# Solenoid

Model electrical characteristics and generated force of solenoid

## Library

Translational Actuators

## Description

The Solenoid block represents the electrical characteristics and generated force for the solenoid in the following figure:

The return spring is optional. To remove the effects of this spring from the model, set the Spring constant parameter to `0`.

The equation of motion for the plunger as a function of position, x, is:

`${F}_{l}+m\stackrel{¨}{x}+\lambda \stackrel{˙}{x}+kx={F}_{e}$`

where Fe is the electromagnetic force, Fl is the load force, λ is the viscous damping term and m is the plunger mass. The electromagnetic force is related to the solenoid current and inductance by:

`${F}_{e}=\frac{1}{2}{i}^{2}\frac{\partial L\left(x\right)}{\partial x}$`

The inductance, which is derived in [1], can be written as:

`$\frac{\partial L\left(x\right)}{\partial x}=\frac{-\beta }{{\left(\alpha +\beta x\right)}^{2}}$`

where α and β are constants. Plugging the preceding equation into the equation for electromagnetic force gives the force-stroke relationship of the solenoid for a current i0:

`$F=\frac{1}{2}{i}_{0}{}^{2}\frac{-\beta }{{\left(\alpha +\beta x\right)}^{2}}$`

The Solenoid block solves for α and β by taking the two specified force and stroke measurements and substituting them into the preceding equation. It solves the resulting equations for α and β.

A positive current from the electrical + to - ports creates a negative force (i.e., a pulling force) from the mechanical C to R ports.

### 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.

## Dialog Box and Parameters

### Magnetic Force Tab

Forces [F1 F2]

A vector of the force values at the two points on the force-stroke curve. The second measurement point must be at a stroke that is greater than that of the first measurement point. When the manufacturer does not provide a force-stroke curve, set F1 to the holding force (when X1 = 0) and F2 to the pull-in force when running the solenoid at the Rated voltage Vdc and Rated current Idc values. The default value is `[7.5 0.75]` N.

Stroke [X1 X2]

A vector of the stroke (plunger distance from the fully closed position) values at the two points on the force-stroke curve. The second measurement point must be at a stroke that is greater than that of the first measurement point. To ensure a finite force value, the points must meet the condition

`$\frac{X2}{X1}>\sqrt{\frac{F1}{F2}}$`

The default value is ```[1 5]``` mm.

Rated voltage Vdc

The voltage at which the solenoid is rated to operate. This voltage value is used to measure the Forces [F1 F2] and Stroke [X1 X2] values. The default value is `50` V.

Rated current Idc

The current that flows when the solenoid is supplied with the Rated voltage Vdc voltage. The default value is `0.05` A.

### Mechanical Tab

Spring constant

Constant representing the stiffness of the spring that acts to retract the plunger when the solenoid is powered off. The force is zero when the plunger is displaced to the Stroke for zero spring force parameter value. The default value is `200` N/m. Set the spring constant to zero if there is no spring.

Stroke for zero spring force

The stroke at which the spring provides no force. The default value is `5` mm.

Damping

The term λ in the equation of motion for the plunger as a function of position that linearly damps the plunger motion. The default value is `1` N/(m/s). The value can be zero.

Plunger mass

The weight of the solenoid plunger. The default value is `0.05` kg. The value can be zero.

Maximum stroke

The maximum amount by which the plunger can be displaced. You can use this parameter to model a hard endstop that limits the stroke. The default value is `Inf` mm, which means no stroke limit.

Initial plunger position

The amount by which the plunger is displaced at the start of the simulation. The default value is `0` m.

Contact stiffness

Stiffness of the plunger contact that models the hard stop at the minimum (x = 0) and maximum (x = Maximum stroke) plunger positions. The default value is `1e+06` N/m.

Contact damping

Damping of the plunger contact that models the hard stop at the minimum (x = 0) and maximum (x = Maximum stroke) plunger positions. The default value is `500` N/(m/s).

## Ports

The block has the following ports:

`+`

Positive electrical input

`-`

Negative electrical input

`C`

Mechanical translational conserving port

`R`

Mechanical translational conserving port

## References

[1] S.E. Lyshevski. Electromechanical Systems, Electric Machines, and Applied Mechatronics. CRC, 1999.