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These solenoids are good examples of how an indirect-acting pilot solenoid functions. The series solenoid that is used is the K-series, which is the smallest solenoid type available from Parker.
All Parker subminiature solenoids consist of KxHxx. The second number in the model defines the valve as being three-way (3) or four-way (4).
The last two digits are used to describe the method that the valve attaches to the pressurized components as well as the attaching electronics. H01 is a manifold base-mount with wire leads. H03 is a manifold base-mount that attaches to a circuit board PCB.

These solenoids are composed of some variety of the acetal polymer Polyphenylene Sulfide, which is an easily-assembled plastic that penetrates to create a clean, air tight seal throughout the surface. The sections of the coil are assembled, then the rectangular housing is simply poured into position around the functional parts. This is a cheap and easy way to produce them, however the coating is certainly known to flake apart relatively easily. Additionally, if there is an internal problem with the coil, it will be unrepairble since it's integrated to its housing. Other manufacturers such as Humphrey have developed their own methods to keep the coil separate from the housing, and in some cases the two can even be separated and replaced if needed. I find these solenoids to be much more reliable and time-tested than the Parker ones, although it makes producing them a bit more expensive.

As stated, these indirect valves are a bit slower than their direct-acting counterparts. This is becuase air pressure has to be routed in and out of the pilot to push the piston, which at the same time is continuously pressing against a gradient of pressure on the other end. These characteristics slow the valves down by a considerable amount, but still leaves room for them to cycle into the speeds of 40 cycles per second and above. Quite nearly all modern pilot-actuated valves can cycle with more than enough speed to actuate a marker using it. The other limiting factor is the small size of the internal porting that must be produces within the pilot, to direct airflow around the valve. Here's a rendering of the internal porting used in the pilot section of a Parker solenoid:

Solenoid Operation:
For the majority of this page, I take the Parker four-way solenoid as an example for diagrams and such. Similar arrangements can be found in other valves as well, with a few slight differences (explained at the bottom of the page). Here are some renderings to show the positions of parts in the valve:

Warning: the third image above is VERY large.

While the solenoid rests idle, the core spring pushes it forward against the pilot section. The purpose of the core is to seal against a very small port facing it, which it can do even though the port has 150+ psi flowing through it. The port is so small that the force of pressure pushing against the core is insignificant enough to be overcome by the tiny core spring.
The pilot has positive pressure flowing into it from the spool section. It travels up around the piston and down to the core (marked red). The other side of the pilot housing is open to the atmosphere (marked black) because the core pushes open the small valve on the other side of the pilot. The second diagram below shows the pressurized sections of the gasket sealing between the pilot and spool.

When the marker fires, the conducing coil around the core is energized and it gets pulled away from the pilot. When this happens, the pressurized port releases air into the space occupied by the core, and flows down into the other side of the pilot housing through the middle. When this happens, the tiny spring coupled against the actuator assembly pushes the actuator forward and seals off the port that allows the pilot piston chamber to vent. The port is closed and pressure fills the chamber and pushes against the pilot piston, which in turn pushes the spool to the "actuated" position, which switches airflow within the marker. The other side of the spool is also pressurized, however the diameter of the pilot is more wide than that of the spool, so it overcomes the pressure holding the spool closed.

Now, once the dwell time has expired and the solenoid coil no longer receives voltage from the battery, the core will move forward and close off the air inlet to the pilot. When this happens, the core opens the pilot actuator again and the pilot chamber is able to vent and shrink in volume as the pilot moves back, due to the pressure pushing at the other end of the spool. The spool airflow is switched back to the idle position, and the solenoid is now ready to cycle again.

The following diagrams depict the movements of parts and pressure within the marker, starting from idle.

A side note; when the solenoid isn't pressurized, the spool sits in the actuated/open position (a factor of the depressurizing of the valve last time the marker was aired up). When you intially pressurize the marker, the solenoid will tend to leak through its vents until pressure increases high enough for the endcap to move the spool to the closed position and seal it up. This is why, occasionally, your marker leaks when you first pressurize it. The exact pressure varies between different solenoids, and different types of solenoids, but is often between 80-100 psi.

Animation: This is an animation I made to show the different pressurizing/depressurizing of the valve.

This is not to scale and some parts of the solenoid aren't shown. But the operation of the core/pilot is explained here. Information for the spool airflow is below.

Some solenoid pilots have a feature known as the manual override as well. This is a small button on the outside of the pilot (between it and the core) which manually pushes the core away from the pilot to fire the marker. The override is fitted with a spring to reset it and prevent it from interfering with the core during normal operation. Here are some renderings of the manual override in the idle and overidden state:

Solenoid Spool Airflow:
Below are some airflow diagrams I made to show where pressure moves when the spool is in different positions. Here is a diagram of the Parker solenoid's pneumatics housing:

Spool diagram

In the rest of these diagrams, I don't show the positions of any springs inside the solenoid. This is because I can't accurately render them "springing" with movements of other parts...sorry. Also note that each side of the solenoid (front and rear) has a pair of exhaust ports, which overlap on these diagrams (one leads to the spool and the other is elbowed to the area surrounding the spool), however the transparent renderings I provide do show them for the most part.

This first diagram is of the solenoid while it rests idle, ready to fire. Air pressure is shunted into the spool's inlet where it is directed to one of the two outputs. At the same time, the other output is allowed to vent whatever pressure stored there to the atmosphere. These actions are marked by red lines. Generally, the output ports connect the solenoid to one of two ends of a pneumatic piston. In this case, output A would lead to the front of the piston in an Impulse, Shocker SFT, Intimidator, or Tribal for example, whereas on other markers it would lead to the rear of the piston. It all depends on the firing assembly used in the rest of the marker.
Idle spool

The other movement of air that travels through the spool is that which is necessary for the spool itself to move (marked as "movement exhaust" on the first diagram). This pressure is created by the movement of the spool by exerting expansion and compression events in different parts of the housing (air gets forced out of one section, and sucked into another). The flow of air pressure caused by the movement of the spool is marked by yellow lines, and includes both sucking air in and pushing it out. When the spool shifts from open to closed, air is drawn into the cavities in front of and behind the spool. You may find it useful to view the animation above to help understand this. Also, the pilot's movement creates venting exhaust as well, as is denoted by the yellow line pointing from off screen and out the spool housing (noted in the animation).

When the solenoid coil is energized and the spool moves to the "open" position, it turns to this:
Firing spool

Air pressure is pushed out from the surrounding areas of the spool when the solenoid opens. Markers are milled to allow the exhaust and venting of the solenoid with small millings made into the surrounding metal. These are sometimes known as exchange grooves.

The two gaskets that separate the spool housing from the pilot on one side and the spool's endcap on the other are specifically designed to allow for the required venting during the movement of the spool. On the side of the pilot, the gasket is also the method by which air pressure is provided for the pilot, and allows it to vent on the bottom. In the case of Parker valves, the only part of the valve that vents is the bottom of the spool housing. Some other manufacturers choose to vent the pilot without routing it back to the spool, and this may create a difference in speed response.

Three-Way Solenoids:
As hinted toward above, the only difference between a four-way and three-way pilot-actuated valve is the porting for the spool. One of the ouptuts is simply removed, and the air travels nowhere. While the spool lies closed (idle), air pressure will enter the solenoid spool but not exit. The air that is trapped in the outlet port is always vented to the outside air through the firing assembly. I would like to make a three-way normally-opne and normally-closed animation, but I'm not sure when that will happen.

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