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I searched for help on an issue and found lots of questions about the 3,5 volts supplied to the solenoid and hope to avoid someone overheat and damage their solenoids by explaining why providing 12 v directly is a bad idea.

Solenoids do need a lot of energy to engage, but not much to maintain an engaged position. If you study solenoids on the car's starting motor, you find two windings, one for engaging and the second for maintaining the position.

In order to avoid overheating the coil, the current (ampere) is electronically controlled using PWM, Pulse With Modulation. The coil is given 12 volts for a short time, enough to move the shifter fork inside the transmission. Then, after a short period, we used 200mS, the electronics reduce the power to 25%, thus preventing overheating. This is all done electronically, either using a timer or a small microcontroller

The reason you do measure 3,5 volts is the multimeter measure the average value. If you hook an oscilloscope to the wires, you will see the voltage is 12 volts for a short period of time, then goes to 0 volts for a period around 4 times longer than the on period. It is a clever solution, and I am confident this is a better solution than a mechanical system that rust and switches that corrode if (when) you get moisture inside.
Warning! In case you use a LED test lamp, you may see some strange results, as the LED is very quick reacting.
 
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