## Selecting and Sizing a Solenoid for Linear Motion

Design and application engineering both require balance. Most engineering challenges are complex.  Rarely do they involve only one variable or mutual exclusivity between variables. Increase size here and lose some over there. More power means more heat. Faster actuation often leads to higher impulse loading and more wear. The list goes on. Carefully tracing down the contours of a multi-variable manifold is the engineer’s bread and butter. Sizing a solenoid is no different. To prove this, let’s walk through sizing and selecting a solenoid for a fun – yet desperately needed – application:

The Application: a breakroom de-scentsitizer:

You’re a diligent engineer. Sometimes you stay late. Sometimes you come in early. You do whatever it takes, even working through lunch. You head to the breakroom shortly after the lunch rush, just after that last bowl of leftovers has left the lunchroom. You have, through no fault of your own, become very familiar with what happens when you mix the smell of burnt popcorn with chicken alfredo and last night’s Thai food and everyone else’s lunch smells. It isn’t pretty! You and your nose wade through that battlefield every day on your way to the refrigerator. And, what’s worse, short of parking an industrial grade kitchen exhaust hood above that microwave, turning the fan speed dial past 10 to typhoon, there really isn’t much you can do.

Or is there?

What if you had a small device, about the size of a soap dispenser, that you could mount somewhere just outside the microwave? What if this little device sprayed food-safe, odor neutralizing spray, eliminating the odor right at its source? Voila!  Problem solved! Right? What if you interlocked it with the microwave door? You could use a solenoid to actuate an aerosol can every time the door interlock closed, and just like that you never have to smell your coworker’s broccoli chicken alfredo again.

It can’t be that easy, can it? It can!  Let’s walk through how to size the solenoid for this application.

It’s tempting to throw in a tl:dr for the zoomers, so here’s that:

TL:DR How to Size a Solenoid.

• Measure necessary force and stroke to actuate your mechanism
• Define the duty cycle of your application.
• Calculate necessary electrical power input and define supply voltage.
• Use solenoid data to find a solenoid that meets force requirements at stroke.
• Pick a suitable coil gauge given your supply voltage.
• Implement a return mechanism and solve whatever other various application-specific problems (vertical orientation, timer circuitry to prevent repeat actuation, etc.)

## Sizing a Solenoid (via basic Energy Balance):

A solenoid is, in its most basic sense, an electromechanical device that receives electrical energy as input to pull the ferromagnetic iron plunger into the coil of wire and do work on some mechanism – in this case, the top of the aerosol can. (If the magic of electromagnetism is still astonishing to you and even if you know Maxwell’s equations like you know your ABCs, you still might enjoy the Feynman Lectures https://feynmanlectures.caltech.edu/II_toc.html).

The first step in sizing a solenoid for an application is to figure out how much mechanical energy is required for motion of the mechanism (aerosol can). As you know:

$W=Fd$

Where, W is work, F is force, and d is distance.

Measure the total travel of the plunger shaft, starting just above the top of the can button and ending the distance the button has traveled after a puff of the appropriate size is dispensed. It’s about 3⁄8 “. You happen to have a mass weighing 1lb and if you set it on top of the can it compresses it without damage. So, you need a 3⁄8 ” stroke length and a force of 1lb.

For the electrical side, a simple energy balance:

$W=Fd=ηIVt$

Where η is an efficiency factor, I is current, V is supplied voltage, and t is actuation time.

You would like to supply a voltage no greater than 12V, which might seem a bit high, but as we’ll see later, a stroke of 3⁄8 ” precludes lower voltages (at least without also using capacitive discharge or a coil with a low gauge, both effectively increase supply current to solenoid).  We ballpark an electrical efficiency factor of 0.1, which is fairly inefficient. Solenoids, while being a practical means to convert electrical energy into mechanical work, are also very effective heaters.

As a result of $I^2 R$ losses and the relatively low efficiency of a solenoid, heat generation is one of the primary concerns. Therefore, it’s important to consider the duty cycle of the solenoid; which is defined as a ratio of the period during which power is supplied over total time of a cycle. Like this:

In this case, we only want the solenoid to actuate long enough to dispense a single puff, so let’s say our solenoid should be on for no longer than 100ms and be off for at least 300ms. This gives a duty cycle of 25%.

As an aside, to affect this kind of duty cycle in some applications where the user is allowed to repeatedly switch on power to the solenoid, it may be necessary to implement timer circuitry to prevent this kind of repeated actuation. For simplicity’s sake, let’s assume that no one will need more than one puff and that all our users are fairly well-behaved (where this is obviously a very rough assumption because we all know that some food smells more than others. This can be both good and bad. I’m looking at you Mr. I-didn’t-mean-to-leave-the-popcorn-in-there-for-that-long).

