# Magnetic Core

**Libraries:**

Simscape /
Electrical /
Passive

## Description

The Magnetic Core block models magnetic reluctance with hysteresis. Use this block to model custom inductances and build transformer models. To connect this block to your electrical network, use a Winding block to represent the interface between the electrical and magnetic domains.

The block uses the Jiles-Atherton model to calculate the magnetic flux density
*B* and magnetic field strength *H*. You can choose to
specify the Jiles-Atherton parameters directly, specify characteristic magnetic parameters, or
tabulate the *B*-*H* saturation curve using the
**Parameterization option** parameter.

To define the geometry for the part of the magnetic network that you are modeling, use the
**Effective length** and **Effective cross-sectional
area** parameters. The block uses this geometry information to map
*B* and *H* to the magnetomotive force and magnetic flux,
which are the Through and Across variables in the magnetic domain.

To model eddy losses, in the **Losses** settings, select the
**Model eddy losses** parameter. To model thermal effects, in the
**Thermal** settings, select the **Model thermal
effects** parameter.

### Equations

These equations define the flux density and magnetomotive force,

$$B={\phi /s}_{eff}$$

$$mmf={l}_{eff}H$$

where:

*B*is the magnetic flux density induced in the core.*φ*is the magnetic flux.*s*is the value of the_{eff}**Effective cross-sectional area**parameter.*mmf*is the magnetomotive force (mmf) across the component.*l*is the value of the_{eff}**Effective length**parameter.*H*is the total field strength.

This equation relates *B* and *H* to the total
magnetization of the core *M*,

$$B={\mu}_{0}\left(H+M\right),$$

where *μ*_{0} is the magnetic
permeability of a vacuum.

The magnetization acts to increase the magnetic flux density, and its value depends on
both the current value and the history of the field strength. The block uses the
Jiles-Atherton [1,2] equations to
determine *M* at any given time. This figure shows a typical plot of the
resulting relationship between *B* and *H*.

In this figure:

*B*is the magnetic flux density at the saturation point._{m}*B*is the magnetic remanence of the saturation cycle._{r}*H*is the magnetic coercivity of the saturation cycle._{c}*H*is the magnetic field intensity at the saturation point._{m}

As the field strength increases from *H* = 0, the plot initially follows an ascending hysteresis curve. At the
saturation point, further increases in field intensity no longer cause significant increases
in magnetic flux.

As you reduce the magnetic field strength from the saturation point, the plot follows a
descending hysteresis curve. The difference between ascending and descending curves is due
to the dependence of *M* on the trajectory history. Physically, the
behavior corresponds to magnetic dipoles in the core aligning as the field strength
increases, but not then fully recovering to their original position as field strength
decreases.

The starting point for the Jiles-Atherton equation is the split of the magnetization
effect into two parts, one part that is purely a function of effective field strength
*H _{eff}* and another, irreversible part that
depends on history. This equation defines the relative contributions of the anhysteretic
magnetisation

*M*and the irreversible magnetization

_{an}*M*to the total magnetization

_{irr}*M*,

$$M=c{M}_{an}+\left(1-c\right){M}_{irr},$$

where *c* is the coefficient for reversible magnetization.

This equation relates *M _{an}* to the equivalent
magnetization

*H*:

_{eff}$${M}_{an}={M}_{s}\left(\mathrm{coth}\left(\frac{{H}_{eff}}{a}\right)-\frac{a}{{H}_{eff}}\right).$$

The function defines a saturation curve with limiting values
±*M _{s}*, where:

*M*_{s}is the*saturation magnetization*.*a*is the*anhysteretic magnetization coefficient*, which determines the point of saturation. It approximately describes the average of the two hysteretic curves.

The Jiles-Atherton model defines the irreversible term by a partial derivative with respect to field strength,

$$\begin{array}{l}\frac{d{M}_{irr}}{dH}=\frac{{M}_{an}-{M}_{irr}}{k\delta -\alpha \left({M}_{an}-{M}_{irr}\right)}\\ \delta =\{\begin{array}{ll}1\hfill & \text{if}H\ge 0\hfill \\ -1\hfill & \text{if}H0\text{}\hfill \end{array}\end{array}$$

where:

*k*is the*bulk coupling coefficient*, which shapes the irreversible characteristic.*α*is the*inter-domain coupling factor*.

Comparison of the equation with a standard first-order differential equation reveals
that as you change the total field strength *H*, the irreversible term
*M _{irr}* attempts to track the reversible term

*M*, but with a variable tracking gain of $$1/\left(k\delta -\alpha \left({M}_{an}-{M}_{irr}\right)\right)$$. The tracking error acts to create the hysteresis at the points where

_{an}*δ*changes sign.

This equation defines the effective field strength of an anhysteretic curve without eddy losses,

$${H}_{eff}=H+\alpha M.$$

The value of *α* affects the shape of the hysteresis curve. Larger
values act to increase the *B*-axis intercepts. However, the stability term $$k\delta -\alpha \left({M}_{an}-{M}_{irr}\right)$$ must be positive for *δ* > 0 and negative for *δ* < 0. There are therefore limits on the values of *α* that you
can provide. A typical maximum value is of the order 1e-3.

You can choose how to parameterize the *B*-*H* curve.
Set the **Parameterization option** parameter to one of these options:

`Jiles-Atherton Model`

— Specify the Jiles-Atherton parameters directly:Saturation magnetization

*M*_{s}Anhysteretic magnetization constant

*a*Inter-domain coupling factor

*α*Coefficient for reversible magnetization

*c*Bulk coupling coefficient

*k*

`Saturated BH Curve`

— Tabulate the ascending or descending*B*-*H*curve. You must specify at least one data point in the first and third quadrant. For ascending curves, you must also specify at least one point in the fourth quadrant. For descending curves, you must also specify at least one point in the second quadrant.`Magnetic Characteristics`

— Specify the magnitude of the*B*and*H*intercepts,*B*_{r}and*H*_{c}, and the magnitude of*B*and*H*at the saturation point,*B*_{m}and*H*_{m}.

