Engee documentation

Nonlinear Inductor

An inductance coil with a nonlinear core.

nonlinear inductor

Description

Unit Nonlinear Inductor is an inductance coil with a non-ideal core. The core can be non-ideal due to its magnetic properties and dimensions. The block provides the following parameterization options:

Single inductance (linear)

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

Where:

  • - terminal voltage.

  • - current through the terminals.

  • - current through the inductor.

  • - parasitic parallel conductivity.

  • - number of winding turns.

  • - magnetic flux.

  • - unsaturated inductance.

One saturation point

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

(up to saturation point)

(after saturation point)

where:

  • - terminal voltage.

  • - current through the terminals.

  • - current through the inductor.

  • - parasitic parallel conductivity.

  • - number of winding turns.

  • - magnetic flux.

  • - magnetic flux saturation bias.

  • - unsaturated inductance.

  • - saturated inductance.

Characterisation of the dependence of magnetic flux on current

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

Where:

  • - terminal voltage.

  • - current through the terminals.

  • - current through the inductor.

  • - parasitic parallel conductivity.

  • - number of winding turns.

  • - magnetic flux.

Magnetic flux is determined using a one-dimensional table consisting of a vector of current values and a vector of corresponding magnetic flux values. Both negative and positive values can be used to specify these vectors, or only positive values can be used. If only positive data are used, the vector must start at 0, while negative data will be automatically calculated by symmetrical mapping with respect to the point (0,0).

Characterisation of the dependence of magnetic induction on magnetic field strength

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

Where:

  • - terminal voltage.

  • - current through the terminals.

  • - current through the inductor.

  • - parasitic parallel conductivity.

  • - number of winding turns.

  • - magnetic flux.

  • - magnetic induction.

  • - magnetic field strength.

  • - effective core length.

  • - 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. Both negative and positive values can be used to specify these vectors, or only positive values can be used. If only positive data are used, the vector must start at 0, while negative data will be automatically calculated by symmetric mapping with respect to the point (0,0).

Characterisation of the dependence of magnetic induction on magnetic field strength with hysteresis

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

μ

Where:

  • - terminal voltage.

  • - current through the terminals.

  • - current through the inductor.

  • - parasitic parallel conductivity.

  • - number of winding turns.

  • - magnetic flux.

  • - magnetic induction.

  • μ - magnetic constant.

  • - magnetic field strength.

  • - core magnetisation.

  • - effective core length.

  • - effective cross-sectional area of the core.

Magnetisation results in an increase in magnetic induction, and its magnitude depends on both the current value of the field strength H and its previous variation in time. The equations of the Giles-Atherton model are used to determine M at any point in time.

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

The member is called the anhysteresis magnetisation because it has no hysteresis. It is described by the following function on the current value of the effective field strength :

This function defines a saturation curve with limit values and a saturation point determined by the value of , the shape factor of the anhysteresis curve. Roughly, it can be considered to describe the average of two hysteresis curves. In the block Nonlinear Inductor values are set at and the points on the angysteresis curve B-H, which are used to determine the values and .

The parameters is the reversible magnetisation coefficient and determines which part of the behaviour is determined by , and which part is determined by the irreversible term . In the Giles-Atherton model, the irreversible term is determined by the partial derivative of the field strength:

δα

δ

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 δα .

The tracking error serves to create hysteresis at points where δ changes sign. The main parameter that forms the irreversible characteristic is K, which is called the bulk coupling coefficient. The parameter α is called the interdomain coupling coefficient and is also used to determine the effective field strength used in plotting the angysteresis curve:

α

The value of α affects the shape of the hysteresis curve: the larger it is, the higher the curve intersects the B-axis. However, it should be noted that the term δα , which must be positive at δ and negative at δ , is necessary for stability. Therefore, not all values of α are acceptable; a typical maximum value is of the order of 1e-3.

Procedure for finding approximate values of the coefficients of the Giles-Atherton equation

The following procedure can be used to determine suitable parameters for the coefficients of the equation:

  1. Specify the value of the parameter Anhysteretic B-H gradient when H is zero ( at ) plus the data point on the anti-hysteresis B-H curve. From these values, the values α and are determined during block initialisation .

  2. Set the value for the Coefficient for reversible magnetisation, c parameters so as to achieve the correct initial B-H derivative when the simulation is run from the point . The value of is approximately equal to the ratio of this initial derivative to the Anhysteretic B-H gradient when H is zero. The value of must be greater than 0 and less than 1.

  3. Set the value for the Bulk coupling coefficient parameters, K, to be approximately equal to the value of , when is on a positive hysteresis curve.

  4. Start with a very small value of α and gradually increase it to adjust the value of when the line crosses . A typical value is in the range of 1e-4 to 1e-3. Values that are too large cause the derivative of the B-H curve to tend to infinity, which is unphysical and results in a run-time assertion error.

It may be necessary to perform these steps several times to get a good match with the predefined B-H curve.

Ports

+ - positive
electricity

Electrical port, represents the positive terminal of the coil.

- is the negative
electricity

Electrical port, represents the positive terminal of the coil.

Parameters

Parameterised by - block parameterization
Single saturation point (by 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 values for number of turns, unsaturated inductance and parasitic parallel conductance.

  • Single inductance (linear) -specify values for number of turns, unsaturated and saturated inductance, saturation magnetic flux and parasitic parallel conductivity. This option is used by default.

