# Inductor core permeability much lower than expected

I am interested in creating an inductor with a large inductance/large mag field in core. However, when empirically measuring the inductance of the wound core, I calculated a relative permeability ~1000x less than expected (2-6 vs 2000). My question is whether this is due to an error in my setup/calculations/understanding, or due to receiving a core not up to spec.

The ferrite rod has stated relative permeability of ~2000 and B_sat of .49T (4900G). The physical dimensions are 25.66mm height x 6.32mm diam. http://www.ebay.com/itm/5-ea-Ferrite-Rod-Core-0-25x1-00-Fair-Rite-4077276011-2000u-Perm-Plain-Slug-77-/251653543001?hash=item3a97b85459

Given that this is ferrite, the permeability should be fairly linear. The datasheet for the material can be found here: http://www.mouser.com/ds/2/150/4077276011-476454.pdf

I wound the core with ~100 turns of 28AWG mag wire (.321mm diameter), with a DC resistance of 1.0 ohm. The result is quite messy, but the length of the coil extends 11.6mm, and the outer diameter goes up to 10.3mm:

The inductance of the coil was measured experimentally by creating an RL series circuit fed by a signal generator, and increasing the frequency until the Vpp across the RL circuit was twice the Vpp across the resistor alone. See http://www.daycounter.com/Articles/How-To-Measure-Inductance.phtml for an explanation of this method.

For a 100 ohm 1% tolerance resistor, the required frequency was 77kHz, resulting in a measured inductance of 358uH. I used the following equations to calculate relative permeability $\mu_r$. The geometry is neither a loop or solenoid, but the equations only differ by a factor of 2-3x in this case.

$B_{loop,center} = \mu_0\mu_r\frac{NI}{2R}$

$B_{solenoid,center} = \mu_0\mu_r\frac{NI}{l}$

$L_{loop} \approx \mu_0\mu_rN^2R(ln(\frac{8R}{r})-2)$, r = wire radius

$L_{solenoid} = \mu_0\mu_r\frac{N^2A}{l}$

Because there is at least 150 ohm in serial with the 2Vpp source, the most current that could flow in one direction is 1/150 = 6.7mA. Assuming the relative permeability is 2000 as stated, there shouldn't be any concern of saturation, as the magnetic field in the center of the loop/coil is at most .28T ($B_{loop,center}$ with R=3mm). (Also, the scoped sine waves were clean.)

Given the dimensions from above and the measured L of 358uH, $\mu_r$ comes out to 2-6, much less than the expected 2000. Why is this?

• You have a huge air gap in your magnetic circuit. It's like having a small resistor (your core) in series with a huge resistor (your air gap due to the core being a rod and not a closed path.) – John D Mar 21 '17 at 19:17
• Do a search on reluctance and magnetomotive force. You will find that geometry has a huge impact on magnetic circuits. Interestingly enough, magnetic equations have a similar form to electrical ones. Continuing with John D's analogy, you are saying the small resistor has a conductivity of A, and you calculate the entire circuit's conductivity of C, you are not accounting for the conductivity of the huge series resistor B. You are not accounting for the permeability and magnetic path of the "air gap". Note, I've tried to do this before by hand and was unable to obtain an accurate answer. – klamb Mar 21 '17 at 19:36
• Here's something that might help with understanding what's going on: info.ee.surrey.ac.uk/Workshop/advice/coils/gap/index.html If you can find a toroid made of the same material and put your windings around that, you will notice a huge difference in your inductance. – John D Mar 21 '17 at 19:38
• @JohnD This is the kind of answer I was looking for; I will have to revisit the theory. This may warrant a new question, but does the existence of an air gap also mean that the magnetic field sensed at a distance (e.g. via magnetometer) will not be amplified by the core material either? I thought the coil would temporarily magnetize the core such that the field due to the magnetized core overpowered the original field due to the coil by ~$\mu_r/1$. – abc Mar 21 '17 at 21:23
• I think your question about sensing the field at a distance does deserve a new question, it shouldn't turn into a discussion in the comments for this question. It would be much better if you describe what you are trying to do and ask how to get there, and for clarification on what you are not sure about. – John D Mar 21 '17 at 21:38