Magnetic Core

Nonlinear reluctance, hysteresis, and optional eddy losses

Since R2024b

Libraries:
Simscape / Electrical / Passive

Description

The Magnetic Core block models magnetic reluctance with hysteresis. Use this block to model custom inductances and build transformer models. To connect this block to your electrical network, use a Winding block to represent the interface between the electrical and magnetic domains.

The block uses the Jiles-Atherton model to calculate the magnetic flux density B and magnetic field strength H. You can choose to specify the Jiles-Atherton parameters directly, specify characteristic magnetic parameters, or tabulate the B-H saturation curve using the Parameterization option parameter.

To define the geometry for the part of the magnetic network that you are modeling, use the Effective length and Effective cross-sectional area parameters. The block uses this geometry information to map B and H to the magnetomotive force and magnetic flux, which are the Through and Across variables in the magnetic domain.

To model eddy losses, in the Losses settings, select the Model eddy losses parameter. To model thermal effects, in the Thermal settings, select the Model thermal effects parameter.

Equations

These equations define the flux density and magnetomotive force,

$B = {φ / s}_{e f f}$

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

where:

B is the magnetic flux density induced in the core.
φ is the magnetic flux.
s_eff is the value of the Effective cross-sectional area parameter.
mmf is the magnetomotive force (mmf) across the component.
l_eff is the value of the Effective length parameter.
H is the total field strength.

This equation relates B and H to the total magnetization of the core M,

$B = μ_{0} (H + M),$

where μ₀ is the magnetic permeability of a vacuum.

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. The block uses the Jiles-Atherton [1,2] equations to determine M at any given time. This figure shows a typical plot of the resulting relationship between B and H.

In this figure:

B_m is the magnetic flux density at the saturation point.
B_r is the magnetic remanence of the saturation cycle.
H_c is the magnetic coercivity of the saturation cycle.
H_m is the magnetic field intensity at the saturation point.

As the field strength increases from H = 0, the plot initially follows an ascending hysteresis curve. At the saturation point, further increases in field intensity no longer cause significant increases in magnetic flux.

As you reduce the magnetic field strength from the saturation point, the plot follows a descending hysteresis curve. The difference between ascending and descending 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 the split of the magnetization effect into two parts, one part that is purely a function of effective field strength H_eff and another, irreversible part that depends on history. This equation defines the relative contributions of the anhysteretic magnetisation M_an and the irreversible magnetization M_irr to the total magnetization M,

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

where c is the coefficient for reversible magnetization.

This equation relates M_an to the equivalent magnetization H_eff:

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

The function defines a saturation curve with limiting values ±M_s, where:

M_s is the saturation magnetization.
a is the anhysteretic magnetization coefficient, which determines the point of saturation. It approximately describes the average of the two hysteretic curves.

The Jiles-Atherton model defines the irreversible term by a partial derivative with respect to field strength,

$\begin{array}{l} \frac{d M_{i r r}}{d H} = \frac{M_{a n} - M_{i r r}}{k δ - α (M_{a n} - M_{i r r})} \\ δ = {\begin{array}{l} 1 & if H \geq 0 \\ - 1 & if H < 0 \end{array} \end{array}$

where:

k is the bulk coupling coefficient, which shapes the irreversible characteristic.
α is the inter-domain coupling factor.

Comparison of the equation with a standard first-order differential equation reveals that as you change the total field strength H, 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.

This equation defines the effective field strength of an anhysteretic curve without eddy losses,

$H_{e f f} = H + α M .$

The value of α affects the shape of the hysteresis curve. Larger values act to increase the B-axis intercepts. However, the stability term $k δ - α (M_{a n} - M_{i r r})$ must be positive for δ > 0 and negative for δ < 0. There are therefore limits on the values of α that you can provide. A typical maximum value is of the order 1e-3.

You can choose how to parameterize the B-H curve. Set the Parameterization option parameter to one of these options:

Jiles-Atherton Model — Specify the Jiles-Atherton parameters directly:
- Saturation magnetization M_s
- Anhysteretic magnetization constant a
- Inter-domain coupling factor α
- Coefficient for reversible magnetization c
- Bulk coupling coefficient k
Saturated BH Curve — Tabulate the ascending or descending B-H curve. You must specify at least one data point in the first and third quadrant. For ascending curves, you must also specify at least one point in the fourth quadrant. For descending curves, you must also specify at least one point in the second quadrant.
Magnetic Characteristics — Specify the magnitude of the B and H intercepts, B_r and H_c, and the magnitude of B and H at the saturation point, B_m and H_m.

Model Losses

The block computes hysteresis losses and eddy current losses.

