# U-Shaped Ferrite Core Inductor question

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1. May 18, 2015

### magnetpedro

I'm using a U-Shaped Ferrite Core with coils in both legs of the U.
I've calculated it's inductance using an analogy between electric/magnetic circuits.

L = N2/Rtotal

where:
N = number of turns of coil in both legs of the U;
Rtotal is the total reluctance of my circuit = R1 + R2

R1 = l1/μ0*μr,core*A
R2 = l2/μ0*μr,air*A

R1 is the reluctance of the circuit within the Ferrite core, where l1 is the mean lenght of the circuit (=0,189m), μr,core is the relative permeability of the core (≈1000) and A is the cross-sectional area (=7e-4 m2).

R2 is the reluctance of the air gap between both legs of the Ferrite core, where l2 is the mean lenght of the magnetic circuit (=0,093m), μr,air. =1, and the A is the same as in R1.

I get a value of L =1,51 mH for a total of 400 turns of coil (N=400).

When I plug it to a LCR meter , with a current of 19,91 mA the inductance L equals 14,21 mH.

Can someone tell me how I can relate both inductances?

Last edited by a moderator: May 18, 2015
2. May 19, 2015

### Jeff Rosenbury

I'm unclear here. Is this an inductor or a transformer?

If the two coils are hooked up in series, this is just an oddly shaped inductor. Yes, the geometry will have some effect on the final value, but not much.

If the coils are hooked up otherwise, you have an oddly shaped transformer. Once again, the geometry matters somewhat, but it's still basically a transformer.

3. May 19, 2015

### magnetpedro

The two coils are hooked up in series, one coil on one leg, connected to the coil on the other leg. It is not a transformer, doesnt have a primary and a secondary. Both coils are in series.
Is there any reluctance parameter lacking in my formula for the inductance?

How can I explain that the LCR meter shows me 14,2mH and the L that I've calculated is 1,5 mH?

Thanks very very much for your help.

4. May 19, 2015

### stedwards

5. May 20, 2015

### magnetpedro

This schematic represents my magnetic U-Shaped Core and the coils.

The inductance L = N2/( R1 + R2)

where

N2 = 400^2 =1,60e5

R1 = l1/μ0*μr,core*A = 0,189 / (4π *10-7 * 1020 * 7,0 * 10-4) ≅ 2,11e5

R2 = l2/μ0*μr,air*A = 0,093 / ( 4π *10-7 * 1 * 7,0 * 10-4) ≅ 1,06e8

L = 1,60e5 / ( 2,11e5 + 1,06e8) = 1,506 mH

This should represent the inductance of my magnetic core away from any magnetic object, or the inductance it has in air or free space.
I believe my calculations are right but would love to hear your opinion.
Am I missing some R, some reluctance of another important magnetic path in my calculation?
Or is this some kind of "Inductance for I=0 or H=0"?

When I measure the inductance of the core in air using a LCR meter with f=20Hz and a current I of 19,91 mA the Inductance is equal to 14,31 mH and is this difference that I don't understand and would like to.

Thank you very much in advance

6. May 20, 2015

### Jeff Rosenbury

Let's go back to basics.

What is the material you are using? I know it's ferrite, but what type?
What is the operational frequency? What test equipment frequency? (What is the test equipment, can you link to a manual?)
What is the cross section of the material? How did you calculate the cross section of the air portion? Did you consider fringing?
Is your wire tightly wound? How many turnss in each winding?

7. May 20, 2015

### magnetpedro

The ferrite type is N47 (Siemens N47) with a relative magnetic permeability of 1020.
I'm using an Agilent E4980AL (link to the manual: http://cp.literature.agilent.com/litweb/pdf/5989-4435EN.pdf [Broken]) with a test frequency of 20Hz. What's the diference between the test frequency and operational frequency?
The cross section of the ferrite core is 28mm*30mm = 8,4 e-4 m2.

I've considered a cross section of air equal to the cross-section of the ferrite but it could have been a bad consideration. And no, I've not considered fringing not any Reluctance related to any leakage flux.
Could this justify the difference between the measured Inductance and the one I've calculated?
My wire is not very tightly wounded, with 200 turns in each leg. It's also important to say that the wire is wounded in 7 layers in each leg (like 7 coaxial solenoids overlapped).

Thanks very much for your help.

Last edited by a moderator: May 7, 2017
8. May 20, 2015

### Jeff Rosenbury

It's my (admittedly weak) understanding that different ferrite mixes are designed to operate at different frequencies, so the specified permeability may be off at 20Hz.

Also your air cross section is way off. I'm not sure of the proper way to figure it, but I'm sure it's bigger than the ferrite. Since this value seemed to be the one driving the equation, I suspect it's your problem child.

Last edited by a moderator: May 7, 2017
9. May 20, 2015

### magnetpedro

I've runned some parametric tests with the LCR meter and the Quality Factor, Q, is maximum at 600 Hz, and the Inductance measured is still 14,21. Could this mean that 600Hz is the best test frequency and consequently the operational frequency?

I've increased the cross section of the air about ten 10 times, and for this value, both Inductances match (the one I calculate and the one I measure). But I was afraid that a cross-section of air 10 times the cross section of the core was out of proportions and not a good approach of reality.

10. May 20, 2015

### Jeff Rosenbury

The N47 material I looked up was tested at 100kHz and had a 20% tolerance for what that is worth. I also found an article that claimed manufacturing stresses could cause permiability shifts. So your numbers for that might be off some as well.

