Nonlinear reluctance with magnetic hysteresis

**Library:**Simscape / Electronics / Passive Devices

The Reluctance with Hysteresis block models a 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** section 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, respectively:

$$\begin{array}{l}B={\Phi /A}_{e}\\ MMF={l}_{e}\cdot H\end{array}$$

where:

*MMF*is magnetomotive force (mmf) across the component.*Φ*is flux through the component.*B*is flux density.*H*is field strength.*A*_{e}is the effective cross-sectional area of the section being modeled.*l*_{e}is the effective length of the section being modeled.

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*_{0}is the magnetic permeability constant.*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 past 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,
*H*_{eff}:

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

This function defines a saturation curve with limiting values
±*M*_{s} and point of saturation determined by
the value of *α*, the anhysteretic shape factor. It can be approximately
thought of as describing the average of the two hysteretic curves. In the block interface,
you provide values for $$d{M}_{an}/d{H}_{eff}$$when *H*_{eff} = 0 and a point [*H*_{1},
*B*_{1}] on the anhysteretic B-H curve, and these
are used to determine values for *α* and
*M*_{s}.

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
*M*_{irr}. The Jiles-Atherton model defines the
irreversible term by a partial derivative with respect to field strength:

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

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
*M*_{an}, 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
*δ* changes sign. The main parameter that shapes the irreversible
characteristic is *K*, which is called the *bulk coupling
coefficient*. The parameter *α* is called the
*inter-domain coupling factor*, and is also used to define the
effective field strength used when defining the anhysteretic curve:

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

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*_{eff}= 0) plus a data point [*H*_{1},*B*_{1}] on the anhysteretic 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 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 will cause the gradient of the B-H curve to tend to infinity, which is nonphysical and generates a run-time assertion error.

Sometimes you need to iterate on these four steps to get a good match against a predefined B-H curve.

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 (Simscape).

[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.

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