### Model Losses

The block computes hysteresis losses and eddy current losses.

The block calculates hysteresis losses using an averaging time. The instantaneous value
of the area inside the *B*-*H* curve is proportional to
the sum of the magnetic energy of the core and the thermal power dissipation. To obtain
accurate results, you must specify an **Averaging period for hysteresis
losses** parameter value that is greater than the
*B*-*H* cycle time. If you specify a smaller value, the
hysteresis loss at some instances is equal to the magnetic energy of the core which is
larger than the real losses. The best option is to specify a value that is an exact multiple
of the *B*-*H* cycle time.

These equations define the energy density dissipated by eddy currents
*W*_{eddy} in the time interval *Δt*
[3-5],

$${W}_{eddy}=\frac{\sigma {d}^{2}}{12\beta}{{\displaystyle {\int}_{\Delta t}\left(\frac{dB}{dt}\right)}}^{2}dt$$

$$\frac{d{W}_{eddy}}{dt}=\frac{\sigma {d}^{2}}{2\beta}{\left(\frac{dB}{dt}\right)}^{2}$$

where:

*σ*is the conductivity.*d*is the value of the**Thickness of laminations**parameter. If the core does not have laminations, this value corresponds to the thickness at the widest cross-section; for example, the diameter of cylindrical or spherical cores.*β*is the value of the**Geometrical coefficient**parameter. For a laminated core,*β*=`6`

. For a cylindrical core,*β*=`16`

. For a spherical core,*β*=`20`

[5].

Eddy currents induce a magnetic field *H _{Eddy}*:

$${H}_{Eddy}=\frac{\sigma {d}^{2}}{12\beta}\frac{dB}{dt}.$$

This field also modifies the effective field strength:

$${H}_{eff}=H+\alpha M-{H}_{Eddy}.$$

### Variables

To set the priority and initial target values for the block variables before simulation,
use the **Initial Targets** section in the block dialog box or Property
Inspector. For more information, see Set Priority and Initial Target for Block Variables.

**Note**

In the **Initial Targets** section, you can specify values for the
**Field strength** and **Flux density** parameters.
The values you provide must approximately correspond to a point on the
*B*-*H* curve.

Use nominal values to specify the expected magnitude of a variable in a model. Using
system scaling based on nominal values increases the simulation robustness. Nominal values
can come from different sources. One of these sources is the **Nominal
Values** section in the block dialog box or Property Inspector. For more
information, see System Scaling by Nominal Values.

## Examples

## Ports

### Conserving

## Parameters

## References

[1] Jiles, D. C. and D. L. Atherton.
"Theory of ferromagnetic hysteresis." *Journal of Magnetism and Magnetic
Materials* 61, no. 1–2 (September 1986): 48–60. https://doi.org/10.1016/0304-8853(86)90066-1.

[2] Jiles, D. C. and D. L. Atherton.
“Ferromagnetic hysteresis.” *IEEE ^{®} Transactions on Magnetics* 19, no. 5 (September 1983): 2183–85.
https://doi.org/10.1109/TMAG.1983.1062594.

[3] Sadowski, N., N. J. Batistela, J. P. A. Bastos, and M. Lajoie-Mazenc. “An Inverse Jiles-Atherton Model to Take into Account Hysteresis in Time-Stepping Finite-Element Calculations.” IEEE 38, no. 2 (March 2002): 797–800. https://doi.org/10.1109/20.996206.

[4] Baghel, A. P. S., and S. V.
Kulkarni. “Dynamic Loss Inclusion in the Jiles–Atherton (JA) Hysteresis Model Using the
Original JA Approach and the Field Separation Approach.” *IEEE Transactions on Magnetics* 50, no. 2 (February 2014): 369–72. https://doi.org/10.1109/TMAG.2013.2284381.

[5] Jiles, D. C. “Frequency Dependence
of Hysteresis Curves in Conducting Magnetic Materials.” *Journal of
Applied Physics* 76, no. 10 (November 1994): 5849–55. https://doi.org/10.1063/1.358399.

[6] Jiles, D. C., J. B. Thoelke, and
M. K. Devine. “Numerical Determination of Hysteresis Parameters for the Modeling of Magnetic
Properties Using the Theory of Ferromagnetic Hysteresis.” *IEEE Transactions on Magnetics* 28, no. 1 (January 1992): 27–35. https://doi.org/10.1109/20.119813.

[7] Jiles, D. C., and J. B. Thoelke.
“Theory of Ferromagnetic Hysteresis: Determination of Model Parameters from Experimental
Hysteresis Loops.” *IEEE Transactions on Magnetics* 25, no. 5 (September 1989): 3928–30.
https://doi.org/10.1109/20.42480.

[8] Li, Z., X. Huang, L. Wu, J. Chen,
Y. Zhong, J. Ma, and Y. Fang. “A Modified Jiles-Atherton Model for Estimating the Iron Loss of
Electrical Steel Considering DC Bias.” In *2018 Thirteenth
International Conference on Ecological Vehicles and Renewable Energies (EVER)*,
1–7. Monte-Carlo: IEEE, 2018. https://doi.org/10.1109/EVER.2018.8362391.

## Extended Capabilities

## Version History

**Introduced in R2024b**