Model inductor with nonideal core

Passive Devices

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:

Magnetic flux density versus magnetic field strength characteristic

Magnetic flux density versus magnetic field strength characteristic with hysteresis

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

$$i={i}_{L}+v{G}_{p}$$

$$v={N}_{w}\frac{d\Phi}{dt}$$

$$\Phi =\frac{L}{{N}_{w}}{i}_{L}$$

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.

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

$$i={i}_{L}+v{G}_{p}$$

$$v={N}_{w}\frac{d\Phi}{dt}$$

$$\Phi =\frac{L}{{N}_{w}}{i}_{L}\text{(forunsaturated)}$$

$$\Phi =\frac{{L}_{sat}}{{N}_{w}}{i}_{L}\pm {\Phi}_{offset}\text{(forsaturated)}$$

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.

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.*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).

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.*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).

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.*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.*μ*_{0}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:

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

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}:

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

This function defines a saturation curve with limiting values
±*M*_{s} 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 *H*_{eff} =
0 and a point [*H*_{1}, *B*_{1}]
on the anhysteretic B-H curve, and these are used to determine values
for *α* 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:

$$\begin{array}{l}\frac{d{M}_{irr}}{dH}=\frac{{M}_{an}-{M}_{irr}}{K\delta -\alpha \left({M}_{an}-{M}_{irr}\right)}\\ \delta =\{\begin{array}{ll}1\hfill & \text{if}H\ge 0\hfill \\ -1\hfill & \text{if}H0\text{}\hfill \end{array}\end{array}$$

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 $$1/\left(K\delta -\alpha \left({M}_{an}-{M}_{irr}\right)\right)$$.
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.

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 ($$d{M}_{an}/d{H}_{eff}$$when*H*_{eff}= 0) plus a data point [*H*_{1},*B*_{1}] on the anhysteretic B-H curve. From these values, the block initialization determines values for*α*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.

**Parameterized by**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**The total number of turns of wire wound around the inductor core. The default value is

`10`

.**Unsaturated inductance**The value of inductance used when the inductor is operating in its linear region. This parameter is visible only when you select

`Single inductance (linear)`

or`Single saturation point`

for the**Parameterized by**parameter. The default value is`2e-4`

H.**Saturated inductance**The value of inductance used when the inductor is operating beyond its saturation point. This parameter is visible only when you select

`Single saturation point`

for the**Parameterized by**parameter. The default value is`1e-4`

H.**Saturation magnetic flux**The value of magnetic flux at which the inductor saturates. This parameter is visible only when you select

`Single saturation point`

for the**Parameterized by**parameter. The default value is`1.3e-5`

Wb.**Current, i**The current data used to populate the magnetic flux versus current lookup table. This parameter is visible only when you select

`Magnetic flux versus current characteristic`

for the**Parameterized by**parameter. The default value is`[ 0 0.64 1.28 1.92 2.56 3.20 ]`

A.**Magnetic flux vector, phi**The magnetic flux data used to populate the magnetic flux versus current lookup table. This parameter is visible only when you select

`Magnetic flux versus current characteristic`

for the**Parameterized by**parameter. The default value is`[0 1.29 2.00 2.27 2.36 2.39 ].*1e-5`

Wb.**Magnetic field strength vector, H**The magnetic field strength data used to populate the magnetic flux density versus magnetic field strength lookup table. This parameter is visible only when you select

`Magnetic flux density versus magnetic field strength characteristic`

for the**Parameterized by**parameter. The default value is`[ 0 200 400 600 800 1000 ]`

A/m.**Magnetic flux density vector, B**The magnetic flux density data used to populate the magnetic flux density versus magnetic field strength lookup table. This parameter is visible only when you select

`Magnetic flux density versus magnetic field strength characteristic`

for the**Parameterized by**parameter. The default value is`[ 0 0.81 1.25 1.42 1.48 1.49 ]`

