Engee documentation

Nonlinear Transformer

Transformer with a non-ideal core.

nonlinear transformer

Description

Unit Nonlinear Transformer is a transformer with a non-ideal core. The core can be non-ideal due to its magnetic properties and dimensions.

The equivalent circuit of a two-winding transformer depends on which of the two options is used for the Winding parameterised by:

  • `Combined primary and secondary values

nonlinear transformer 1 s

  • Separate primary and secondary values

nonlinear transformer 2 s

where:

  • Req is the total winding resistance.

  • Leq - total inductance of the scattering.

  • R1 - resistance of the primary winding.

  • L1 - scattering inductance of the primary winding.

  • R2 - resistance of the secondary winding.

  • L2 - scattering inductance of the secondary winding.

  • Rm - magnetising resistance.

  • Lm - magnetising inductance.

The figure below shows the equivalent circuit of a three-winding transformer:

three winding nonlinear transformer 1 s

Where:

  • R1 - resistance of the primary winding.

  • L1 - scattering inductance of the primary winding.

  • R2 - resistance of the first secondary winding.

  • L2 - scattering inductance of the first secondary winding.

  • R2 - resistance of the second secondary winding.

  • L2 - scattering inductance of the second secondary winding.

  • Rm - magnetising resistance.

  • Lm - magnetising inductance.

The block provides the following options for parameterization of nonlinear magnetising inductance:

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 transformer.

  • - parasitic parallel conductance.

  • - 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 transformer.

  • - parasitic parallel conductance.

  • - 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 transformer.

  • - parasitic parallel conductance.

  • - 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 transformer.

  • - parasitic parallel conductance.

  • - 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 transformer.

  • - parasitic parallel conductance.

  • - 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 Transformer 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, K, A/m parameters 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 it crosses the line . 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

Non-directional

1+ - positive terminal of the primary winding
electricity

Electricity port, represents the primary winding terminal with positive polarity.

1- is the negative terminal of the primary winding
electricity

Electricity port, represents the primary winding terminal with negative polarity.

2+ represents the positive terminal of the secondary winding
electricity

Electricity port, represents the secondary winding terminal with positive polarity.

2- is the negative terminal of the secondary winding
electricity

Electricity port, represents the secondary winding terminal with negative polarity.

3+ is the positive terminal of the second secondary winding
electricity

Electricity port, represents the terminal of the second secondary winding with positive polarity.

Dependencies

To use this port, set the Number of windings parameters to Three.

3- is the negative terminal of the second secondary winding
electricity

Electricity port, represents the terminal of the second secondary winding with negative polarity.

Dependencies

To use this port, set the Number of windings parameters to Three.

Parameters

Number of windings - Switching between two and three winding transformer.
Two (By default) | Three.

Switching between two and three windings.

  • Two - the block simulates a two-winding transformer.

  • Three - the block simulates a three-winding transformer.

Primary number of turns - number of turns of the primary winding
`100 (By default).

The number of turns of wire on the primary winding of the transformer.

Secondary number of turns - number of turns of the secondary winding
`200 (By default).

The number of turns of wire on the secondary winding of the transformer.

Tertiary number of turns - number of turns of the second secondary winding
200 (By default)

The number of turns of wire on the second secondary winding of the transformer.

Dependencies

To use this parameter, set the Number of windings parameter to Three.

Winding parameterized by - winding dissipation type
Combined primary and secondary values (by default) | `Separate primary and secondary values `

Select one of the following methods to specify the winding dissipation:

  • Combined primary and secondary values - use concentrated resistance and inductance values representing the combined leakage in the primary and secondary windings.

  • Separate primary and secondary values - use separate resistance and inductance values to represent primary and secondary leakage.

*Combined winding resistance` - Combined winding resistance
0.01 (By default).

The combined equivalent resistance Req, which represents the combined power losses of the primary and secondary windings.

Dependencies

This parameter is only used when Combined primary and secondary values is selected for the Winding parameterised by parameter.

Combined leakage inductance - Combined leakage inductance
0.0001 (By default).

The combined equivalent inductance Leq, which is the combined magnetic flux losses of the primary and secondary windings.

Dependencies

This parameter is only used when Combined primary and secondary values is selected for the Winding parameterised by parameter.

Primary winding resistance - primary winding resistance
0.01 (By default)

Resistance R1, which represents the power loss of the primary winding.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by parameter.

Primary leakage inductance - primary leakage inductance
0.0001 (By default)

The L1 inductance, which represents the magnetic flux losses of the primary winding.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by parameter.

Secondary winding resistance - secondary winding resistance
0.01 (by default)

Resistance R2, which represents the power loss of the secondary winding.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by parameter.

Secondary leakage inductance - secondary leakage inductance
0.0001 (By default)

The L2 inductance, which represents the magnetic flux losses of the secondary winding.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by parameter.

Tertiary winding resistance, Ohm - resistance of the tertiary winding
0.01 (By default).

Resistance R3, which represents the power loss of the second secondary winding.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by parameter and Three for the Number of windings.

Tertiary leakage inductance, H - scattering inductance on the tertiary winding
0.0001 (by default)

L3 inductance, which represents the magnetic flux losses of the second secondary winding.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by and Three for the Number of windings.

Magnetisation resistance - magnetization resistance
100 (By default)

The Rm resistance representing the magnetic losses in the transformer core.

Magnetisation inductance parameterized 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 the number of turns, unsaturated inductance and parasitic parallel conductance.

