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Model electrical characteristics and generated force of solenoid

Translational Actuators

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:

$${F}_{l}+m\ddot{x}+\lambda \dot{x}+kx={F}_{e}$$

where *F _{e}* is
the electromagnetic force,

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

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

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

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* i _{0}*:

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

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.

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.

**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.

**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 (

= 0) and maximum (*x*=*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 (

= 0) and maximum (*x*=*x***Maximum stroke**) plunger positions. The default value is`500`

N/(m/s).

The block has the following ports:

`+`

Positive electrical input

`-`

Negative electrical input

`C`

Mechanical translational conserving port

`R`

Mechanical translational conserving port

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

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