Lawn Mower / Starter Solenoid questions

I've got a walker 52" with a twenty horse kohler. Got done mowing the other day and when I cut the key off, it sounded like the starter was still engaged and spinning the flywheel. It kept doing this until it ran the battery down. Talked to a friend of mine who said it sounded like the solenoid was sticking. Got a new starter/solenoid today and put it on to no avail.i also changed out the ignition switch. The starter tested good albeit without a load. I've got 12 volts going to the starter solenoid and everything is tight. The little traveler wire is what my question is about. What kind of voltage and when is it supposed to have voltage? When the ignition is in the run position or in the start position?

TL;DR : How does a starter solenoid work?