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I am an undergraduate who is currently designing an actuation system for a CubeSat, more specifically a PCB magnetorquer. (with 2 ferromagnetic core coils and one air core)

I made an excel sheet that helps me determine the magnetic dipole moment and more from some sets of data including the size and material the core is made of. The material of choice is permalloy.

I supposed it is known that the magnetic dipole moment is a function of the intensity of the current, the area of the core , the number of turns AND the gain factor $K$:

${\mu} = k N I A$

where $N$ is the number of turns, $I$ is the current value and $A$ vector cross-sectional area of the solenoid.

Unfortunately, I found little information online about this gain factor which is taken as $K=1$ for an air core coil, but it varies between values of 100 and 300 for a ferromagnetic core, depending on the length and diameter shape factor and permeability of the material.

That is all that I've found online about this gain factor and it is the reason why I am asking you guys to provide a general formula of this factor or at least its value for permalloy.

Another thing that is worth mentioning is that I had a conversation last year about this value with an academic researcher and confirmed me the existence of it, but we did not further developed on the subject.

To be noted, there is a typo problem in the pic attached, it should be $k=100$ to $300$.

The thesis where I found this "K" value: Design optimization of the CADRE Magnetorquers by Duncan Miller at U of M Ann Arbor, May 2, 2013

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    $\begingroup$ +1 This reminds me that I'd meant to revisit and improve my answer to a different but related question Ways to measure dipole moment of magnetorquer and there might be some helpful information in Are there any modern cubesats or smallsats that have relied only on magnetotorquers for attitude control?. $\endgroup$
    – uhoh
    Aug 6, 2020 at 0:45
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    $\begingroup$ Unfortunately, I found no information about a gain factor or a similar value in the documents linked above. And to answer the second question quickly, there are a bunch of CubeSat satellites up to 6U that are being actuated only by the use of magnetorquer and the hysterises desaturation problem is becoming less and less harmful as the technology upgrades. $\endgroup$
    – Vlad Ilincăi
    Aug 6, 2020 at 15:19
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    $\begingroup$ Oh I see now, ya $K$ is just a "bucket" representing a complicated calculation based on Litho's link. It's usually done numerically, but there may be a way to approximate it with an equation by assuming uniform magnetization, though that's not a very realistic assumption. $\endgroup$
    – uhoh
    Aug 8, 2020 at 11:12
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    $\begingroup$ Okay, so I've asked around and I received an answer: Yes, $K$ has no general formula. I do not have the exact steps, but it goes as follow: the local demagnetization factor must be integrated with respect to the volume. This factor depends on the position of each element of the core reported to the winding, therefore there is no simple analytical answer. For an approximation, as @uhoh stated, there is a general formula assuming uniform magnetization that I used in my calculations, as the core is a cylinder. The equation in its general form is: $K=(1+({\mu}-1))/[1+({\mu}-1)*Nd]$ $\endgroup$
    – Vlad Ilincăi
    Aug 10, 2020 at 5:54
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    $\begingroup$ Where $Nd$ represents the "total demagnetization factor" and ${\mu}$ is the relative permeability. For a cylindrical-type core with a length of $L$ and radius $R$, then the $Nd$ is: $Nd=4* [ln(L/R) -1] / [(L/R)^2 - 4* ln(L/R)]$ This formula is valid only when $L$ is much bigger than $R$ ($L>>R$) $\endgroup$
    – Vlad Ilincăi
    Aug 10, 2020 at 6:00

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Okay, so I've asked around and I received an answer: Yes, $K$ has no general formula. I do not have the exact steps, but it goes as follow: the local demagnetizing factor must be integrated with respect to the volume. This factor depends on the position of each element of the core reported to the winding, therefore there is no simple analytical answer.

For an approximation, as @uhoh stated in the comments, there is a general formula assuming uniform magnetization that I used in my calculations. That equation in its general form is:

$$K=\frac{1 + (μ−1)}{1 + N_d(μ−1)} \ \text{(new formatting)}$$

Where $N_d$ represents the "total demagnetizing factor" and $μ$ is the relative permeability of the material.

For a cylindrical-type core with a length of $L$ and radius $R$, the $Nd$ is:

$$ N_d = \frac{4 \ln(L/R)−1}{(L/R)^2 − 4 \ln(L/R)} \ \text{(new formatting)}$$

This formula is valid only when $L$ is much bigger than $R$ --> $L>>R$

Demagnetizing general information

Demagnetizing Factors for Various Geometries research + more references

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