# dipole moment vector of solenoid with core

I have already found some useful information about how to calculate the magnetic dipole moment vector $\vec m$ of a solenoid with ferromagnetic core.

By adding the moment vector of the core and the moment vector of the solenoid the result is $$\vec m_{total} = \vec m_{core} + \vec m_{solenoid}=NI \left( \frac{\mu}{\mu_0}-1\right)A+NIA$$ with $N$ being the number of turns of the solenoid, $I$ the current flowing through the solenoid, $A$ the cross sectional area of the solenoid, $\mu$ the permeability of the ferromagnetic material of the core and $\mu_0$ the permeability constant.

This equation completely neglects the hysteresis of the core. If the current of the solenoid is shut down ($I = 0$) the magnetic moment vector is 0.0. The remanent B field inside the core might be very small, but it is still $> 0$ and therefore the magnetic moment vector should also be $>0$.

I think the moment vector of the core can be rewritten to $$\vec m_{core} = \frac{2 \pi r^3 B}{\mu_0}$$ with $B$ being the magnetic flux density of the core.

By using a hysteresis model (preisach model; already implemented) an algorithm for computing the magnetic moment vector could be:

At each time point calculate:

1. magnetic field strength of the solenoid $$H=I \frac{N}{l}$$ with $l$ being the length of the solenoid
2. Use this calculated $H$ as external field input into the hysteresis model and get the resulting internal $B$
3. Using the second equation for the moment vector of the core, applying the result of the hysteresis model ($B$ inside the core) and putting $r=l_{core}$ as the length of the core obtain the moment vector $\vec m_{core}$
4. add the moment vector of the core and the moment vector of the solenoid to get the $\vec m_{total}$

My questions are:

1. Can the magnetized state of the core be represented as $\vec m_{core} = \frac{2 \pi r^3 B}{\mu_0}$?
2. Can someone verify that the algorithm stated above would give a more realistic representation of a solenoid with ferromagnetic core compared to assumptions that the core behaves linearly with the current?
• Commented Dec 29, 2016 at 11:39

When the current in the solenoid is zero then the H-field is zero. As $$\vec{B} = \mu_0 (\vec{H} + \vec{M}),$$ where $\vec{M}$ is the magnetisation, then $\vec{M} = \vec{B}/\mu_0$.
As magnetisation is dipole moment per unit volume, then the total magnetic moment of the core would be $m_{\rm core}=(l_{\rm core} A/\mu_0)\vec{B}$.
As for the algorithm, I don't follow step 3. In general, the dipole moment of the core will be $m_{\rm core}=l_{\rm core} A \vec{M}$, where $\vec{M}$ is a function of the applied H-field and its history. i.e. $\vec{M} = \vec{B}/\mu_0 - \vec{H}$, where I guess your B-field comes from the Preisach model.
• The input into the preisach model ist the external field strength $H$ and the result is the internal flux density $B$. With $\mu_r = \frac{\vec B} {\vec H}$ we have a division by zero if the external H is zero. Commented Dec 29, 2016 at 15:06
• @alaeX Yes, bad choice for my last sentence. $\mu_r \mu_0 \neq B/H$ if the material has a permanent or even temporary magnetisation when there is no H-field. Commented Dec 29, 2016 at 16:58