Next, evaluate the earlier energy balance equations. From a force perspective, in SI units:

$W=Fd=(4.4N)(.0095m)= .0418J$

Now, solve the electrical half of this equation for the power term:

The calculation tells us that we need a solenoid with 41 watts of input power to generate the desired force at stroke.

### Force/Stroke Graphs:

Now we need data on how much force a solenoid can produce at a particular stroke and input power. Using the JE catalog as an example, you’ll see the STA Tubular line is organized by size (diameter x length) from ½”x½” up to 1 ½”x 2 ½” and separated into both push and pull models.  We need a push for our application. The force vs stroke and actuation speed graphs can be used for a first pass selection. Note that force decreases exponentially with increasing stroke and that strokes longer than 2” are difficult for a solenoid. We need a stroke of 3⁄8 “. The force/stroke graphs, show four curves that represent the force/stroke profiles of the solenoid at a particular duty cycle. Here’s an example:

A quick word on the relationship between force, current and duty cycle. A solenoid’s force (at a given stroke) is roughly proportional to the square of its amp-turns.  An amp-turn is a unit derived from the number of times the wire is wound around the bobbin multiplied by the current supplied to the coil. For a given coil, holding the number of turns constant and the geometry of the coil (i.e., diameter, length, cross sectional area), an increase in power will increase the force generated by the solenoid. But as mentioned earlier, the heat generated in the solenoid increases with the square of the current. So we find ourselves with a bit of a balancing act. Current, force, and heat are all directly proportional with different ratios of proportionality.  Force goes like the square of a coil’s amp-turns. Heat varies with the square of current. Increase any one of these variables and the others must increase along with it.

Along with the force/stroke graphs are tables separated into four columns by duty cycle. So, too, are the four curves on the force/stroke graphs. These are divided into families of curves ordered by duty cycle or input power. This family of curves is not a discrete set but is continuous as a function of input power. The more power is supplied to the solenoid, the more force is developed at stroke.  However, doing this requires a trade off time-on to prevent thermal damage.

For example, if we select a coil rated for 10W at 4.4 volts and instead give it 8V we may encounter thermal failure like melted coil bobbin or fusing of magnet wire if we keep it powered for an equivalent amount of time.

Back to our selection. Having established a 25% duty cycle with an input power of about 40W, we look through the force/stroke curves to find that the 1”x2” 100-STA (part number: 195207-XXX) is a good fit (see force/stroke graph above and coil gauge chart below for the 100-STA). It’s a good fit from a force/stroke perspective: it generates about 2lbs of force at 25% duty.  And it’s a good fit from a power input perspective, with a 25% duty cycle rated at 40W. The 102-STA would also be a reasonable selection, but it generates almost exactly 1lb of force at stroke, leaving no factor of safety (remember the 102-STA is smaller than the 100). Note that if in application, our duty cycle is lower than 25% then the 102-STA might be a good choice.

## Sizing Coil Gauge:

Almost done! After selecting the 100-STA, we need to select a coil gauge based on the input voltage constraint. We previously decided upon supply voltage of no greater than 12V.  The table above shows that the 23-gauge coil requires 8.9V to generate the 40W needed to provide 1lb force at stroke.  We’ve arrived at the 100-STA with a flat face plunger and a 23-gauge coil (part number 195207-123). Note that if we went with the 60° conical plunger, we might be able to do 20W input power instead and this sort of insight is something engineers are familiar with and the reason why iterative design is so important.

This design envisions the solenoid mounted vertically with shaft down, so something to keep the plunger in place against gravity is needed. The plunger also needs to return to its original position after actuation. A return spring can solve both problems.

We also see that the extra force we sized into the solenoid (2lbs at 25% duty, 1lb over what is actually needed) can be used in part to overcome the spring force during actuation. Return springs are commonly seen in solenoid applications. NB: Recall that the spring force varies linearly with stroke and the force developed by the solenoid varies nonlinearly with stroke. This is a complicated way of saying that at longer strokes you might run into situations where the spring force is greater than the force developed by the solenoid, but at shorter strokes this is rarely an issue.

Since our crash course is becoming more of a slow burn, we’ll assume that we can pick a return spring force low enough that the extra 1lb force will be enough to overcome the linear spring force (a fair assumption).

## In Summary:

Let’s revisit the steps we’ve taken to select our solenoid (TL:DR):

• Measure necessary force and stroke to actuate your mechanism
• Define the duty cycle of your application.
• Calculate necessary electrical power input and define supply voltage.
• Use JE solenoid catalog to find a solenoid that meets force requirements at stroke.
• Pick a suitable coil gauge given your supply voltage.
• Implement a return mechanism and solve other application-specific problems: vertical orientation, timer circuitry to prevent repeat actuation, etc.

While this was a fun exercise for those of us that do this every day, if you find yourself scratching your head and wondering where to start, remember: You do not have to do all this on your own. Johnson Electric employs Application Engineers that will help choose the right solenoid for your specific design – and we’ll have fun doing it!