But the cross section of the air core should be much larger than the N47. Ten times the radius does seem a bit much, but possibly not. You need to check fringe effects of air gaps. I know the equations exist somewhere. They are used in magnetic recording and electric motor calculations.

11. May 20, 2015

### magnetpedro

Could you please send me the link of those tests at 100kHz? I've also read that article about the manufacturing stresses, and I've taken my μr from there
At that frequency (100kHz), I measure an Inductance of 2,47mH, and with a cross section of area of about 1,6 times the cross section of the core both Inductances match.

12. May 20, 2015

### Jeff Rosenbury

It was just an inductor data sheet I got from googling the N47.

From the looks of it, your problem is now solved? 1.6 looks like a realistic number. You might find the equations if you need them.

Good luck.

13. May 20, 2015

### magnetpedro

Sounds like it is.

Thank you very much.

14. May 20, 2015

### stedwards

I don't know if you're still there, however...

The permeability of the core material is so much greater than air, that (other than saturation possibility) it can be ignored for a rough estimate.

With the relatively high permeability, the coil winding are well coupled so that n2 can be used without much lose in accuracy. Most of the flux from each winding passes through the others.

As Jeff allude to, the air gap is the problem child. Other than the turns coupling, to good approximation, everything else happens in the air gap. The majority of the MMF drives flux through the gap.

By examination of your drawing the air gap, in your case, is 6.4 cm. (9.3 cm minus the core width of about 2.9 cm)

Dimensional analysis says the effective area of the air gap, for small pole areas, is proportional to the square of the air gap. $A_e=k {l_g}^2$ The value of k unknown to us.
$$R_g = \frac{l_g 10^7}{A_e 4 \pi} = \frac{10^7}{k {l_g} 4 \pi}$$ Using a value of $k=1$,
$$L = 400^2/R_g=12.9mH$$ Not bad, estimating k=1, and all else considering. How to deal better with air gaps has long been something off a mystery to me.

Last edited: May 20, 2015
15. May 21, 2015

### magnetpedro

This seems much more like it and I thank you deeply.

I've some questions though. I've measured my air gap, and it's 3.7 cm (9,3 cm minus both core widths of 2.8 cm <=> 9,3 - 5,6 = 3,7). Am I doing this wrong?

Using your process, with a k=2, and $l_g = 3,7$ I get an L = 14,8. I would like to read more about this k factor and this dimensional analysis for small pole areas. Could you also redirect me for some links?

Thanks again

16. May 21, 2015

### stedwards

The value of k is a half educated guess. You probably won't find it anywhere on the internet. At one time I designed magnetic circuits.

http://www.kgmagnetics.org/APNOTES-06/An-115.pdf [Broken] goes over the basics nicely, complete with many drawings. It has an equation to account for flux fringing, but where the gap length is small compared to the width of a pole face. You could try it, but I think it isn't applicable in your case.

Solutions for flux fringing are nonanalytical for fields in three dimensions as far as I know.

Once you had the field solution, you would need solve the double integral over flux tubes, $\int \int \Phi_H dA dl$, where $\Phi_H$is the H field density to get an effective A/l.

Below is an example in two dimensions that is analytical and subject to conformal transformations of the plane. It should give you a rough idea of what large gap flux fringing would look like over a volume if spun around a vertical axis. The orange and blue dots would be the pole faces.

I'm curious. Why do you want a calculated value of inductance?

Last edited by a moderator: May 7, 2017
17. May 21, 2015

### magnetpedro

This was a really good help and I thank greatly you for it.
I think I will try to estimate a value of k by ajusting it to the inductance I measure.

I'm developing a magnetic "probe" to measure the change of inductance when I lean the probe (my U-shaped magnetic probe) in a surface with ferromagnetic materials on it. This surface is composed of a small quantity of ferromagnetic fibers but they are not uniformly distributed through the surface. By changing the direction of my probe during the measurements I can estimate wich part of the surface has more fibers and also what is the preferential direction of their allignment. It's thus important for me to have a good description of my probe and be sure that the changes in the inductance that I measure are real and relatively precise.

There's also another question I would like to ask.
I've done a parametric test changing the frequency test of my LCR from 20Hz to 300kHz. The higher Q factor I've encountered is at f=600Hz. This means that this is the preferencial frequency test for my process? Or should I be looking for another parameter to a better calibration of my method?

Thanks again.

Last edited by a moderator: May 7, 2017
18. May 21, 2015

### Jeff Rosenbury

If this is for use in test equipment, you might consider replacing the ferrite with something more stable. As I understand it, a good, physical jolt could change the value of your probe. Or you could recalibrate for each session?

Perhaps soft iron with a bias current? Perhaps degaussing it regularly? Or perhaps there's a more stable ferrite mix?

Also, the shape of the ends of the bars will have a big effect on the fringing. You might consider various shapes.

19. May 21, 2015

### stedwards

On that, I have no idea. And good luck.

I read the write-up on the core material that I think you linked to. There are plots of inductance changed under pressure. This is due to the gap between core halves. If the interface is not dead parallel, pressure will tend to close the gap changing the inductance. An initial 250 micro gap can have a big effect under pressure variations. In the OPs case, this isn't an issue, though.

20. May 22, 2015

### magnetpedro

My funds are limited so I would like to keep my probe.
I have also a smaller probe made of soft iron. When you say degaussing it what's the meaning? I beg your pardon but I'm not familiar with the concept.

Thanks