  • Single inductance (linear) - specifies the values of 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, the current vector and magnetic flux vector are specified 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 parasitic parallel conductivity, the effective length and cross-sectional area of the core, as well as the magnetic field strength vector and the magnetic induction vector are given 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, the magnetic induction and field strength at a particular point on the B-H curve, as well as the reversible magnetisation coefficient, the bulk coupling coefficient and the interdomain coupling coefficient to determine the magnetic induction as a function of the current value and history of the magnetic field strength are given.

Unsaturated inductance - unsaturated inductance
2e-4 (by default).

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

Dependencies

This parameter is used if Single inductance (linear) or Single saturation point is selected for the parameter Magnetisation inductance parameterized by.

Saturated inductance - saturated inductance
1e-4 (by default)

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

Dependencies

This parameter is only used when Single saturation point is selected for the Magnetisation inductance parameterized by.

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

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

Dependencies

This parameter is only used when Single saturation point is selected for the Magnetisation inductance parameterized by.

Current vector, i - vector of current values
[0, .4, .8, 1.2, 1.6, 2] (by default).

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

Dependencies

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

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 if `Magnetic flux versus current characteristic' is selected for the Magnetisation inductance parameterized by.

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

Magnetic flux density vector, B, T - vector of magnetic induction values
[0, .81, 1.25, 1.42, 1.48, 1.49] (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 Magnetisation inductance parameterized by.

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

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

Dependencies

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

Effective cross-sectional area, m² - effective cross-sectional area
1.6e-5 (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 Magnetisation inductance parameterized by.

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 (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 parameter Magnetisation inductance parameterized by.

Flux density point on anhysteretic B-H curve - value of magnetic induction at a point on anhysteresis B-H curve
`1.49 (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 if `Magnetic flux density versus magnetic field strength characteristic with hysteresis' is selected for the parameter Magnetisation inductance parameterized by.

Corresponding field strength is the corresponding field strength
1000 (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 Magnetisation inductance parameter parameter parameterized by.

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 Magnetisation inductance parameter parameter parameterized by.

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

Parameters 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 Magnetisation inductance parameter parameterised by.

Inter-domain coupling factor, α is the 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 parameter Magnetisation inductance parameterized by.

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.

Combined leakage inductance initial current - Combined leakage inductance initial current
0 (By default).

The initial current value used to calculate the leakage inductance at zero time for both transformer windings together.

Dependencies

This parameter is only used when Combined primary and secondary values is selected for the parameter Winding parameterised by.

Primary leakage inductance initial current - initial leakage inductance current for the primary winding
0 (By default)

The initial current value used to calculate the leakage inductance at zero time for the transformer primary winding.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by parameter.

Secondary leakage inductance initial current - initial current for the secondary winding
0 (By default)

The initial current value used to calculate the dissipation inductance at zero time for the transformer secondary winding.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by parameter.

Second secondary leakage inductance initial current - initial leakage inductance current for the second secondary winding
0 (By default)

The initial current value used to calculate the dissipation inductance at zero time for the second secondary winding of the transformer.

Dependencies

This parameter is only used when selecting Separate primary and secondary values for the Winding parameterised by parameter and Three for the Number of windings.

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

Select the appropriate option for specifying the initial state:

  • Current - setting the initial state of the transformer by the initial current through the transformer. This option is used by default.

  • Magnetic flux - setting the initial state of the transformer 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 parameter Magnetisation inductance parameterized by.

Magnetisation inductance initial current - magnetization inductance initial current
0 (By default)

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

Dependencies

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

Magnetisation inductance initial magnetic flux - initial magnetic flux of magnetization inductance
0 (By default)

The value of magnetic flux at zero moment of time.

Dependencies

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

Magnetisation inductance initial magnetic flux density - initial magnetic induction of magnetization inductance
0 (By default)

Magnetic induction value 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 Magnetisation inductance parameter parameter parameterized by.

Magnetisation inductance initial field strength - initial field strength of magnetization inductance
0 (By default)

The value of the magnetic field strength at zero time.

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

Combined leakage inductance parasitic parallel conductance - Combined leakage inductance parasitic parallel conductance
`1e-9 (By default).

This parameters is used to represent small parasitic effects in parallel to the aggregate leakage inductance. A small parallel conductance may be required for modelling some circuit topologies.

Dependencies

This parameter is only used when Combined primary and secondary values is selected for the Winding parameterised by parameter.

Primary leakage inductance parasitic parallel conductance - scattering inductance, parasitic parallel conductance of the primary winding
1e-9 (By default)

This parameter is used to represent small parasitic effects parallel to the scattering inductance on the primary winding. A small parallel conductance may be required for modelling some circuit topologies.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by parameter.

Secondary leakage inductance parasitic parallel conductance - scattering inductance parasitic parallel conductance of secondary winding
1e-9 (By default)

This parameter is used to represent small parasitic effects in parallel to the scattering inductance on the secondary winding.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by parameter.

Tertiary leakage inductance parasitic parallel conductance, 1/Ohm - scattering inductance, parasitic parallel conductance of the second secondary winding
1e-9 (By default)

This parameter is used to represent small parasitic effects parallel to the scattering inductance on the second secondary winding. The small parallel conductance may be required for modelling some circuit topologies.

Dependencies

This parameter is only used when Separate primary and secondary values is selected for the Winding parameterised by parameter and Three for the Number of windings.