Rotary actuator defined in terms of magnetic flux

**Library:**Simscape / Electrical / Electromechanical / Mechatronic Actuators

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:

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

See the Finite Element Parameterized
Solenoid example model and its initialization file `ee_fem_solenoid_ini.m`

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.

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.

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.