well be misinterpreting the question.
It asks what the MMF is in each part of the path of the circuit to achieve B in the specimen plate of 0.5T.
I have magnetization curves for the materials concerned.
It looks as if I have to work out MMF for the specimen plate with known dimensions and B and H values from the relevant curve.
Taking this answer as one path.
I think the remaining three paths (legs on the electromagnet) should all be the same and could be calculated using their dimensions and B and H values from their relevant curve.
I think this makes sense?
I believe that I have an equation here to calculate MMF, but it looks a wee bit complicated to me at the moment.
Thanks again.

#### Noodler

Joined Dec 29, 2015
5
Hi,

The mean magnetic path is just a measurement of length of the magnetic path, no matter what it goes through. There are other properties that are considered separately when all the items are not of the same material, such as the normal air gap (not this particular air gap however). The other properties would definitely include the width of the air gap (if it was a simple air gap) and for this problem the length of the steel plate. Those lengths would also need to be considered, but the mean path is still just the total distance a magnetic field line takes if it was located at the very center of the three dimensional sides. This single line is at the very center so for a side that has a square cross section 6mm by 6mm and any length the line is 3mm in from all surfaces of that one side. If the cross section was 6mm by 8mm then the line would be 3mm in from one surface and 4mm in from the other surface. But usually these problems only use 2d drawings so we only see 6mm or 8mm and then we assume that the other dimension is also uniform, so we just divide by 2 and get either 3mm or 4mm respectively.

So the first thing to do is calculate the mean magnetic path, then later we would have to know the length of the steel plate and perhaps the length of the core that is not the steel plate, but only if the permeability for each material was different, which they usually are. So it does make sense to note that there could be more than one length involved, but this problem seems to be asking for that one mean length.

The mean length that has already been noted follows a line at the very center of each arm of the construct, and runs the length of an arm from the center of the butt at one end to the butt of the other end. This has already been drawn out nicely in the attached pdf.

For a circular core the mean path is the sum of the outside circumference and the inside circumference, then divided by 2. For a square or rectangular construct it would be the same i would think, although we could prove this algebraically.

In any case though, the mean path here is 73 because no matter how you calculate it, the result is the same.

I'll see if i can work up a proof if i have time. For example, if the width of one side is A and outside length L, and the width of one of the butts is B and the other C, then the mean length of the side A is:
Lm=L-B/2-C/2

and we would just need an expression for the whole thing and another expression for the other method and then equate the two. If they equate algebraically, then they are the same. For that one side though we would also know the inside length K is:
K=L-B-C

so the mean this second way would be:
Lm=(L+K)/2
so:
Lm=(L+L-B-C)/2=(2*L-B-C)/2
so:
Lm=L-B/2-C/2

so it does seem to prove that both methods are the same because every side would be calculated using the same procedure. We might have to look at angled sides too, but that's probably the same since the circular shape works the same.
Thank you, it looks like I have the mean path length calculated correctly.
Your confirmation and the explanation of this is much appreciated.

I believe that the two different material types are related to another part of the question regarding calculation of MMF.
It asks what the MMF is in each part of the path of the circuit to achieve a certain B in the specimen plate.
I have magnetization curves for the materials concerned.
It looks as if I have to work out MMF for the specimen plate using known dimensions and B and H values from the relevant curve.
Taking this answer as one path.
I looks like the remaining three paths (on the electromagnet) should all be the same and could be calculated using their dimensions and B and H values from their relevant curve.
I think this makes sense.
I believe that I have an equation here to calculate MMF, but it looks a wee bit complicated to me at the moment.
Thanks again.

#### mildred00

Joined Dec 8, 2015
1
Noodler,

Have you had any luck with calculating the MMF for each part of the circuit?