  • Magnetic flux versus current characteristic - in addition to the number of turns and parasitic parallel conductance value, specify the current vector and magnetic flux vector to complete the magnetic flux versus current characteristic table.

  • `Magnetic flux density versus magnetic field strength characteristic' - in addition to the number of turns and the value of the parasitic parallel conductivity, give the values of the effective length and the cross-sectional area of the core, as well as the magnetic field strength vector and the magnetic induction vector to complete the magnetic induction versus magnetic field strength table.

  • `Magnetic flux density versus magnetic field strength characteristic with hysteresis' - in addition to the number of turns, effective length and cross-sectional area of the core, the values of the initial derivative of the B-H anhysteresis curve, magnetic induction and field strength at a particular point on the B-H curve, and the reversible magnetisation coefficient, volume coupling coefficient and interdomain coupling coefficient to determine the magnetic induction as a function of the current value and history of the magnetic field strength are also given.

Number of turns - total number of turns
10 (By default)

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

Unsaturated inductance - unsaturated inductance
2e-4 H (By default).

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

Dependencies

This parameter is used when Single inductance (linear) or Single saturation point is selected for the Parameterised by parameters.

Saturated inductance - saturated inductance
1e-4 H (By default).

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

Dependencies

This parameter is only used when Single saturation point is selected for the Parameterised by parameter.

Saturation magnetic flux - saturation magnetic flux
1.3e-05 Wb (By default).

The value of magnetic flux at which saturation of the inductor occurs.

Dependencies

This parameter is only used when Single saturation point is selected for the Parameterised by parameter.

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

Current values used to populate the magnetic flux-current dependence table.

Dependencies

This parameter is used when `Magnetic flux versus current characteristic' is selected for the Parameterised by parameters.

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

Magnetic flux values used to fill in the table of magnetic flux-current dependence.

Dependencies

This parameter is used when `Magnetic flux versus current characteristic' is selected for the Parameterised by parameters.

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

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

Dependencies

This parameter is used if `Magnetic flux versus current characteristic' is selected for the Parameterised by parameter.

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

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

Dependencies

This parameter is used if `Magnetic flux versus current characteristic' is selected for the Parameterised by parameter.

Effective length - effective length of the core
0.032 m (By default).

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

Dependencies

This parameter is used when `Magnetic flux density versus magnetic field strength characteristic' or `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.

Effective cross-sectional area, m² - effective cross-sectional area
1.6e-5 m^2 (by default).

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

Dependencies

This parameter is used when `Magnetic flux density versus magnetic field strength characteristic' or `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.

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

The derivative of the anhysteresis (no hysteresis) B-H curve near zero field strength. Set as the average of the derivative of the positive and negative hysteresis curves.

Dependencies

This parameter is used if `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.

Flux density point on anhysteretic B-H curve - value of magnetic induction at a point on anhysteresis B-H curve
`1.49 T (By default)'.

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

Dependencies

This parameter is used when `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.

Corresponding field strength - Corresponding field strength
1000 A/m (By default).

Corresponding field strength for the point given by the parameters Flux density point on anhysteretic B-H curve.

Dependencies

This parameter is used if `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.

Coefficient for reversible magnetisation, c - coefficient of reversible magnetization
0.1 (by default).

The fraction of magnetisation that is reversible. The value must be greater than zero and less than one.

Dependencies

This parameter is used if `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.

Bulk coupling coefficient, K - Bulk coupling coefficient
200 A/m (By default).

A parameter of the Giles-Atherton model, primarily determining the value of field strength at which the B-H curve crosses the line of zero magnetic induction.

Dependencies

This parameter is used if `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.

Inter-domain coupling factor, α - inter-domain coupling factor
1e-4 (By default).

Giles-Atherton parameters, affecting primarily the points of intersection of the B-H curves with the zero field strength line. Typical values range from 1e-4 to 1e-3.

Dependencies

This parameter is used if `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.

Parasitic parallel conductance - parasitic parallel conductance
1e-9 1/Ohm (by default)

This parameters is used to represent small parasitic effects. Small parallel conductance may be required for modelling some circuit topologies.

Interpolation option - interpolation option
Linear(By default) | Smooth

Search table interpolation option. Select one of the following interpolation methods:

  • Linear - select this option for best performance.

  • Smooth - select this option to obtain a continuous curve with continuous first order derivatives.

Specify state by - option to specify initial state
Current (By default) | Magnetic flux.

Select the appropriate option for specifying the initial state:

  • Current - set the initial state of the inductor by the initial current through the inductor (iL). This option is used by default.

  • Magnetic flux - set the initial state of the inductor by the magnetic flux.

Dependencies

This parameter is not used if `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.

Initial current - initial current
0 A (By default).

Initial current value used to calculate the magnetic flux value at zero time. This is the current flowing through the inductor. It consists of the current through the inductor and the current through the parasitic parallel conductance.

Dependencies

This parameter is only used when Current is selected for the Specify initial state by parameter.

Initial magnetic flux - initial magnetic flux
0 Wb (By default).

Value of magnetic flux at zero moment of time.

Dependencies

This parameter is only used when Magnetic flux is selected for the Specify initial state by parameter.

Initial magnetic flux density - initial magnetic induction
0 T (By default).

Value of magnetic induction at zero moment of time.

Dependencies

This parameter is used if `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.

Initial field strength - initial field strength
0 A/m (By default).

Value of magnetic field strength at zero moment of time.

This parameter is used if `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the Parameterised by parameter.