Nonlinear Inductor

Inductor with nonideal core

Libraries:
Simscape / Electrical / Passive

Description

The Nonlinear Inductor block represents an inductor with a nonideal core. A core may be nonideal due to its magnetic properties and dimensions. The block provides the following parameterization options:

Single Inductance (Linear)

The relationships between voltage, current and flux are defined by the following equations:

$i = i_{L} + v G_{p}$

$v = N_{w} \frac{d Φ}{d t}$

$Φ = \frac{L}{N_{w}} i_{L}$

where:

v is the terminal voltage.
i is the terminal current.
i_L is the current through inductor.
G_p is the parasitic parallel conductance.
N_w is the number of winding turns.
Φ is the magnetic flux.
L is the unsaturated inductance.

Single Saturation Point

The relationships between voltage, current and flux are defined by the following equations:

$i = i_{L} + v G_{p}$

$v = N_{w} \frac{d Φ}{d t}$

$Φ = \frac{L}{N_{w}} i_{L} (for unsaturated)$

$Φ = \frac{L_{s a t}}{N_{w}} i_{L} \pm Φ_{o f f s e t} (for saturated)$

where:

v is the terminal voltage.
i is the terminal current.
i_L is the current through inductor.
G_p is the parasitic parallel conductance.
N_w is the number of winding turns.
Φ is the magnetic flux.
Φ_offset is the magnetic flux saturation offset.
L is the unsaturated inductance.
L_sat is the saturated inductance.

Magnetic Flux Versus Current Characteristic

The relationships between voltage, current and flux are defined by the following equations:

$i = i_{L} + v G_{p}$

$v = N_{w} \frac{d Φ}{d t}$

$Φ = f (i_{L})$

where:

v is the terminal voltage.
i is the terminal current.
i_L is the current through inductor.
G_p is the parasitic parallel conductance.
N_w is the number of winding turns.
Φ is the magnetic flux.

Magnetic flux is determined by one-dimensional table lookup, based on the vector of current values and the vector of corresponding magnetic flux values that you provide. You can construct these vectors using either negative and positive data, or positive data only. If using positive data only, the vector must start at 0, and the negative data will be automatically calculated by rotation about (0,0).

Magnetic Flux Density Versus Magnetic Field Strength Characteristic

The relationships between voltage, current and flux are defined by the following equations:

$i = i_{L} + v G_{p}$

$v = N_{w} \frac{d Φ}{d t}$

$Φ = B \cdot A_{e}$

$B = f (H)$

$H = \frac{N_{w}}{l_{e}} i_{L}$

where:

v is the terminal voltage.
i is the terminal current.
i_L is the current through inductor.
G_p is the parasitic parallel conductance.
N_w is the number of winding turns.
Φ is the magnetic flux.
B is the magnetic flux density.
H is the magnetic field strength.
l_e is the effective core length.
A_e is the effective core cross-sectional area.

Magnetic flux density is determined by one-dimensional table lookup, based on the vector of magnetic field strength values and the vector of corresponding magnetic flux density values that you provide. You can construct these vectors using either negative and positive data, or positive data only. If using positive data only, the vector must start at 0, and the negative data will be automatically calculated by rotation about (0,0).

Magnetic Flux Density Versus Magnetic Field Strength Characteristic with Hysteresis

The relationships between voltage, current and flux are defined by the following equations:

$i = i_{L} + v G_{p}$

$v = N_{w} \frac{d Φ}{d t}$

$Φ = B \cdot A_{e}$

$B = μ_{0} (H + M)$

$H = \frac{N_{w}}{l_{e}} i_{L}$

where:

v is the terminal voltage.
i is the terminal current.
i_L is the current through inductor.
G_p is the parasitic parallel conductance.
N_w is the number of winding turns.
Φ is the magnetic flux.
B is the magnetic flux density.
μ₀ is the magnetic constant, permeability of free space.
H is the magnetic field strength.
M is the magnetization of the inductor core.
l_e is the effective core length.
A_e is the effective core cross-sectional area.

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 Jiles-Atherton [1, 2] equations are used 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:

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

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 Nonlinear Inductor block, 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 a 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:

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

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 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 a 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.

Examples

Nonlinear Inductor Characteristics

A comparison of nonlinear inductor behavior for different parameterizations. Starting with fundamental parameter values, the parameters for linear and nonlinear representations are derived. These parameters are then used in a Simscape™ model and the simulation outputs compared.

