FEM-Parameterized Linear Actuator

Linear actuator defined in terms of magnetic flux

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• Simscape / Electrical / Electromechanical / Mechatronic Actuators

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Description

The FEM-Parameterized Linear Actuator block implements a model of a linear actuator defined in terms of magnetic flux. Use this block to model custom solenoids and linear motors where magnetic flux depends on both distance asond current. You parameterize the block using data from a third-party magnetic finite-element method (FEM) package.

Note

To use this block to represent a solenoid with air gap x between case C and plunger R, then the tabulated force should be negative to represent the pulling force bringing R towards C.

The block has two options for the electrical equation. The first, Define in terms of dPhi(i,x)/dx and dPhi(i,x)/di, defines the current in terms of partial derivatives of the magnetic flux (Φ) with respect to distance (x) and current (i), the equations for which are:

$$\frac{di}{dt}=\left(v-iR-\frac{\partial \Phi }{\partial x}\frac{dx}{dt}\right)/\frac{\partial \Phi }{\partial i}$$

The second option, Define in terms of Phi(i,x), defines the voltage across the component directly in terms of the flux, the equation for which is:

$$v=iR+\frac{d}{dt}\Phi \left(x,i\right)$$

Numerically, defining the electrical equation in terms of flux partial derivatives is better because the back-emf is piecewise continuous. If using the flux directly, using a finer grid size for current and position will improve results, as will selecting cubic or spline interpolation.

In both cases, you have an option to either directly specify the force as a function of current and position, by using the Force matrix, F(i,x) parameter, or have the block automatically calculate the force matrix.

If entering the electromagnetic force data directly, you can either use data supplied by the finite element magnetic package (which you used to determine the flux) or calculate the force from the flux with following equation:

$$F={\displaystyle \underset{0}{\overset{i}{\int }}\frac{\partial \Phi \left(x,i\right)}{\partial x}}di$$

For an example that shows how to implement this type of integration in MATLAB^®, see Solenoid Parameterized with FEM Data. The related file ee_solenoid_fem_params.m contains the code that calculates and plots the flux data.

Alternatively, the block can automatically calculate the force matrix from the flux information that you provide. To select this option, set the Calculate force matrix? parameter to Yes. The force matrix calculation occurs at model initialization based on current block flux linkage information. The force is calculated by numerically integrating the rate of change of flux linkage with respect to position over current, according to the preceding equation. If the Electrical model parameter is set to Define in terms of Phi(i,x), then the block must first estimate the Flux partial derivative wrt displacement, dPhi(i,x)/dx parameter value from the flux linkage data. When doing this, the block uses the interpolation method specified by the Interpolation method parameter. Typically, the Smooth option is most accurate, but the Linear option is most robust.

You can define Φ and its partial derivatives for just positive, or positive and negative currents. If defining for just positive currents, then the block assumes that Φ(–i,x) = –Φ(i,x). Therefore, if the current vector is positive only:

• The first current value must be zero.

• The flux corresponding to zero current must be zero.

• The partial derivative of flux with respect to displacement must be zero for zero current.

To model a linear motor with a repeated flux pattern, set the Flux dependence on displacement parameter to Cyclic. When selecting this option, the force and flux (or force and flux partial derivatives depending on the option chosen) must have identical first and last columns.

Note

The actuated motion direction of this block can be swapped by flipping the block and swapping the R and C connections.

Thermal Port

The block has an optional thermal port, hidden by default. To expose the thermal port, right-click the block in your model, and then from the context menu select Simscape > Block choices > Show thermal port. This action displays the thermal port H on the block icon, and exposes the Temperature Dependence and Thermal Port parameters.

Use the thermal port to simulate the effects of copper resistance losses that convert electrical power to heat. For more information on using thermal ports and on the Temperature Dependence and Thermal Port parameters, see Simulating Thermal Effects in Rotational and Translational Actuators.

Assumptions and Limitations

• You must supply a consistent set of force and flux data. There is no check that ensures that the force matrix is consistent with the flux data.

• When driving the FEM-Parameterized Linear Actuator block via a series inductor, you may need to include a parallel conductance in the inductor component.

Ports

Conserving

+ — Positive terminal

electrical

Electrical conserving port associated with the actuator positive terminal.

- — Negative terminal

electrical

Electrical conserving port associated with the actuator negative terminal.

C — Case

mechanical

Mechanical translational conserving port associated with the actuator case.

R — Plunger

mechanical

Mechanical translational conserving port associated with the plunger.

H — Thermal port

thermal

Thermal port. For more information, see Thermal Port.

Parameters

Magnetic Force

Electrical model — Electrical model parameterization

Define in terms of dPhi(i,x)/dx and dPhi(i,x)/di (default) | Define in terms of Phi(i,x)

Select one of the following parameterization options, based on the underlying electrical model:

• Define in terms of dPhi(i,x)/dx and dPhi(i,x)/di — Define the current through the block in terms of partial derivatives of the magnetic flux with respect to distance and current.

• Define in terms of Phi(i,x) — Define the voltage across the block terminals directly in terms of the flux.

