# FEM-Parameterized Linear Actuator

Model linear actuator defined in terms of magnetic flux

## Library

Translational Actuators

## 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 and current. You parameterize the block using data from a third party Finite Element Magnetic (FEM) package.

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:

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

See the Finite Element Parameterized SolenoidFinite Element Parameterized Solenoid example model and its initialization file `elec_fem_solenoid_ini.m``elec_fem_solenoid_ini.m` for an example of how to implement this type of integration in MATLAB®.

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 `Spline` option is most accurate, but the `Linear` option is most robust. The potential risk of the `Spline` and `Cubic` methods is that nonphysical local torque reversals may occur if the flux data is too sparse or non-smooth.

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.

### 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 adds the Temperature Dependence and Thermal Port tabs to the block dialog box.

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 tab parameters, see Simulating Thermal Effects in Rotational and Translational Actuators.

## Basic Assumptions and Limitations

This block has the following limitations:

• It is imperative that you supply a consistent set of force and flux data. There is no checking 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.

## Dialog Box and Parameters

### Magnetic Force Tab

Electrical model

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. This is the default method.

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

Current vector, i

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. The default value is ```[ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ]``` A.

Displacement vector, x

Specify a vector of monotonically increasing displacement values corresponding to your force-flux data. The default value is ```[ 0 0.05 0.1 0.15 0.2 ]``` m/m.

Flux partial derivative wrt current, Phi(i,x)/di

Specify a matrix of the flux partial derivatives with respect to current. This parameter is visible only if Electrical model is set to ```Define in terms of dPhi(i,x)/dx and dPhi(i,x)/di```. 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 ]```
Flux partial derivative wrt displacement, Phi(i,x)/dx

Specify a matrix of the flux partial derivatives with respect to displacement. This parameter is visible only if Electrical model is set to ```Define in terms of dPhi(i,x)/dx and dPhi(i,x)/di```. 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 ]```

Specify a matrix of the total flux linkage, that is, flux times the number of turns. This parameter is visible only if Electrical model is set to `Define in terms of Phi(i,x)`. 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 ]```

Calculate force matrix?

Specify the way of providing the electromagnetic force data:

• `No — specify directly` — 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)

Specify a matrix of the electromagnetic force applied to the plunger or moving part. This parameter is visible only if Calculate force matrix? is set to ```No — specify directly```. 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 ]```
Flux dependence on displacement

Specify the flux pattern:

• `Unique` — No flux pattern present. This is the default option.

• `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

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

• `Linear` — Uses a bilinear interpolation algorithm, which is an extension of linear interpolation for functions in two variables.

• `Cubic` — Uses the bicubic interpolation algorithm.

• `Spline` — Uses the bicubic spline interpolation algorithm.

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

Extrapolation method

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:

• `From last 2 points` — Extrapolates using the linear method (regardless of the interpolation method specified), based on the last two output values at the appropriate end of the range. That is, the block uses the first and second specified output values if the input value is below the specified range, and the two last specified output values if the input value is above the specified range.

• `From last point` — Uses the last specified output value at the appropriate end of the range. That is, the block uses the last specified output value for all input values greater than the last specified input argument, and the first specified output value for all input values less than the first specified input argument.

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

This parameter is not available if you set the Flux dependence on displacement parameter to `Cyclic`.

Winding resistance

Total resistance of the electrical winding. The default value is `14` Ohm.

### Mechanical Tab

Damping

Linear damping. The default value is `1` N/(m/s). The value can be zero.

Plunger mass

Mass of the moving part, which corresponds to mechanical translational port R. The default value is `0.05` kg. The value can be zero.

Minimum stroke

The stroke at which the lower mechanical end stop is applied. The default value is `0`. The value can be `-Inf`.

Maximum stroke

The stroke at which the upper mechanical end stop is applied. The default value is `0.2` mm. The value can be `Inf`.

Initial plunger position

Position of the plunger at the start of the simulation. The default value is `0` mm.

Initial plunger velocity

Speed of the plunger at the start of the simulation. The default value is `0` mm/s.

Contact stiffness

Contact stiffness between plunger and end stops. The default value is `1e8` N/m.

Contact damping

Contact damping between plunger and end stops. The default value is `1e4` N/(m/s).

## Ports

This block has the following ports:

`+`

Positive electrical conserving port

`-`

Negative electrical conserving port

`C`

Mechanical translational conserving port connected to the actuator case

`R`

Mechanical translational conserving port connected to the plunger

## Examples

The Finite Element Parameterized Solenoid example illustrates the use and parameterization options of this block.