The block calculates hysteresis losses using an averaging time. The instantaneous value of the area inside the B-H curve is proportional to the sum of the magnetic energy of the core and the thermal power dissipation. To obtain accurate results, you must specify an Averaging period for hysteresis losses parameter value that is greater than the B-H cycle time. If you specify a smaller value, the hysteresis loss at some instances is equal to the magnetic energy of the core which is larger than the real losses. The best option is to specify a value that is an exact multiple of the B-H cycle time.

These equations define the energy density dissipated by eddy currents W_eddy in the time interval Δt [3-5],

$W_{e d d y} = \frac{σ d^{2}}{12 β} {\int_{Δ t} (\frac{d B}{d t})}^{2} d t$

$\frac{d W_{e d d y}}{d t} = \frac{σ d^{2}}{2 β} {(\frac{d B}{d t})}^{2}$

where:

σ is the conductivity.
d is the value of the Thickness of laminations parameter. If the core does not have laminations, this value corresponds to the thickness at the widest cross-section; for example, the diameter of cylindrical or spherical cores.
β is the value of the Geometrical coefficient parameter. For a laminated core, β = 6. For a cylindrical core, β = 16. For a spherical core, β = 20 [5].

Eddy currents induce a magnetic field H_Eddy:

$H_{E d d y} = \frac{σ d^{2}}{12 β} \frac{d B}{d t} .$

This field also modifies the effective field strength:

$H_{e f f} = H + α M - H_{E d d y} .$

Variables

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

Note

In the Initial Targets section, you can specify values for the Field strength and Flux density parameters. The values you provide must approximately correspond to a point on the B-H curve.

Use nominal values 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 these sources is the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

Examples

New

Parameterize a Magnetic Core Block Using B-H Curve Data

Parameterize a magnetic core block using B-H curve data. This block represents a magnetic core with magnetic hysteresis implementing a Jiles-Atherton model.

Since R2024b
Open Live Script

New

Model Hysteresis and Eddy Current Losses in an Iron Core Using BH Curve Data

Model hysteresis and eddy current losses of an iron core using the magnetic core block. This block represents a magnetic core with magnetic hysteresis and implements the Jiles-Atherton model. You can parameterize the block using BH curve data or Jiles-Atherton parameters.

Since R2024b
Open Live Script

Ports

Conserving

expand all

N — North terminal
magnetic

Magnetic conserving port associated with the north terminal.

S — South terminal
magnetic

Magnetic conserving port associated with the south terminal.

H — Thermal mass
thermal

Thermal port associated with the thermal mass.

Dependencies

To enable this port, select the Model thermal effects parameter.

Parameters

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To edit block parameters interactively, use the Property Inspector. From the Simulink^® Toolstrip, on the Simulation tab, in the Prepare gallery, select Property Inspector.

Main

Effective length — Effective core length
`0.032` `m` (default) | positive finite scalar

Effective core length. This parameter represents the average length of the magnetic path around the core.

Effective cross-sectional area — Effective core cross-sectional area
`1.6e-5` `m^2` (default) | positive finite scalar

Effective core cross-sectional area. This parameter represents the average area of the magnetic path around the core.

Parameterization option — Parameterization method
`Magnetic Characteristics` (default) | `Jiles-Atherton Model` | `Saturated BH Curve`

Parameterization method that you use to describe the B-H curve. Choose one of these options:

Magnetic Characteristics — Specify the magnitude of the B and H intercepts, B_r and H_c, and the magnitude of B and H at the saturation point, B_m and H_m.
Jiles-Atherton Model — Specify the Jiles-Atherton parameters directly.
Saturated BH Curve — Tabulate the ascending or descending B-H curve.

Magnetic remanence of saturation cycle — Residual magnetism
`0.68` `T` (default) | positive scalar

Residual magnetism left in the core after it magnetizes to its saturation point and the external magnetic field reduces to zero, B_r. This parameter is the magnitude of the flux density where the B-H curve intersects the B-axis.

Dependencies

To enable this parameter, set Parameterization option to Magnetic Characteristics.

Magnetic coercivity of saturation cycle — Magnitude of field strength to demagnetize core
`151` `A/m` (default) | positive scalar

Magnitude of the magnetic field strength that you need to demagnetize the core, H_c. This parameter is the magnitude of the field intensity where the B-H curve intersects the H-axis.

To enable this parameter, set Parameterization option to Magnetic Characteristics.

Magnetic flux density at saturation point — Maximum flux density magnitude
`1.25` `T` (default) | positive scalar

Maximum flux density magnitude that the core can achieve, B_m.

To enable this parameter, set Parameterization option to Magnetic Characteristics or Saturated BH Curve.

Magnetic field intensity at saturation point — Field intensity at saturation point
`3000` `A/m` (default) | positive scalar

Field intensity at the saturation point, H_m. This parameter is the magnitude of the field intensity at which further increases in the magnitude of H no longer cause significant increases in the magnitude of B.

To enable this parameter, set Parameterization option to Magnetic Characteristics.