# FEM-Parameterized Rotary Actuator

Rotary actuator defined in terms of magnetic flux

Libraries:
Simscape / Electrical / Electromechanical / Mechatronic Actuators

## Description

The FEM-Parameterized Rotary Actuator block implements a model of a rotary actuator defined in terms of magnetic flux. Use this block to model custom rotary actuators and motors where magnetic flux depends on both rotor angle 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,theta)/dtheta and dPhi(i,theta)/di```, defines the current in terms of partial derivatives of the magnetic flux (Φ) with respect to rotor angle (θ) and current (i), the equations for which are:

`$\frac{di}{dt}=\left(v-iR-\frac{\partial \Phi }{\partial \theta }\frac{d\theta }{dt}\right)/\frac{\partial \Phi }{\partial i}$`

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

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

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 torque as a function of current and rotor angle, by using the Torque matrix, T(i,theta) parameter, or have the block automatically calculate the torque matrix.

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

`$T=\underset{0}{\overset{i}{\int }}\frac{\partial \Phi \left(\theta ,i\right)}{\partial \theta }di$`

See the Solenoid Parameterized with FEM Data example model for an example of how to implement this type of integration in MATLAB®.

Alternatively, the block can automatically calculate the torque matrix from the flux information that you provide. To select this option, set the Calculate torque matrix? parameter to `Yes`. The torque matrix calculation occurs at model initialization based on current block flux linkage information. The torque is calculated by numerically integrating the rate of change of flux linkage with respect to angle over current, according to the preceding equation. If the Electrical model parameter is set to `Define in terms of Phi(i,theta)`, then the block must first estimate the Flux partial derivative wrt angle, Phi(i,theta)/dtheta 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 rotor angle must be zero for zero current.

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

### Model Thermal Effects

You can expose the thermal port to model the effects of losses that convert power to heat. To expose the thermal port, set the Modeling option parameter to either:

• `No thermal port` — The block does not contain a thermal port.

• `Show thermal port` — The block contains one thermal conserving port.

For more information about using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

## Assumptions and Limitations

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

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

## Ports

### Conserving

expand all

Electrical conserving port associated with the actuator positive terminal.

Electrical conserving port associated with the actuator negative terminal.

Mechanical rotational conserving port associated with the actuator case.

Mechanical rotational conserving port associated with the rotor.

Thermal port.

#### Dependencies

To enable this port, set Modeling option to ```Show thermal port```.

## Parameters

expand all

Whether to enable the thermal port of the block and model the effects of losses that convert power to heat.

### Magnetic Force

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

• ```Define in terms of dPhi(i,theta)/dtheta and dPhi(i,theta)/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,theta)``` — Define the voltage across the block terminals directly in terms of the flux.

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

Specify a vector of monotonically increasing rotor angle values corresponding to your torque-flux data.

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,theta)/dtheta and dPhi(i,theta)/di```. The default value, in Wb/A, is:

```[ 0.002 0.0024 0.0035 0.0052 0.0074 0.0096 0.0118 0.0135 0.0146 ... 0.015 0.0146 0.0135 0.0118 0.0096 0.0074 0.0052 0.0035 0.0024 0.002; 0.002 0.0024 0.0035 0.0052 0.0074 0.0096 0.0118 0.0135 0.0146 ... 0.015 0.0146 0.0135 0.0118 0.0096 0.0074 0.0052 0.0035 0.0024 0.002; 0.002 0.0024 0.0035 0.0052 0.0074 0.0096 0.0118 0.0135 0.0146 ... 0.015 0.0146 0.0135 0.0118 0.0096 0.0074 0.0052 0.0035 0.0024 0.002; 0.002 0.0024 0.0035 0.0052 0.0074 0.0096 0.0118 0.0135 0.0146 ... 0.015 0.0146 0.0135 0.0118 0.0096 0.0074 0.0052 0.0035 0.0024 0.002; 0.002 0.0024 0.0035 0.0052 0.0074 0.0096 0.0118 0.0135 0.0146 ... 0.015 0.0146 0.0135 0.0118 0.0096 0.0074 0.0052 0.0035 0.0024 0.002; 0.002 0.0024 0.0035 0.0052 0.0074 0.0096 0.0118 0.0135 0.0146 ... 0.015 0.0146 0.0135 0.0118 0.0096 0.0074 0.0052 0.0035 0.0024 0.002; ]```

#### Dependencies

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

Specify a matrix of the flux partial derivatives with respect to rotor angle. This parameter is visible only if Electrical model is set to ```Define in terms of dPhi(i,theta)/dtheta and dPhi(i,theta)/di```. The default value, in Wb/rad, is:

```[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 9e-4 0.0017 0.0023 0.0026 0.0026 0.0023 0.0017 9e-4 ... 0 -9e-4 -0.0017 -0.0023 -0.0026 -0.0026 -0.0023 -0.0017 -9e-4 0; 0 0.0018 0.0033 0.0045 0.0051 0.0051 0.0045 0.0033 0.0018 ... 0 -0.0018 -0.0033 -0.0045 -0.0051 -0.0051 -0.0045 -0.0033 -0.0018 0; 0 0.0027 0.005 0.0068 0.0077 0.0077 0.0068 0.005 0.0027 ... 0 -0.0027 -0.005 -0.0068 -0.0077 -0.0077 -0.0068 -0.005 -0.0027 0; 0 0.0036 0.0067 0.009 0.0102 0.0102 0.009 0.0067 0.0036 ... 0 -0.0036 -0.0067 -0.009 -0.0102 -0.0102 -0.009 -0.0067 -0.0036 0; 0 0.0044 0.0084 0.0113 0.0128 0.0128 0.0113 0.0084 0.0044 ... 0 -0.0044 -0.0084 -0.0113 -0.0128 -0.0128 -0.0113 -0.0084 -0.0044 0 ]```

#### Dependencies

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

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,theta)```. The default value, in Wb, is:

```[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 4e-4 4.8e-4 7e-4 0.00105 0.00147 0.00193 0.00235 0.0027 0.00292 ... 0.003 0.00292 0.0027 0.00235 0.00193 0.00147 0.00105 7e-4 4.8e-4 4e-4; 8e-4 9.6e-4 0.00141 0.0021 0.00295 0.00385 0.0047 0.00539 0.00584 ... 0.006 0.00584 0.00539 0.0047 0.00385 0.00295 0.0021 0.00141 9.6e-4 8e-4; 0.0012 0.00144 0.00211 0.00315 0.00442 0.00578 0.00705 0.00809 0.00876 ... 0.009 0.00876 0.00809 0.00705 0.00578 0.00442 0.00315 0.00211 0.00144 0.0012; 0.0016 0.00191 0.00282 0.0042 0.0059 0.0077 0.0094 0.01078 0.01169 ... 0.012 0.01169 0.01078 0.0094 0.0077 0.0059 0.0042 0.00282 0.00191 0.0016; 0.002 0.00239 0.00352 0.00525 0.00737 0.00963 0.01175 0.01348 0.01461 ... 0.015 0.01461 0.01348 0.01175 0.00963 0.00737 0.00525 0.00352 0.00239 0.002 ]```

#### Dependencies

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

Specify the way of providing the electromagnetic torque data:

• `No` — Enter the electromagnetic torque data directly, by using the Torque matrix, T(i,theta) parameter.

• Yes — The block calculates the torque from the flux linkage information, as a function of current and rotor angle.

Specify a matrix of the electromagnetic torque applied to the rotor. This parameter is visible only if Calculate torque matrix? is set to `No`. The default value, in mN*m, is:

```[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0.0889 0.1671 0.2252 0.2561 0.2561 0.2252 0.1671 0.0889 ... 0 -0.0889 -0.1671 -0.2252 -0.2561 -0.2561 -0.2252 -0.1671 -0.0889 0; 0 0.3557 0.6685 0.9007 1.0242 1.0242 0.9007 0.6685 0.3557 ... 0 -0.3557 -0.6685 -0.9007 -1.0242 -1.0242 -0.9007 -0.6685 -0.3557 0; 0 0.8003 1.5041 2.0265 2.3045 2.3045 2.0265 1.5041 0.8003 ... 0 -0.8003 -1.5041 -2.0265 -2.3045 -2.3045 -2.0265 -1.5041 -0.8003 0; 0 1.4228 2.674 3.6027 4.0968 4.0968 3.6027 2.674 1.4228 ... 0 -1.4228 -2.674 -3.6027 -4.0968 -4.0968 -3.6027 -2.674 -1.4228 0; 0 2.2231 4.1781 5.6292 6.4013 6.4013 5.6292 4.1781 2.2231 ... 0 -2.2231 -4.1781 -5.6292 -6.4013 -6.4013 -5.6292 -4.1781 -2.2231 0 ]```

#### Dependencies

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

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.

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.

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.

#### Dependencies

This parameter is visible only when you set the Flux dependence on displacement parameter to `Unique`.

Total resistance of the electrical winding.

Magnetizing resistance. The value must be greater than zero. The default value is `inf`, which implies that there are no iron losses.

### Mechanical

Rotary damping. The value can be zero.

Inertia of the rotor attached to mechanical translational port R. The value can be zero.

The rotor angle at which the lower mechanical end stop is applied.

The rotor angle at which the upper mechanical end stop is applied.

Contact stiffness between rotor and end stops.

Contact damping between rotor and end stops.

### Temperature Dependence

To enable these parameters, set Modeling option to `Show thermal port`.

Resistance temperature coefficient.

Temperature for which the actuator parameters are defined.

### Thermal Port

To enable this parameter, set Modeling option to `Show thermal port`.

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

## Version History

Introduced in R2010a