# Nonlinear Inductor

Inductor with nonideal core

**Libraries:**

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. To choose between these parameterization options, use the
**Parameterized by** parameter:

### Single Inductance (Linear)

These equations define the relationships between voltage, current, and flux,

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

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

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

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.

### Single Saturation Point

These equations define the relationships between voltage, current, and flux,

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

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

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

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

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.

### Magnetic Flux Versus Current Characteristic

These equations define the relationships between voltage, current, and flux,

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

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

$$\Phi =f\left({i}_{L}\right)$$

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.

The block determines magnetic flux by using a one-dimensional table lookup, based on the vector of electrical current values and the vector of corresponding magnetic flux values that you provide. You can construct these vectors using negative and positive data, or positive data only. If you use positive data only, the first element of the vector must be 0. The block calculates negative values by rotating the positive data by 180° about (0,0).

### Magnetic Flux Density Versus Magnetic Field Strength Characteristic

These equations define the relationships between voltage, current, and flux,

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

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

$$\Phi =B\cdot {A}_{e}$$

$$B=f\left(H\right)$$

$$H=\frac{{N}_{w}}{{l}_{e}}{i}_{L}$$

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.

The block determines the magnetic flux density 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 you use positive data only, the first element of the vector must be 0. The block calculates negative values by rotating the positive data by 180° about (0,0).

### Magnetic Flux Density Versus Magnetic Field Strength Characteristic with Hysteresis

These equations define the relationships between voltage, current, and flux,

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

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

$$\Phi =B\cdot {A}_{e}$$

$$B={\mu}_{0}\left(H+M\right)$$

$$H=\frac{{N}_{w}}{{l}_{e}}{i}_{L}$$

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 permeability of a vacuum.*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. The block
uses the Jiles-Atherton 1, 2] equations to
determine *M* at any given time. This figure shows a typical plot
of the resulting relationship between *B* and *H*.

As the field strength increases from *H* = 0, the plot initially follows an ascending hysteresis curve. At the
saturation point, further increases in field intensity no longer cause significant
increases in magnetic flux.

As you reduce the magnetic field strength from the saturation point, the plot
follows a descending hysteresis curve. The difference between ascending and
descending 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 the split of the
magnetization effect into two parts, one part that is purely a function of effective
field strength *H _{eff}* and another,
irreversible part that depends on history. This equation defines the relative
contributions of the anhysteretic magnetisation

*M*and the irreversible magnetization

_{an}*M*to the total magnetization

_{irr}*M*,

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

where *c* is the coefficient for reversible magnetization.

This equation relates *M _{an}* to the
equivalent magnetization

*H*:

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

The function defines a saturation curve with limiting values
±*M _{s}*, where:

*M*_{s}is the*saturation magnetization*.*a*is the*anhysteretic magnetization coefficient*, which determines the point of saturation. It approximately describes the average of the two hysteretic curves.

In the Nonlinear Inductor block mask, 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. The block uses these values to
determine values for *a* and
*M*_{s}.

The Jiles-Atherton model defines the irreversible term by a partial derivative with respect to field strength,

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

where:

*K*is the*bulk coupling coefficient*, which shapes the irreversible characteristic.*α*is the*inter-domain coupling factor*.

Comparison of the equation with a standard first-order differential equation
reveals that as you change the total field strength *H*, the
irreversible term *M _{irr}* attempts to track
the reversible term

*M*, 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

_{an}*δ*changes sign.

This equation defines the effective field strength of an anhysteretic curve,

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

The value of *α* affects the shape of the hysteresis curve. Larger
values act to increase the *B*-axis intercepts. However, the
stability term $$K\delta -\alpha \left({M}_{an}-{M}_{irr}\right)$$ must be positive for *δ* > 0 and negative for *δ* < 0. There are therefore limits on the values of *α*
that you can provide. A typical maximum value is of the order 1e-3.

**Procedure for Finding Approximate Values for Jiles-Atherton (JA) Equation Coefficients**

You can determine representative values for the equation coefficients using this procedure:

Provide a value for the

**Anhysteretic B-H gradient when H is zero**parameter ($$d{M}_{an}/d{H}_{eff}$$ when*H*= 0) and a data point [_{eff}*H*_{1},*B*] on the anhysteretic_{1}*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 ascending hysteresis curve.Start with a very small value for the

**Inter-domain coupling factor, alpha**parameter, and gradually increase it to tune the value of*B*when crossing the*H*= 0 line. A typical value is in the range of 1e-4 to 1e-3. Values that are too large cause the gradient of the*B*-*H*curve to tend to infinity, which is nonphysical and generates a run-time assertion error.

To get a good match against a predefined
*B*-*H* curve, iterate on these four
steps.

## Examples

## Ports

### Conserving

## Parameters

## References

[1] Jiles, D. C. and D. L. Atherton. "Theory of ferromagnetic
hysteresis." *Journal of Magnetism and Magnetic Materials* 61, no.
1–2 (September 1986): 48–60.https://doi.org/10.1016/0304-8853(86)90066-1.

[2] Jiles, D. C. and D. L. Atherton. “Ferromagnetic
hysteresis.” *IEEE ^{®} Transactions on Magnetics* 19, no. 5 (September 1983):
2183–2185. https://doi.org/10.1109/TMAG.1983.1062594.

## Extended Capabilities

## Version History

**Introduced in R2012b**