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Nonlinear reluctance with magnetic hysteresis

**Library:**Simscape / Electrical / Passive

The Nonlinear Reluctance block models linear or nonlinear reluctance with magnetic hysteresis. Use this block to build custom inductances and transformers that exhibit magnetic hysteresis.

The length and area parameters in the **Geometry** settings let you
define the geometry for the part of the magnetic circuit that you are modeling. The
block uses the geometry information to map the magnetic domain Through and Across
variables to flux density and field strength.

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 magnetic flux.

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

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 unstaurated relative magnetic permeability.

For the reluctance (B-H Curve) parameterization, specify the material property by B-H curve.

The flux density and magnetomotive force equations are:

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

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

where:

*B*is flux density.φ is magnetic flux.

*s*is the effective cross-sectional area of the section being modeled._{eff}*mmf*is magnetomotive force (mmf) across the component.*l*is the effective length of the section being modeled._{eff}*H*is field strength.

The block then implements the relationship between *B* and
*H* according to the Jiles-Atherton [1, 2] equations. The
equation that relates *B* and *H* to the
magnetization of the core is:

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

where:

*μ*is the magnetic permeability constant._{0}*M*is magnetization of the core.

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
*H*. The block uses the Jiles-Atherton equations to determine
*M* at any given time.

The figure below shows a typical plot of the resulting relationship between
*B* and *H*.

In this case, the magnetization starts as zero, and hence the plot starts at *B* = *H* = 0. As the field strength increases, the plot tends to the
positive-going hysteresis curve; then on reversal the rate of change of
*H*, it follows the negative-going hysteresis curve. The
difference between positive-going and negative-going 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 to split the
magnetization effect into two parts, one that is purely a function of effective
field strength (*H _{eff}*) and the other an
irreversible part that depends on history:

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

The *M _{an}* term is called the
anhysteretic magnetization because it exhibits no hysteresis. It is described by
the following function of the current value of the effective field strength,

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

This function defines a saturation curve with limiting values
±*M _{s}* and point of
saturation determined by the value of

The parameter *c* is the coefficient for reversible
magnetization, and dictates how much of the behavior is defined by
*M _{an}* and how much by the
irreversible term

$$\frac{d{M}_{irr}}{dH}=\frac{{M}_{an}-{M}_{irr}}{K\delta -\alpha \left({M}_{an}-{M}_{irr}\right)}$$

For $$H\ge 0$$, $$\delta =1$$.

For $$H<0$$, $$\delta =-1$$.

Comparison of this equation with a standard first order differential equation
reveals that as increments in field strength, *H*, are made,
the irreversible term *M _{irr}* attempts to
track the reversible term

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

The value of *α* affects the shape of the hysteresis curve,
larger values acting to increase the B-axis intercepts. However, notice that for
stability the term $$K\delta -\alpha \left({M}_{an}-{M}_{irr}\right)$$ must be positive for *δ* > 0 and negative for *δ* < 0. Therefore not all values of *α* are
permissible, a typical maximum value being of the order 1e-3.

You can determine representative parameters for the equation coefficients by using the following procedure:

Provide a value for the

**Anhysteretic B-H gradient when H is zero**parameter ($$d{M}_{an}/d{H}_{eff}$$when*H*= 0) plus a data point [_{eff}*H*_{1},*B*] on the anhysteretic B-H curve. From these values, the block initialization determines values for_{1}*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 positive-going hysteresis curve.Start with

*α*very small, and gradually increase to tune the value of*B*when crossing*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, you may have to iterate on these four steps

Use the **Variables** section of the block
interface to set the priority and initial target values for the block
variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables.

[1] Jiles, D. C. and D. L.
Atherton. “Theory of ferromagnetic hysteresis.” *Journal of
Magnetism and Magnetic Materials*. Vol. 61, 1986, pp.
48–60.

[2] Jiles, D. C. and D. L.
Atherton. “Ferromagnetic hysteresis.” *IEEE ^{®} Transactions on Magnetics*. Vol. 19, No. 5, 1983, pp.
2183–2184.