Here's how it works.... but I have to warn up front that "duty cycle" and "pulse width modulation" may appear to be the same thing - they are not... It is nigh on impossible to accurately position the pintle of a solenoid anywhere in between all the way in and all the way out. To overcome this, the engineers "dither" the solenoid - turn it on and off rapidly so that the pintle moves around it's desired position. We call this "duty cycling".
Not all duty cycled solenoids are 12 volt solenoids.... some will burn out very quickly with the application of 12 volts.
Let's start with one cycle... the time period is set by the device (in this case it is the PCM) controlling the circuit. Just to make the math easy, let's say the frequency of the circuit is 100Hz (100 cycles per second). One cycle would last 10 milliseconds. At 0% duty cycle, the circuit is off all the time. At 100% duty cycle, the circuit is full on. At 50% duty cycle, the circuit would be turned on for one half of the cycle (in our example, 50 milliseconds) and off for the other half of the duty cycle.
More complications.... Some of these circuits are considered "full on" at
60% duty cycle... applying a steady 12 volts to one of these solenoids will kill the poor little beastie quite suddenly. Others that do get cycled to
100% wont last long with a steady 12 volts applied but, in their normal life, they will occasionally get 100% for very brief periods.
These types of circuits are extremely light duty.... It takes time for the magnetic field to build and decay yet we still need the action to be rapid to avoid a loping or porpoising effect (if one cycle lasts less time, it becomes easier to position the pintle more accurately but we have less time to build the magnetic field. The resulting weaker field can only move so much pintle). Since these circuits are light duty, it doesn't take much to "trap" the pintle.... very slight mechanical resistance can make these controls fail.
HTH.