Nonlinear Inductor

An inductor with a non-linear core.

blockType: AcausalElectricPowerSystems.Passive.NonlinearInductor

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Description

Block Nonlinear Inductor It is an inductor coil with an imperfect core. The core may be imperfect due to its magnetic properties and size. The block provides the following parameterization options:

Single inductance (linear)
One saturation point
Characteristics of the dependence of the magnetic flux on the current
Characterization of the dependence of magnetic induction on the magnetic field strength
Characterization of the dependence of magnetic induction on the strength of a magnetic field with hysteresis

Single inductance (linear)

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

where

— terminal voltage;
— current through the terminals;
— current through the inductor;
— parasitic parallel conduction;
— the number of turns of the winding;
— magnetic flux;
— unsaturated inductance.

One saturation point

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

(up to saturation point),

(after the saturation point),

where

— terminal voltage;
— current through the terminals;
— current through the inductor;
— parasitic parallel conduction;
— the number of turns of the winding;
— magnetic flux;
— displacement of the saturation of the magnetic flux;
— unsaturated inductance;
— saturated inductance.

Characteristics of the dependence of the magnetic flux on the current

The relationships between voltage, current, and flow are determined by the following equations:

where

— terminal voltage;
— current through the terminals;
— current through the inductor;
— parasitic parallel conduction;
— the number of turns of the winding;
— magnetic flux.

The magnetic flux is determined using a one-dimensional table consisting of a vector of current values and a vector of corresponding magnetic flux values. To set these vectors, you can use both negative and positive values, or only positive ones. If only positive data is used, then the vector should start from 0, and negative data will be automatically calculated by symmetrical mapping relative to the point (0,0).

Characterization of the dependence of magnetic induction on the magnetic field strength

The relationships between voltage, current, and flow are determined by the following equations:

where

— terminal voltage;
— current through the terminals;
— current through the inductor;
— parasitic parallel conduction;
— the number of turns of the winding;
— magnetic flux;
— magnetic induction;
— magnetic field strength;
— effective core length;
— the effective cross-sectional area of the core.

Magnetic induction is determined using a one-dimensional table consisting of a vector of magnetic field strength values and a vector of corresponding magnetic induction values. To set these vectors, you can use both negative and positive values, or only positive ones. If only positive data is used, then the vector should start from 0, and negative data will be automatically calculated by symmetrical mapping relative to the point (0,0).

Characterization of the dependence of magnetic induction on the strength of a magnetic field with hysteresis

The relationships between voltage, current, and flow are determined by the following equations:

where

— terminal voltage;
— current through the terminals;
— current through the inductor;
— parasitic parallel conduction;
— the number of turns of the winding;
— magnetic flux;
— magnetic induction;
— magnetic constant;
— magnetic field strength;
— core magnetization;
— effective core length;
— the effective cross-sectional area of the core.

Magnetization leads to an increase in magnetic induction, and its magnitude depends both on the current value of the field strength H and on its previous change over time. The equations of the Giles-Atherton model are used to determine M at any given time.

The starting point for the Giles-Atherton equation is to divide the magnetization effect into two parts, one of which is purely a function of the effective field strength ( ), and the other is an irreversible part that depends on past history:

Member It is called anhysteresis magnetization because it does not have hysteresis. It is described by the following function of the current value of the effective field strength :

This function sets the saturation curve with the limit values. and the saturation point, determined by the value — coefficient of the shape of the anhysteresis curve. Approximately, it can be assumed that it describes the average value of two hysteresis curves. In the block Nonlinear Inductor values are set by and the dots on the anhysteresis curve B-H, which are used to determine the values of and .

Parameter it is the coefficient of reversible magnetization and determines which part of the behavior determines And which one is an irreversible member . In the Giles-Atherton model, the irreversible term is determined by the partial derivative of the field strength:

A comparison of this equation with the standard first-order differential equation shows that as the field strength H increases, the irreversible term follows the reversible term , but with a variable gain .

