Nonlinear Reluctance

Nonlinear reluctance with magnetic hysteresis

Libraries:
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:

$B = μ_{0} μ_{r} H$

$m m f = l_{e f f} H$

$φ = s_{e f f} B$

where:

B is the flux density.
μ₀ 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_eff is the effective length of the section being modeled.
φ is magnetic flux.
s_eff 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

$m m f = l_{e f f} H$

$φ = s_{e f f} B$

$m m f = R φ$

If $B < B_{s a t}$ .

$B = μ_{0} μ_{r_u n s a t} H$

Otherwise,

$B = B_{s a t} + μ_{0} (H - \frac{B_{s a t}}{μ_{0} μ_{r_u n s a t}})$

where:

mmf is the magnetomotive force (mmf) across the component.
l_eff is the effective length of the section being modeled.
H is the field strength.
φ is magnetic flux.
s_eff is the effective cross-sectional area of the section being modeled.
B is the flux density.
B_sat is the flux density at saturation.
R_sat is the magnetic reluctance at saturation.
μ₀ is the permeability in a vacuum.
μ_r is the relative magnetic permeability.
μ_{r_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:

$B = {φ / s}_{e f f}$

$m m f = l_{e f f} \cdot H$

where:

B is flux density.
φ is magnetic flux.
s_eff is the effective cross-sectional area of the section being modeled.
mmf is magnetomotive force (mmf) across the component.
l_eff 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:

$B = μ_{0} (H + M)$

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 (H_eff) and the other an irreversible part that depends on history:

$M = c M_{a n} + (1 - c) M_{i r r}$

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:

$M_{a n} = M_{s} (\coth (\frac{H_{e f f}}{a}) - \frac{a}{H_{e f f}})$

This function defines a saturation curve with limiting values ±M_s and point of saturation determined by the value of a, 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 $d M_{a n} / d H_{e f f}$ when H_eff = 0 and a point [H₁, B₁] 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:

$\frac{d M_{i r r}}{d H} = \frac{M_{a n} - M_{i r r}}{K δ - α (M_{a n} - M_{i r r})}$

For $H \geq 0$ , $δ = 1$ .

For $H < 0$ , $δ = - 1$ .

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 M_an, but with a variable tracking gain of $1 / (K δ - α (M_{a n} - M_{i r r}))$ . 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:

$H_{e f f} = H + α M$

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 $K δ - α (M_{a n} - M_{i r r})$ 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:

Provide a value for the Anhysteretic B-H gradient when H is zero parameter ( $d M_{a n} / d H_{e f f}$ when H_eff = 0) plus a data point [H₁, B₁] 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 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

To set the priority and initial target values for the block variables prior to 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.

Nominal values provide a way 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 which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

Examples

Custom Solenoid with Magnetic Hysteresis

A limited travel solenoid with return spring. Magnetic hysteresis is modeled using the Reluctance with Hysteresis library block.

Open Model

Electrical Transformer with Hysteresis

Model a custom transformer that exhibits hysteresis by using the Non Linear Reluctance block in a magnetic circuit. The transformer is rated for a 50W load and steps down from 120V to 12V rms. The magnetizing resistance Rm is modeled in the magnetic domain using an Eddy Loss block.

Open Model

Inductor with Hysteresis

How modifying the equation coefficients of the Jiles-Atherton magnetic hysteresis equations affects the resulting B-H curve. The simulation parameters are configured to run four complete AC cycles with initial field strength (H) and magnetic flux density (B) both set to zero.

Open Model

Ports

Conserving

expand all

N — North terminal
magnetic

Magnetic conserving port associated with the block North terminal.

S — South terminal
magnetic

Magnetic conserving port associated with the block South terminal.

Parameters

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Main

Effective length — Effective length of the section being modeled
0.032 m (default) | positive noninfinite scalar

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 — Effective cross-sectional area of the section being modeled
`1.6e-5` `m^2` (default) | positive noninfinite scalar

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 power logging — Excitation period used for averaging
`0` `s` (default) | scalar

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

Parameterized by — B-H Curve parameterization method
`Reluctance (B-H curve)` (default) | `Linear reluctance` | `Reluctance with single saturation point` | `Nonlinear reluctance with hysteresis (JA model)`

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 — Relative magnetic permeability
`5000` (default) | scalar

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.

Unsaturated relative magnetic permeability — Relative magnetic permeability
`5000` (default) | scalar

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 at saturation — Magnetic flux density at saturation
`1.5` `T` (default) | positive noninfinite scalar

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 vector, H — Magnetic field strength vector
`[0, 200, 400, 600, 800, 1000]` `A/m` (default) | noninfinite vector

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 vector, B — Magnetic flux density vector
`[0, 1.25, 1.35, 1.44, 1.48, 1.49]` `T` (default) | vector

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).

Anhysteretic B-H gradient when H is zero — Gradient of the anhysteretic B-H curve around zero field strength
`0.0063` `m*T/A` (default) | scalar

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.