Current vector, i — Current vector

[0, .1, .2, .3, .4, .5, .6, .7, .8, .9, 1] A (default)

Specify a vector of monotonically increasing current values corresponding to your force-flux data. If you specify positive currents only, the first element must be zero.

Displacement vector, x — Displacement vector

[0, .05, .1, .15, .2] mm (default)

Specify a vector of monotonically increasing displacement values corresponding to your force-flux data.

Flux partial derivative wrt current, dPhi(i,x)/di — Flux partial derivative with respect to current

[.104, .098, .091, .085, .078; .095, .089, .084, .079, .073; .085, .081, .077, .073, .069; .076, .073, .07, .067, .064; .067, .065, .063, .061, .06; .057, .057, .056, .056, .055; .048, .049, .049, .05, .05; .038, .04, .042, .044, .046; .029, .032, .035, .038, .041; .02, .024, .028, .033, .037; .01, .016, .021, .027, .032] Wb/A (default)

Specify a matrix of the flux partial derivatives with respect to current. The default value, in Wb/A, is:

[ 0.104 0.098 0.091 0.085 0.078; 
0.095 0.089 0.084 0.079 0.073; 
0.085 0.081 0.077 0.073 0.069; 
0.076 0.073 0.07 0.067 0.064; 
0.067 0.065 0.063 0.061 0.06; 
0.057 0.057 0.056 0.056 0.055; 
0.048 0.049 0.049 0.05 0.05; 
0.038 0.04 0.042 0.044 0.046; 
0.029 0.032 0.035 0.038 0.041; 
0.02 0.024 0.028 0.033 0.037; 
0.01 0.016 0.021 0.027 0.032 ]

Dependencies

This parameter is visible only when you set the Electrical model parameter to Define in terms of dPhi(i,x)/dx and dPhi(i,x)/di.

Flux partial derivative wrt displacement, dPhi(i,x)/dx — Flux partial derivative with respect to displacement

[0, 0, 0, 0, 0; -11.94, -10.57, -9.19, -7.81, -6.43; -21.17, -19.92, -18.67, -17.42, -16.16; -27.99, -26.87, -25.75, -24.62, -23.5; -32.42, -31.43, -30.43, -29.43, -28.44; -34.46, -33.59, -32.72, -31.85, -30.98; -34.09, -33.35, -32.61, -31.87, -31.12; -31.33, -30.72, -30.1, -29.49, -28.87; -26.17, -25.68, -25.2, -24.71, -24.22; -18.62, -18.26, -17.9, -17.54, -17.18; -8.66, -8.43, -8.2, -7.97, -7.73] Wb/m (default)

Specify a matrix of the flux partial derivatives with respect to displacement. The default value, in Wb/m, is:

[ 0 0 0 0 0; 
-11.94 -10.57 -9.19 -7.81 -6.43; 
-21.17 -19.92 -18.67 -17.42 -16.16; 
-27.99 -26.87 -25.75 -24.62 -23.5; 
-32.42 -31.43 -30.43 -29.43 -28.44; 
-34.46 -33.59 -32.72 -31.85 -30.98; 
-34.09 -33.35 -32.61 -31.87 -31.12; 
-31.33 -30.72 -30.1 -29.49 -28.87; 
-26.17 -25.68 -25.2 -24.71 -24.22; 
-18.62 -18.26 -17.9 -17.54 -17.18; 
-8.66 -8.43 -8.2 -7.97 -7.73 ]

Dependencies

This parameter is visible only when you set the Electrical model parameter to Define in terms of dPhi(i,x)/dx and dPhi(i,x)/di.

Flux linkage matrix, Phi(i,x) — Flux linkage matrix

[0, 0, 0, 0, 0; .0085, .0079, .0075, .0071, .0067; .0171, .016, .0151, .0143, .0137; .0254, .0239, .0226, .0215, .0206; .033, .0312, .0297, .0283, .0271; .0396, .0377, .036, .0345, .0331; .0452, .0433, .0415, .0399, .0384; .0495, .0478, .0461, .0446, .0431; .0526, .0512, .0498, .0485, .0472; .0545, .0537, .0528, .0519, .0508; .0554, .0553, .0551, .0548, .0542] Wb (default)

Specify a matrix of the total flux linkage, that is, flux times the number of turns. The default value, in Wb, is:

[ 0 0 0 0 0; 
0.0085 0.0079 0.0075 0.0071 0.0067; 
0.0171 0.016 0.0151 0.0143 0.0137; 
0.0254 0.0239 0.0226 0.0215 0.0206; 
0.033 0.0312 0.0297 0.0283 0.0271; 
0.0396 0.0377 0.036 0.0345 0.0331; 
0.0452 0.0433 0.0415 0.0399 0.0384; 
0.0495 0.0478 0.0461 0.0446 0.0431; 
0.0526 0.0512 0.0498 0.0485 0.0472; 
0.0545 0.0537 0.0528 0.0519 0.0508; 
0.0554 0.0553 0.0551 0.0548 0.0542 ]

Dependencies

This parameter is visible only when you set the Electrical model parameter to Define in terms of Phi(i,x).