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

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+ — Positive terminal
electrical

Electrical conserving port associated with the inductor positive voltage.

- — Negative terminal
electrical

Electrical conserving port associated with the inductor negative voltage.

Parameters

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Main

Parameterized by — Block parameterization
`Single saturation point` (default) | `Single inductance (linear)` | `Magnetic flux versus current characteristic` | `Magnetic flux density versus magnetic field strength characteristic` | `Magnetic flux density versus magnetic field strength characteristic with hysteresis`

Select one of the following methods for block parameterization:

Single inductance (linear) — Provide the values for number of turns, unsaturated inductance, and parasitic parallel conductance.
Single saturation point — Provide the values for number of turns, unsaturated and saturated inductances, saturation magnetic flux, and parasitic parallel conductance. This is the default option.
Magnetic flux versus current characteristic — In addition to the number of turns and the parasitic parallel conductance value, provide the current vector and the magnetic flux vector, to populate the magnetic flux versus current lookup table.
Magnetic flux density versus magnetic field strength characteristic — In addition to the number of turns and the parasitic parallel conductance value, provide the values for effective core length and cross-sectional area, as well as the magnetic field strength vector and the magnetic flux density vector, to populate the magnetic flux density versus magnetic field strength lookup table.
Magnetic flux density versus magnetic field strength characteristic with hysteresis — In addition to the number of turns and the effective core length and cross-sectional area, provide the values for the initial anhysteretic B-H curve gradient, the magnetic flux density and field strength at a certain point on the B-H curve, as well as the coefficient for the reversible magnetization, bulk coupling coefficient, and inter-domain coupling factor, to define magnetic flux density as a function of both the current value and the history of the field strength.

Number of turns — Total number of turns
`10` (default)

The total number of turns of wire wound around the inductor core.

Unsaturated inductance — Unsaturated inductance
`2e-4` `H` (default)

The value of inductance used when the inductor is operating in its linear region.

Dependencies

This parameter is visible only when you select Single inductance (linear) or Single saturation point for the Parameterized by parameter.

Saturated inductance — Saturated inductance
`1e-4` `H` (default)

The value of inductance used when the inductor is operating beyond its saturation point.

Dependencies

This parameter is visible only when you select Single saturation point for the Parameterized by parameter.

Saturation magnetic flux — Saturation magnetic flux
`1.3e-05` `Wb` (default)

The value of magnetic flux at which the inductor saturates.

Dependencies

This parameter is visible only when you select Single saturation point for the Parameterized by parameter.

Current vector, i — Current data vector
`[0, .64, 1.28, 1.92, 2.56, 3.2]` `A` (default)

The current data used to populate the magnetic flux versus current lookup table.

Dependencies

This parameter is visible only when you select Magnetic flux versus current characteristic for the Parameterized by parameter.

Magnetic flux vector, phi — Magnetic flux data
`[0, 1.29, 2, 2.27, 2.36, 2.39] .* 1e-5` `Wb` (default)

The magnetic flux data used to populate the magnetic flux versus current lookup table.

Dependencies

This parameter is visible only when you select Magnetic flux versus current characteristic for the Parameterized by parameter.

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

The magnetic field strength data used to populate the magnetic flux density versus magnetic field strength lookup table.

Dependencies

This parameter is visible only when you select Magnetic flux density versus magnetic field strength characteristic for the Parameterized by parameter.

Magnetic flux density vector, B — Magnetic flux density data
`[0, .81, 1.25, 1.42, 1.48, 1.49]` `T` (default)

The magnetic flux density data used to populate the magnetic flux density versus magnetic field strength lookup table.

Dependencies

This parameter is visible only when you select Magnetic flux density versus magnetic field strength characteristic for the Parameterized by parameter.

Effective length — Effective core length
`0.032` `m` (default)

The effective core length, that is, the average distance of the magnetic path.

Dependencies

This parameter is visible only when you select Magnetic flux density versus magnetic field strength characteristic or Magnetic flux density versus magnetic field strength characteristic with hysteresis for the Parameterized by parameter.

Effective cross-sectional area — Effective cross-sectional area
`1.6e-5` `m^2` (default)

The effective core cross-sectional area, that is, the average area of the magnetic path.

Dependencies

Anhysteretic B-H gradient when H is zero — Anhysteretic B-H gradient around zero field strength
`0.005` `m*T/A` (default)

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.