Reluctance with Hysteresis

Nonlinear reluctance with magnetic hysteresis

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Description

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.

• Ae is the effective cross-sectional area of the section being modeled.

• le 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:

• μ0 is the 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 (Heff) and the other an irreversible part that depends on past history:

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

The Man 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, Heff:

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

This function defines a saturation curve with limiting values ±Ms 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 Heff = 0 and a point [H1, B1] on the anhysteretic B-H curve, and these are used to determine values for α and Ms.

The parameter c is the coefficient for reversible magnetization, and dictates how much of the behavior is defined by Man and how much by the irreversible term Mirr. The Jiles-Atherton model defines the irreversible term by a partial derivative with respect to field strength:

Comparison of this equation with a standard first order differential equation reveals that as increments in field strength, H, are made, the irreversible term Mirr attempts to track the reversible term Man, 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.

Procedure for Finding Approximate Values for Jiles-Atherton Equation Coefficients

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

1. Provide a value for the Anhysteretic B-H gradient when H is zero parameter ($d{M}_{an}/d{H}_{eff}$when Heff = 0) plus a data point [H1, B1] on the anhysteretic B-H curve. From these values, the block initialization determines values for α and Ms.

2. 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.

3. Set the Bulk coupling coefficient, K parameter to the approximate magnitude of H when B = 0 on the positive-going hysteresis curve.

4. 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.

Variables

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).

Ports

Conserving

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Magnetic conserving port associated with the block North terminal.

Magnetic conserving port associated with the block South terminal.

Parameters

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Geometry

Effective length of the section being modeled, that is, the average distance of the magnetic path.

Effective cross-sectional area of the section being modeled, that is, the average area of the magnetic path.

B-H Curve

The gradient of the anhysteretic (no hysteresis) B-H curve around zero field strength. Set it to the average gradient of the positive-going and negative-going hysteresis curves.

Specify a point on the anhysteretic curve by providing its flux density value. Picking a point at high field strength where the positive-going and negative-going hysteresis curves align is the most accurate option.

The corresponding field strength for the point that you define by the Flux density point on anhysteretic B-H curve parameter.

The proportion of the magnetization that is reversible. The value must be greater than zero and less than one.

The Jiles-Atherton parameter that primarily controls the field strength magnitude at which the B-H curve crosses the zero flux density line.

The Jiles-Atherton parameter that primarily affects the points at which the B-H curves intersect the zero field strength line. Typical values are in the range of 1e-4 to 1e-3.

References

[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.