# Nonlinear Reluctance

Nonlinear reluctance with magnetic hysteresis

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• Simscape / Electrical / Passive

## Description

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.

### Equations for Linear Reluctance Parameterization

The equations for the linear reluctance parameterization are:

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

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

where:

• B is the flux density.

• μ0 is the permeability in a vacuum.

• μr is the relative magnetic permeability.

• H is the field strength.

• mmf is the magnetomotive force (mmf) across the component.

• leff is the effective length of the section being modeled.

• φ is magnetic flux.

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

### 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 𝜇0.

The equations for reluctance with single saturation point are

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

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

`$mmf=R\phi$`

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

Otherwise,

where:

• mmf is the magnetomotive force (mmf) across the component.

• leff is the effective length of the section being modeled.

• H is the field strength.

• φ is magnetic flux.

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

• B is the flux density.

• Bsat is the flux density at saturation.

• Rsat is the magnetic reluctance at saturation.

• μ0 is the permeability in a vacuum.

• μr is the relative magnetic permeability.

• μr_unsat is the unstaurated relative magnetic permeability.

### Reluctance (B-H Curve)

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

### Equations for Reluctance with Hysteresis Parameterization

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.

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

• mmf is magnetomotive force (mmf) across the component.

• leff is the effective length of the section being modeled.

• 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:

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

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

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 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 (JA) 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 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

### 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.

## 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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### Main

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

The value must be positive and noninfinite.

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

The value must be positive and noninfinite.

Averaging period for the hysteresis losses calculation. These losses are proportional to the area enclosed by the B-H trajectory. If the block is excited at a known, fixed frequency, you can set this value to the corresponding excitation period to calculate the hysteresis loss. In this case, the block logs the hysteresis loss once per AC cycle to the variable `power_dissipated`. If you are using a fixed-step solver, this value must be an integer multiple of the simulation step size.

If the block is not excited at a known, fixed frequency, set this parameter to `0`. In this case, the block sets `power_dissipated` to zero, and you can calculate the actual hysteresis loss by post-processing the logged variable `power_instantaneous`.

#### Dependencies

This parameter is visible only when, in the B-H Curve settings, the Parameterized by parameteris set to ```Nonlinear reluctance with hysteresis (JA model)```.

### B-H Curve

B-H Curve parameterization method.

#### Dependencies

Selecting a parameterization method makes related parameters in the B-H Curve settings. If the ```Nonlinear reluctance with hysteresis (JA model)``` is the selected parameterization method, in the Main settings, the Averaging period for power logging is visible.

Relative magnetic permeability.

The value must be positive and noninfinite.

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Linear reluctance```.

Relative magnetic permeability for an unsaturated inductor.

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Reluctance with single saturation point```.

Magnetic flux density for a saturated inductor.

The value must be positive and noninfinite.

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Reluctance with single saturation point```.

Magnetic field strength, H, specified as a vector with the same number of elements as the magnetic flux density, B. The vector must starts with zero and increase monotonically.

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Reluctance (B-H curve)```.

Magnetic flux density, B, specified as a vector with the same number of elements as the magnetic field strength vector, H. The vector must starts with zero and increase monotonically.

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Reluctance (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.

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Nonlinear reluctance with hysteresis (JA model)```.

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.

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Nonlinear reluctance with hysteresis (JA model)```.

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

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Nonlinear reluctance with hysteresis (JA model)```.

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

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Nonlinear reluctance with hysteresis (JA model)```.

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

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Nonlinear reluctance with hysteresis (JA model)```.

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.

#### Dependencies

This parameter is visible only when the Parameterization method parameter is set to ```Nonlinear reluctance with hysteresis (JA model)```.

Select the method that the block uses to determine the output current values at intermediate time points that are not specified in the preceding vectors:

• `Linear` — Prioritize performance by using a linear function.

• `Smooth` — Prioritize accuracy by producing a continuous curve with continuous first-order derivatives.

## 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.