T.**Effective length**The effective core length, that is, the average distance of the magnetic path. 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 default value is`0.032`

m.**Effective cross-sectional area**The effective core cross-sectional area, that is, the average area of the magnetic path. 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 default value is`1.6e-5`

m^2.**Anhysteretic B-H gradient when H is zero**The gradient of the anhyseretic (no hysteresis) B-H curve around zero field strength. Set it to the average gradient of the positive-going and negative-going hysteresis curves. This parameter is visible only when you select

`Magnetic flux density versus magnetic field strength characteristic with hysteresis`

for the**Parameterized by**parameter. The default value is`0.005`

m*T/A.**Flux density point on anhysteretic B-H curve**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. This parameter is visible only when you select

`Magnetic flux density versus magnetic field strength characteristic with hysteresis`

for the**Parameterized by**parameter. The default value is`1.49`

T.**Corresponding field strength**The corresponding field strength for the point that you define by the

**Flux density point on anhysteretic B-H curve**parameter. This parameter is visible only when you select`Magnetic flux density versus magnetic field strength characteristic with hysteresis`

for the**Parameterized by**parameter. The default value is`1000`

A/m.**Coefficient for reversible magnetization, c**The proportion of the magnetization that is reversible. The value should be greater than zero and less than one. This parameter is visible only when you select

`Magnetic flux density versus magnetic field strength characteristic with hysteresis`

for the**Parameterized by**parameter. The default value is`0.1`

.**Bulk coupling coefficient, K**The Jiles-Atherton parameter that primarily controls the field strength magnitude at which the B-H curve crosses the zero flux density line. This parameter is visible only when you select

`Magnetic flux density versus magnetic field strength characteristic with hysteresis`

for the**Parameterized by**parameter. The default value is`200`

A/m.**Inter-domain coupling factor, alpha**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. This parameter is visible only when you select

`Magnetic flux density versus magnetic field strength characteristic with hysteresis`

for the**Parameterized by**parameter. The default value is`1e-4`

.**Parasitic parallel conductance**Use this parameter to represent small parasitic effects. A small parallel conductance may be required for the simulation of some circuit topologies. The default value is

`1e-9`

1/Ω.**Interpolation option**The lookup table interpolation option. 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. Select one of the following interpolation methods:`Linear`

— Uses a linear interpolation function.`Cubic`

— Uses the Piecewise Cubic Hermite Interpolation Polynomial (PCHIP).

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

**Specify initial state by**Select the appropriate initial state specification option:

`Current`

— Specify the initial state of the inductor by the initial current through the inductor (*i*_{L}). This is the default option.`Magnetic flux`

— Specify the initial state of the inductor by the magnetic flux.

This parameter is not visible when you select

`Magnetic flux density versus magnetic field strength characteristic with hysteresis`

for the**Parameterized by**parameter on the**Main**tab.**Initial current**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. This parameter is visible only when you select

`Current`

for the**Specify initial state by**parameter. The default value is`0`

A.**Initial magnetic flux**The value of magnetic flux at time zero. This parameter is visible only when you select

`Magnetic flux`

for the**Specify initial state by**parameter. The default is`0`

Wb.**Initial magnetic flux density**The value of magnetic flux density at time zero. 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 default is`0`

T.**Initial field strength**The value of magnetic field strength at time zero. 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 default is`0`

A/m.

The block has the following ports:

`+`

Positive electrical port

`-`

Negative electrical port

For comparison of nonlinear inductor behavior with different parameterization options, see the Nonlinear Inductor Characteristics example.

The Inductor With Hysteresis example shows how modifying the equation coefficients of the Jiles-Atherton magnetic hysteresis equations affects the resulting B-H curve.

[1] Jiles, D.C and D.L. Atherton. "Theory
of ferromagnetic hysteresis." *Journal of Magnetism
and Magnetic Materials*. Vol. 61, 1986, pp. 48–60.

[2] 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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