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# Nonlinear Inductor

Inductor with nonideal core

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• 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:

`$i={i}_{L}+v{G}_{p}$`
`$v={N}_{w}\frac{d\Phi }{dt}$`
`$\Phi =\frac{L}{{N}_{w}}{i}_{L}$`

where:

• v is the terminal voltage.

• i is the terminal current.

• iL is the current through inductor.

• Gp is the parasitic parallel conductance.

• Nw 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:

`$i={i}_{L}+v{G}_{p}$`
`$v={N}_{w}\frac{d\Phi }{dt}$`

where:

• v is the terminal voltage.

• i is the terminal current.

• iL is the current through inductor.

• Gp is the parasitic parallel conductance.

• Nw is the number of winding turns.

• Φ is the magnetic flux.

• Φoffset is the magnetic flux saturation offset.

• L is the unsaturated inductance.

• Lsat is the saturated inductance.

### Magnetic Flux Versus Current Characteristic

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

`$i={i}_{L}+v{G}_{p}$`
`$v={N}_{w}\frac{d\Phi }{dt}$`
`$\Phi =f\left({i}_{L}\right)$`

where:

• v is the terminal voltage.

• i is the terminal current.

• iL is the current through inductor.

• Gp is the parasitic parallel conductance.

• Nw 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:

`$i={i}_{L}+v{G}_{p}$`
`$v={N}_{w}\frac{d\Phi }{dt}$`
`$\Phi =B\cdot {A}_{e}$`
`$B=f\left(H\right)$`
`$H=\frac{{N}_{w}}{{l}_{e}}{i}_{L}$`

where:

• v is the terminal voltage.

• i is the terminal current.

• iL is the current through inductor.

• Gp is the parasitic parallel conductance.

• Nw is the number of winding turns.

• Φ is the magnetic flux.

• B is the magnetic flux density.

• H is the magnetic field strength.

• le is the effective core length.

• Ae 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:

`$i={i}_{L}+v{G}_{p}$`
`$v={N}_{w}\frac{d\Phi }{dt}$`
`$\Phi =B\cdot {A}_{e}$`
`$B={\mu }_{0}\left(H+M\right)$`
`$H=\frac{{N}_{w}}{{l}_{e}}{i}_{L}$`

where:

• v is the terminal voltage.

• i is the terminal current.

• iL is the current through inductor.

• Gp is the parasitic parallel conductance.

• Nw 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.

• le is the effective core length.

• Ae 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 (Heff) and the other an irreversible part that depends on past history:

`$M=c{M}_{an}+\left(1-c\right){M}_{irr}$`

The Man 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, Heff:

`${M}_{an}={M}_{s}\left(\mathrm{coth}\left(\frac{{H}_{eff}}{\alpha }\right)-\frac{\alpha }{{H}_{eff}}\right)$`

This function defines a saturation curve with limiting values ±Ms and point of saturation determined by the value of α, 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 $d{M}_{an}/d{H}_{eff}$when Heff = 0 and a point [H1, B1] on the anhysteretic B-H curve, and these are used to determine values for α and Ms.

The parameter c is the coefficient for reversible magnetization, and dictates how much of the behavior is defined by Man and how much by the irreversible term Mirr. The Jiles-Atherton model defines the irreversible term by a partial derivative with respect to field strength:

Comparison of this equation with a standard first order differential equation reveals that as increments in field strength, H, are made, the irreversible term Mirr attempts to track the reversible term Man, but with a variable tracking gain of $1/\left(K\delta -\alpha \left({M}_{an}-{M}_{irr}\right)\right)$. 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:

`${H}_{eff}=H+\alpha M$`

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 $K\delta -\alpha \left({M}_{an}-{M}_{irr}\right)$ 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:

1. Provide a value for the Anhysteretic B-H gradient when H is zero parameter ($d{M}_{an}/d{H}_{eff}$when Heff = 0) plus a data point [H1, B1] on the anhysteretic B-H curve. From these values, the block initialization determines values for α and Ms.

2. 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.

3. Set the Bulk coupling coefficient, K parameter to the approximate magnitude of H when B = 0 on the positive-going hysteresis curve.

4. 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.

## Ports

### Conserving

expand all

Electrical conserving port associated with the inductor positive voltage.

Electrical conserving port associated with the inductor negative voltage.

## Parameters

expand all

### Main

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

#### Dependencies

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

Specify a point on the anhysteretic curve by providing its flux density value. Picking a point at high field strength where the positive-going and negative-going hysteresis curves align is the most accurate option.

#### Dependencies

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

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

#### Dependencies

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

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

#### Dependencies

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

The Jiles-Atherton parameter that primarily controls the field strength magnitude at which the B-H curve crosses the zero flux density line.

#### Dependencies

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

The Jiles-Atherton parameter that primarily affects the points at which the B-H curves intersect the zero field strength line. Typical values are in the range of 1e-4 to 1e-3.

#### Dependencies

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

Averaging period for the hysteresis losses calculation. These losses are proportional to the area enclosed by the B-H trajectory. If the block is excited at a known, fixed frequency, you can set this value to the corresponding excitation period to calculate the hysteresis loss. In this case, the block logs the hysteresis loss once per AC cycle to the variable `power_dissipated`. If you are using a fixed-step solver, this value must be an integer multiple of the simulation step size.

If the block is not excited at a known, fixed frequency, set this parameter to `0`. In this case, the block sets `power_dissipated` to zero, and you can calculate the actual hysteresis loss by post-processing the logged variable `power_instantaneous`.

#### Dependencies

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

Use this parameter to represent small parasitic effects. A small parallel conductance may be required for the simulation of some circuit topologies.

The lookup table interpolation option. Select one of the following interpolation methods:

• `Linear` — Select this option to get the best performance.

• `Smooth` — Select this option to produce a continuous curve with continuous first-order derivatives.

For more information on interpolation algorithms, see the PS Lookup Table (1D) block reference page.

#### Dependencies

This parameter is visible only when you select ```Magnetic flux versus current characteristic``` or ```Magnetic flux density versus magnetic field strength characteristic``` for the Parameterized by parameter on the Main tab.

### Initial Conditions

Select the appropriate initial state specification option:

• `Current` — Specify the initial state of the inductor by the initial current through the inductor (iL). This is the default option.

• `Magnetic flux` — Specify the initial state of the inductor by the magnetic flux.

#### Dependencies

This parameter is not visible when you select ```Magnetic flux density versus magnetic field strength characteristic with hysteresis``` for the Parameterized by parameter.

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

#### Dependencies

This parameter is visible only when you select `Current` for the Specify initial state by parameter.

The value of magnetic flux at time zero.

#### Dependencies

This parameter is visible only when you select `Magnetic flux` for the Specify initial state by parameter.

The value of magnetic flux density at time zero.

#### Dependencies

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

The value of magnetic field strength at time zero.

#### Dependencies

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

 Jiles, D. C. and D. L. Atherton. “Theory of ferromagnetic hysteresis.” Journal of Magnetism and Magnetic Materials . Vol. 61, 1986, pp. 48–60.

 Jiles, D. C. and D. L. Atherton. “Ferromagnetic hysteresis.” IEEE® Transactions on Magnetics . Vol. 19, No. 5, 1983, pp. 2183–2184.

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