# Nonlinear Reluctance

Linear or nonlinear reluctance with optional hysteresis

**Libraries:**

Simscape /
Electrical /
Passive

## Description

The Nonlinear Reluctance block models linear or nonlinear reluctance with optional magnetic 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.

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. Choose
how to parameterize the *B*-*H* curve using the
**Parameterized by** parameter.

### Equations for Linear Reluctance Parameterization

The equations for the linear reluctance parameterization are

$$B={\mu}_{0}{\mu}_{r}H$$

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

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

where:

*B*is the flux density.μ

_{0}is the permeability in a vacuum.μ

_{r}is the relative magnetic permeability.*H*is the field strength.*mmf*is the magnetomotive force (mmf) across the component.*l*is the effective length of the section being modeled._{eff}φ is the magnetic flux.

*s*is the effective cross-sectional area of the section being modeled._{eff}

### Equations for Reluctance with Single Saturation Point Parameterization

This parameterization models a switch-linear reluctance. In the unsaturated state,
the material has a specified relative magnetic permeability. In the saturated state,
the relative permeability is 𝜇_{0}.

The equations for reluctance with single saturation point are

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

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

$mmf=R\phi $

If $B<{B}_{sat}$,

$B={\mu}_{0}{\mu}_{r\_unsat}H.$

Otherwise,

$B={B}_{sat}+{\mu}_{0}(H-\frac{{B}_{sat}}{{\mu}_{0}{\mu}_{r\_unsat}})$

where:

*mmf*is the magnetomotive force (mmf) across the component.*l*is the effective length of the section being modeled._{eff}*H*is the field strength.φ is magnetic flux.

*s*is the effective cross-sectional area of the section being modeled._{eff}*B*is the flux density.*B*is the flux density at saturation._{sat}*R*is the magnetic reluctance at saturation._{sat}μ

_{0}is the permeability in a vacuum.μ

_{r}is the relative magnetic permeability.μ

_{r_unsat}is the unsaturated relative magnetic permeability.

### Reluctance (B-H Curve)

To model reluctance without hysteresis by tabulating the
*B*-*H* curve, set the **Parameterized
by** parameter to ```
Reluctance (B-H
curve)
```

.

### Equations for Reluctance with Hysteresis Parameterization

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*.

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.

In the Nonlinear Reluctance block mask, you provide
values for $$d{M}_{an}/d{H}_{eff}$$ when *H _{eff}* = 0 and a point [

*H*,

_{1}*B*] on the anhysteretic

_{1}*B*-

*H*curve. The block uses these values to determine values for

*α*and

*M*.

_{s}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,

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

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.

**Note**

You can also use the Magnetic
Core block to model a magnetic core that exhibits
nonlinear reluctance and hysteresis. Both blocks use the Jiles-Atherton
model to determine the relationship between *B*,
*H*, and *M*. However, the
Magnetic Core block gives you additional options:

In addition to specifying the Jiles-Atherton parameters directly, you can parameterize the

*B*-*H*curve using these options:Specify the magnitude of the

*B*and*H*intercepts and the magnitude of*B*and*H*at the saturation point.Tabulate the ascending or descending

*B*-*H*curves.

You can model eddy losses and thermal effects.

**Procedure for Finding Approximate Values for Jiles-Atherton (JA) Equation Coefficients**

You can determine representative values for the equation coefficients using this procedure:

Provide a value for the

**Anhysteretic B-H gradient when H is zero**parameter ($$d{M}_{an}/d{H}_{eff}$$ when*H*= 0) and a data point [_{eff}*H*_{1},*B*] on the anhysteretic_{1}*B*-*H*curve. From these values, the block initialization determines values for*α*and*M*_{s}.Set the

**Coefficient for reversible magnetization, c**parameter to achieve correct initial*B*-*H*gradient when starting a simulation from [*H**B*] = [0 0]. The value of*c*is approximately the ratio of this initial gradient to the**Anhysteretic B-H gradient when H is zero**. The value of*c*must be greater than 0 and less than 1.Set the

**Bulk coupling coefficient, K**parameter to the approximate magnitude of*H*when*B*= 0 on the ascending hysteresis curve.Start with a very small value for the

**Inter-domain coupling factor, alpha**parameter, and gradually increase it to tune the value of*B*when crossing the*H*= 0 line. A typical value is in the range of 1e-4 to 1e-3. Values that are too large cause the gradient of the*B*-*H*curve to tend to infinity, which is nonphysical and generates a run-time assertion error.

To get a good match against a predefined
*B*-*H* curve, iterate on these four
steps.

### 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.

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):
2183–2185. https://doi.org/10.1109/TMAG.1983.1062594.

## Extended Capabilities

## Version History

**Introduced in R2017b**

## See Also

Magnetic Core | Nonlinear Inductor | Nonlinear Transformer | Winding