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,

$i = i_{L} + v G_{p}$

$v = N_{w} \frac{d Φ}{d t}$

$Φ = \frac{L}{N_{w}} i_{L}$

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,

$i = i_{L} + v G_{p}$

$v = N_{w} \frac{d Φ}{d t}$

$Φ = \frac{L}{N_{w}} i_{L} (for unsaturated)$

$Φ = \frac{L_{s a t}}{N_{w}} i_{L} \pm Φ_{o f f s e t} (for saturated)$

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,

$i = i_{L} + v G_{p}$

$v = N_{w} \frac{d Φ}{d t}$

$Φ = f (i_{L})$

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.

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,

$i = i_{L} + v G_{p}$

$v = N_{w} \frac{d Φ}{d t}$

$Φ = B \cdot A_{e}$

$B = f (H)$

$H = \frac{N_{w}}{l_{e}} i_{L}$

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,

$i = i_{L} + v G_{p}$

$v = N_{w} \frac{d Φ}{d t}$

$Φ = B \cdot A_{e}$

$B = μ_{0} (H + M)$

$H = \frac{N_{w}}{l_{e}} i_{L}$

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.
μ₀ 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_an and the irreversible magnetization M_irr to the total magnetization M,

$M = c M_{a n} + (1 - c) M_{i r r},$

where c is the coefficient for reversible magnetization.

This equation relates M_an to the equivalent magnetizationH_eff:

$M_{a n} = M_{s} (\coth (\frac{H_{e f f}}{a}) - \frac{a}{H_{e f f}}) .$

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 $d M_{a n} / d H_{e f f}$ when H_eff = 0 and a point [H₁, B₁] 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,

$\begin{array}{l} \frac{d M_{i r r}}{d H} = \frac{M_{a n} - M_{i r r}}{K δ - α (M_{a n} - M_{i r r})} \\ δ = {\begin{array}{l} 1 & if H \geq 0 \\ - 1 & if H < 0 \end{array} \end{array}$

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_an, but with a variable tracking gain of $1 / (K δ - α (M_{a n} - M_{i r r}))$ . The tracking error acts to create the hysteresis at the points where δ changes sign.

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

$H_{e f f} = H + α M .$

The value of α affects the shape of the hysteresis curve. Larger values act to increase the B-axis intercepts. However, the stability term $K δ - α (M_{a n} - M_{i r r})$ 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 ( $d M_{a n} / d H_{e f f}$ when H_eff = 0) and a data point [H₁, B₁] 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 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

Nonlinear Inductor Characteristics

A comparison of nonlinear inductor behavior for different parameterizations. Starting with fundamental parameter values, the parameters for linear and nonlinear representations are derived. These parameters are then used in a Simscape™ model and the simulation outputs compared.

Open Model

Inductor with Hysteresis

How modifying the equation coefficients of the Jiles-Atherton magnetic hysteresis equations affects the resulting B-H curve. The simulation parameters are configured to run four complete AC cycles with initial field strength (H) and magnetic flux density (B) both set to zero.

Open Model

Ports

Conserving

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+ — Positive terminal
electrical

Electrical conserving port associated with the positive terminal.

- — Negative terminal
electrical

Electrical conserving port associated with the negative terminal.

Parameters

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Main

Parameterized by — Block parameterization
`Single saturation point` (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`

Parameterization method that you use to describe the B-H curve. Choose one of these options:

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.
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 — Total number of turns
`10` (default)

Total number of turns of wire wound around the inductor core.

Unsaturated inductance — Unsaturated inductance
`2e-4` `H` (default)

Inductance when the inductor operates in its linear region.

Dependencies

To enable this parameter, set Parameterized by to Single inductance (linear) or Single saturation point.

Saturated inductance — Saturated inductance
`1e-4` `H` (default)

Inductance when the inductor operates beyond its saturation point.

Dependencies

To enable this parameter, set Parameterized by to Single saturation point.

Saturation magnetic flux — Saturation magnetic flux
`1.3e-05` `Wb` (default)

Magnetic flux at which the inductor saturates.

Dependencies

To enable this parameter, set Parameterized by to Single saturation point.

Current vector, i — Current data vector
`[0, .64, 1.28, 1.92, 2.56, 3.2]` `A` (default)

Current data that the block uses to populate the magnetic flux versus current lookup table.

Dependencies

To enable this parameter, set Parameterized by to Magnetic flux versus current characteristic.

Magnetic flux vector, phi — Magnetic flux data
`[0, 1.29, 2, 2.27, 2.36, 2.39] .* 1e-5` `Wb` (default)

Magnetic flux data that the block uses to populate the magnetic flux versus current lookup table.

Dependencies

To enable this parameter, set Parameterized by to Magnetic flux versus current characteristic.

Magnetic field strength vector, H — Magnetic field strength vector
`[0, 200, 400, 600, 800, 1000]` `A/m` (default) | vector

Magnetic field intensity H, specified as a vector with the same number of elements as the magnetic flux density vector B.

Dependencies

To enable this parameter, set Parameterized by to Magnetic flux density versus magnetic field strength characteristic.

Magnetic flux density vector, B — Magnetic flux density data
`[0, .81, 1.25, 1.42, 1.48, 1.49]` `T` (default) | vector

Magnetic flux density B, specified as a vector with the same number of elements as the magnetic field strength vector H.

Dependencies

To enable this parameter, set Parameterized by to Magnetic flux density versus magnetic field strength characteristic.

Effective length — Effective core length
`0.032` `m` (default) | positive finite scalar

Effective core length. This parameter represents the average length of the magnetic path around the core.

Dependencies

To enable this parameter, set Parameterized by to Magnetic flux density versus magnetic field strength characteristic or Magnetic flux density versus magnetic field strength characteristic with hysteresis.

Effective cross-sectional area — Effective core cross-sectional area
`1.6e-5` `m^2` (default) | positive finite scalar

Effective core cross-sectional area. This parameter represents the average area of the magnetic path around the core.

Dependencies

Anhysteretic B-H gradient when H is zero — Gradient of anhysteretic B-H curve around zero field strength
`0.005` `m*T/A` (default) | scalar

Gradient of the anhysteretic B-H curve around zero field strength. Set this parameter to the average gradient of the ascending and descending hysteresis curves.