The tracking error is used to create hysteresis at the points where δ changes sign. The main parameter forming the irreversible characteristic is K, which is called the volumetric coupling coefficient. Parameter It is called the inter-domain coupling coefficient and is also used to determine the effective field strength used in constructing the anhysteresis curve.:

Meaning affects the shape of the hysteresis curve: the larger it is, the higher the curve intersects with the B-axis. However, it should be noted that for stability it is necessary to use a member which should be positive when and negative when . Therefore, not all values of α are acceptable, and the typical maximum value is of the order of 1e−3.

The procedure for finding approximate values of the coefficients of the Giles-Atherton equation

The appropriate parameters for the coefficients of the equation can be determined using the following procedure:

Specify the value of the parameter Anhysteretic B-H gradient when H is zero ( by ) plus a data point on the anti-hysteresis curve B-H. The values are determined from these values during block initialization and .
Set the value for the parameter Coefficient for reversible magnetization, c, so as to achieve the correct initial derivative B-H when starting the simulation from a point . Meaning approximately equal to the ratio of this initial derivative to the Anhysteretic B-H gradient when H is zero. Meaning There should be more 0 And less 1.
Set the value for the parameter Bulk coupling coefficient, K, approximately equal to When on the positive hysteresis curve.
Start with a very small value and gradually increase it to adjust the value. when crossing the line . A typical value is in the range of 1e−4 before 1e−3. Too large values lead to the fact that the derivative of the B-H curve tends to infinity, which is unphysical and leads to a statement error during program execution.

To get a good match with the predefined B-H curve, you may need to perform these steps several times.

Ports

+ — positive
electricity

The electrical port represents the positive terminal of the coil.

− — negative
electricity

The electrical port represents the positive terminal of the coil.

Parameters

Parameterized by — parameterization of the pass block:q[<br>] 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 block parameterization methods:

Single saturation point (default) — specify the values of the number of turns, unsaturated inductance and parasitic parallel conductivity.
Single inductance (linear) — the values of the number of turns, unsaturated and saturated inductors, saturation magnetic flux and parasitic parallel conductivity are indicated. This option is used by default.
Magnetic flux versus current characteristic — in addition to the number of turns and the value of the parasitic parallel conductivity, specify the current vector and the magnetic flux vector to fill in the table of the dependence of the magnetic flux on the current.
Magnetic flux density versus magnetic field strength characteristic — in addition to the number of turns and the value of parasitic parallel conductivity, specify the values of the effective length and cross-sectional area of the core, as well as the vector of magnetic field strength and the vector of magnetic induction to fill in the table of dependence of magnetic induction on magnetic field strength.
Magnetic flux density versus magnetic field strength characteristic with hysteresis — in addition to the number of turns, the effective length and the cross-sectional area of the core, the values of the initial derivative of the anhysteresis curve B-H, magnetic induction and field strength at a certain point of the curve B-H are indicated, as well as the coefficient of reversible magnetization, the volume coupling coefficient and the inter-domain coupling coefficient for determining magnetic induction depending on the current value and history changes in magnetic field strength.

Number of turns — total number of turns of the
10 (default)

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

Unsaturated induction — unsaturated inductance
2e−4H (default)

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

Dependencies

This parameter is used if the Parameterized by parameter is set to Single inductance (linear) or Single saturation point.

Saturated induction — saturated inductance
1e−4 H (default)

The value of the inductance used when the inductor is operating in the saturation zone.

Dependencies

This parameter is used only when selecting a value. Single saturation point for the Parameterized by parameter.

Saturation magnetic flux — saturated magnetic flux
1.3e−05 Wb (default)

The value of the magnetic flux at which the inductor is saturated.

Dependencies

This parameter is used only when selecting a value. Single saturation point for the Parameterized by parameter.

Current vector, i — vector of current values
[0, .64, 1.28, 1.92, 2.56, 3.2] A (default)

The current values used to fill in the table of magnetic flux versus current.

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux versus current characteristic.

Magnetic flux vector, Φ, Wb is a vector of values of the magnetic flux
[0, 1.29, 2, 2.27, 2.36, 2.39] .* 1e−5 Wb (default)

The values of the magnetic flux used to fill in the table of the dependence of the magnetic flux on the current.

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux versus current characteristic.

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

The values of the magnetic field strength used to fill in the table of the dependence of magnetic induction on the magnetic field strength.

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux versus current characteristic.