Calculate force matrix? — Electromagnetic force data specification

No (default) | Yes

Specify the way of providing the electromagnetic force data:

• No — Enter the electromagnetic force data directly, by using the Force matrix, F(i,x) parameter. This is the default option.

• Yes — The block calculates the force from the flux linkage information, as a function of current and displacement.

Force matrix, F(i,x) — Force matrix

[0, 0, 0, 0, 0; -.6, -.5, -.4, -.3, -.3; -2.3, -2, -1.7, -1.4, -1.2; -4.9, -4.3, -3.7, -3.2, -2.7; -8.3, -7.3, -6.4, -5.5, -4.7; -12.2, -10.7, -9.4, -8.2, -7.2; -16.2, -14.4, -12.7, -11.3, -10; -20, -17.9, -15.9, -14.3, -12.9; -23.3, -20.9, -18.8, -17.1, -15.7; -25.7, -23.1, -21.1, -19.4, -18.2; -26.5, -24.1, -22.2, -20.9, -20.1] N (default)

Specify a matrix of the electromagnetic force applied to the plunger or moving part. The default value, in N, is:

[ 0 0 0 0 0; 
-0.6 -0.5 -0.4 -0.3 -0.3; 
-2.3 -2 -1.7 -1.4 -1.2; 
-4.9 -4.3 -3.7 -3.2 -2.7; 
-8.3 -7.3 -6.4 -5.5 -4.7; 
-12.2 -10.7 -9.4 -8.2 -7.2; 
-16.2 -14.4 -12.7 -11.3 -10; 
-20 -17.9 -15.9 -14.3 -12.9; 
-23.3 -20.9 -18.8 -17.1 -15.7; 
-25.7 -23.1 -21.1 -19.4 -18.2; 
-26.5 -24.1 -22.2 -20.9 -20.1 ]

Dependencies

This parameter is visible only when you set the Calculate force matrix? parameter to No.

Flux dependence on displacement — Flux pattern

Unique (default) | Cyclic

Specify the flux pattern:

• Unique — No flux pattern present.

• Cyclic — Select this option to model a linear motor with a repeated flux pattern. The force and flux (or force and flux partial derivatives, depending on the Electrical model option chosen) must have identical first and last columns.

Interpolation method — Interpolation method

Linear (default) | Smooth

Select one of the following interpolation methods for approximating the output value when the input value is between two consecutive grid points:

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

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

For more information on interpolation algorithms, see the PS Lookup Table (2D) block reference page.

Extrapolation method — Extrapolation method

Linear (default) | Nearest | Error

Select one of the following extrapolation methods for determining the output value when the input value is outside the range specified in the argument list:

• Linear — Select this option to produce a surface with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.

• Nearest — Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data.

• Error — Issues an error if the input signal is outside the range of the table. Select this option to avoid going into the extrapolation mode when you want your data to be within the table range.

For more information on extrapolation algorithms, see the PS Lookup Table (2D) block reference page.

Winding resistance — Electrical winding total resistance

14 Ohm (default)

Total resistance of the electrical winding.

Mechanical

Damping — Damping

1 N/(m/s) (default)

Linear damping. The value can be zero.

Plunger mass — Mass of the moving part

0.05 kg (default)

Mass of the moving part, which corresponds to mechanical translational port R. The value can be zero.

Minimum stroke — Minimum stroke

0 mm (default)

The stroke at which the lower mechanical end stop is applied. The value can be -Inf.

Maximum stroke — Maximum stroke

0.2 mm (default)

The stroke at which the upper mechanical end stop is applied. The value can be Inf.

Contact stiffness — Contact stiffness

1e8 N/m (default)

Contact stiffness between plunger and end stops.

Contact damping — Contact damping

1e4 N/(m/s) (default)

Contact damping between plunger and end stops.

Temperature Dependence

This tab appears only for blocks with exposed thermal port. For more information, see Thermal Port.

Resistance temperature coefficient — Resistance temperature coefficient

3.93e-3 1/K (default)

Resistance temperature coefficient.

Measurement temperature — Measurement temperature

25 degC (default)

The temperature for which the actuator parameters are defined.

Thermal Port

This tab appears only for blocks with exposed thermal port. For more information, see Thermal Port.

Thermal mass — Thermal mass

100 J/K (default)

The thermal mass is the energy required to raise the temperature by one degree.

Model Examples

Solenoid Parameterized with FEM Data

A limited travel solenoid with return spring. When unpowered, the spring holds the plunger at a distance of 0.1mm from the fully energized position. At 0.1 seconds, the solenoid is powered on and the displacement goes to zero. At 0.06s a force higher than the holding force is applied, and the plunger moves to its maximum travel of 0.2mm. The solenoid force and back emf characteristics are defined by the FEM-Parameterized Linear Actuator block. This block takes data in the format typically provided by a finite element magnetic field modeling tool. There are two parameterization options, one which works directly with flux data, and one which uses partial derivatives of flux with respect to current and displacement. The latter option is usually the better choice, it giving more accurate results for a given density of current and position data points. However, it requires more data pre-processing.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.