Inductor with nonideal core

**Library:**Simscape / Electrical / Passive

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:

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.

[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.