Magnetic flux density vector, B — vector of values of magnetic induction
[0, .81, 1.25, 1.42, 1.48, 1.49] T (default)

The values of magnetic induction used to fill in the table of dependence of magnetic induction on the strength of the magnetic field.

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux versus current characteristic.

Effective length — effective length of the pass core:q[<br>] 0.032 m (default)

The effective length of the core, i.e. the average length of the magnetic flux path.

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux density versus magnetic field strength characteristic or Magnetic flux density versus magnetic field strength characteristic with hysteresis.

Effective cross-sectional area, m2 — effective cross-sectional area of
1.6e−5 m^2 (default)

The effective cross-sectional area of the core, i.e. the average area of the magnetic flux path.

Dependencies

Anhysteretic B-H gradient when H is zero, T*m/A is the derivative of the anhysteretic curve B—H near zero field strength
0.005 m*T/A (default)

The derivative of the anhysteretic (without hysteresis) B-H curve is near zero field strength. It is set as the average value of the derivative of the positive and negative hysteresis curves.

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux density versus magnetic field strength characteristic with hysteresis.

Flux density point on anhysteretic B-H curve — the value of magnetic induction at a point on the anhysteretic curve B-H
1.49 T (default)

Specify the magnetic induction value at a point on the anhysteresis curve. The most accurate option is to select a point at high field strength, when the positive and negative hysteresis curves coincide.

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux density versus magnetic field strength characteristic with hysteresis.

Corresponding field strength — corresponding field strength
1000 A/m (default)

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

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux density versus magnetic field strength characteristic with hysteresis.

Coefficient for reversible magnetization, c is the coefficient of reversible magnetization
0.1 (default)

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

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux density versus magnetic field strength characteristic with hysteresis.

Bulk coupling coefficient, K is the volume coupling coefficient of
200 A/m (default)

A parameter of the Giles-Atherton model, which primarily determines the magnitude of the field strength at which the B-H curve intersects the line of zero magnetic induction.

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux density versus magnetic field strength characteristic with hysteresis.

Inter-domain coupling factor, α is the coefficient of inter—domain coupling
1e−4 (default)

The Giles-Atherton parameter, which primarily affects the intersection points of the B-H curves with the zero field strength line. Typical values range from 1e−4 before 1e−3.

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux density versus magnetic field strength characteristic with hysteresis.

Parasitic parallel conduction — parasitic parallel conduction
1e−9 1/Ohm (default)

This parameter is used to represent small spurious effects. A small amount of parallel conduction may be required to simulate some circuit topologies.

Interpolation option — pass interpolation option:q[<br>] Linear(by default) | Smooth

The option to interpolate the search table. Choose one of the following interpolation methods:

Linear — Select this option to get the best performance.
Smooth — select this option to obtain a continuous curve with continuous first-order derivatives.

Specify state by — option for setting the initial state of
Current(by default) | Magnetic flux

Select the appropriate option for setting the initial state:

Current — setting the initial state of the inductor by the initial current through the inductor (iL). This option is used by default.
Magnetic flux — setting the initial state of the inductor by magnetic flux.

Dependencies

This parameter is not used if the Parameterized by parameter is set to Magnetic flux density versus magnetic field strength characteristic with hysteresis.

Initial current — initial current
0 A (default)

The initial value of the current used to calculate the value of the magnetic flux at time zero. This is the current passing through the inductor. It consists of a current passing through an inductor and a current passing through a parasitic parallel conduction.

Dependencies

This parameter is used only when selecting a value. Current for the Specify initial state by parameter.

Initial magnetic flux — initial magnetic flux
0 Wb (default)

The value of the magnetic flux at time zero.

Dependencies

This parameter is used only when selecting a value. Magnetic flux for the Specify initial state by parameter.

Initial magnetic flux density — initial magnetic induction
0 T (default)

The value of magnetic induction at time zero.

Dependencies

This parameter is used if the Parameterized by parameter is set to Magnetic flux density versus magnetic field strength characteristic with hysteresis.

Initial field strength — initial field strength
0 A/m (default)

The value of the magnetic field strength at time zero.

This parameter is used if the Parameterized by parameter is set to Magnetic flux density versus magnetic field strength characteristic with hysteresis.